Teaching & Learning Plan Inferential Statistics for Proportions Leaving Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve. Prior Knowledge points to relevant knowledge students may already have and also to knowledge which may be necessary in order to support them in accessing this new topic. Learning Outcomes outline what a student will be able to do, know and understand having completed the topic. Relationship to Syllabus refers to the relevant section of either the Junior and/ or Leaving Certificate Syllabus. Resources Required lists the resources which will be needed in the teaching and learning of a particular topic. Introducing the topic (in some plans only) outlines an approach to introducing the topic. Lesson Interaction is set out under four sub-headings: i. Student Learning Tasks – Teacher Input: This section focuses on possible lines of inquiry and gives details of the key student tasks and teacher questions which move the lesson forward. ii. Student Activities – Possible Responses: Gives details of possible student reactions and responses and possible misconceptions students may have. iii. Teacher’s Support and Actions: Gives details of teacher actions designed to support and scaffold student learning. iv. Assessing the Learning: Suggests questions a teacher might ask to evaluate whether the goals/learning outcomes are being/have been achieved. This evaluation will inform and direct the teaching and learning activities of the next class(es). Student Activities linked to the lesson(s) are provided at the end of each plan. 2 Teaching & Learning Plans: Inferential Statistics for Proportions Aims1 The aim of this series of lessons is: • To understand why sampling is important. • To identify that there is a link between statistics and probability. • To understand the phrase “inferential statistics”. • To understand the link between the 95% confidence and the empirical rule. • To recognise how sampling variability influences the use of sample information to make statements about the population. • To understand what factors must be kept in mind when sample information is used to make statements about the population. • To apply the idea of a confidence interval. • To understand that a sample proportion may not be the same as the population proportion. • To evaluate margin of error for a population proportion. • To analyse that increasing the sample size decreases the size or radius of the margin of error. • To observe that doubling the sample size does not halve the size or radius of the margin of error. • To analyse the idea of hypothesis testing. • To understand how to conduct a hypothesis test on a population proportion using the margin of error. formalises the intuitive notion about the size of a • To understand that 95% confidence interval for a population proportion. • To apply knowledge and skills relating to statistics to solve problems. • To use mathematical language, both written and spoken, to communicate understanding effectively. 1 This Teaching & Learning Plan illustrates a number of strategies to support the implementation of Literacy and Numeracy for Learning and Life: the National Strategy to Improve Literacy and Numeracy among Children and Young People 2011-2020 (Department of Education & Skills 2011). Attention to the recommended strategies will be noted at intervals within the Lesson Interaction Section of this Teaching and Learning Plan. © Project Maths Development Team 2014 www.projectmaths.ie 3 Teaching & Learning Plan: Inferential Statistics for Proportions Prior Knowledge Students have prior knowledge of: • Quantifying probabilities from Teaching and Learning Plan 1: Introduction to Probability • Task on Household Sizes from page 2 of the Workshop 10 booklet on www.projectmaths.ie • The Empirical Rule • Sampling Variability • The difference between a population and a sample. • Simple random sampling • Describing the shape, centre and spread of distributions • The Data Handling Cycle. Learning Outcomes As a result of studying this topic, students will be able to: • Calculate the margin of error for a 95% confidence interval for a population proportion using . • Make a statement about the population proportion using a 95% confidence interval. • Conduct a hypothesis test on a population proportion using the margin of error. • Understand how inferential statistics might be applied in every-day situations. Catering for Learner Diversity In class, the needs of all students, whatever their level of ability level, are equally important. In daily classroom teaching, teachers can cater for different abilities by providing students with different activities and assignments graded according to levels of difficulty so that students can work on exercises that match their progress in learning. Less able students, may engage with the activities in a relatively straightforward way while the more able students should engage in more open-ended and challenging activities. In interacting with the whole class, teachers can make adjustments to meet the needs of all of the students. Apart from whole-class teaching, teachers can utilise pair and group work to encourage peer interaction and to facilitate discussion. The use of different grouping arrangements in these lessons should help ensure that the needs of all students are met and that students are encouraged to articulate their mathematics openly and to share their learning. © Project Maths Development Team 2014 www.projectmaths.ie 4 Teaching & Learning Plan: Inferential Statistics for Proportions Relationship to Leaving Certificate Syllabus Sub-Topic Learning Outcomes Students learn about Students working at OL should be able to 1.4 Statistical reasoning with an aim to becoming a statistically aware consumer • discuss populations and samples Students working at HL should be able to • decide to what extent conclusions can be generalised • construct 95% 1.7 Analysing, • recognise how sampling confidence intervals for variability influences the use interpreting the population mean of sample information to and drawing from a large sample make statements about the inferences and for the population population from data proportion, in both cases • use appropriate tools to using z tables describe variability drawing inferences about the population from the sample • interpret the analysis and relate the interpretation to the original question • make decisions based on the empirical rule • recognise the concept of a hypothesis test • calculate the margin of error ( ) for a population proportion* • conduct a hypothesis test on a population proportion using the margin of error * The margin of error referred to here is the maximum value of the radius of the 95% confidence interval. Resources Required Formulae and Tables, whiteboards, rulers, GeoGebra and calculators. 60 yellow unifix cubes, blocks or pieces of card. 140 non-yellow unifix cubes, blocks or pieces of card. © Project Maths Development Team 2014 www.projectmaths.ie 5 Teaching & Learning Plan: Inferential Statistics for Proportions Lesson Interaction Student Learning Tasks: Teacher Input Student Activities: Possible Teacher’s Supports and and Expected Responses Actions Checking Understanding Teacher Reflections Section A – Sampling variability and confidence intervals »» In today’s lesson we are going to carry out a statistical investigation. From the investigation we would like to answer the following question: “What proportion of Irish post-primary students keep their mobile phone under their pillow at night?” »» On one half of the board write the question “What proportion of Irish post-primary students keep their mobile phone under their pillow at night?” »» When we say “Irish post‑primary • Many. students” how many Irish post‑primary students do we mean? • 50,000 students. »» Do students understand that when we say “Irish postprimary students” we mean all of them? • 300,000 students. • All of the post-primary students. • All post-primary students in Ireland. »» In statistics when we refer to “all” or “everybody”, what name do we give to this group? »» So we would like to know what proportion of the population of Irish post-primary students keep their mobile phone under their pillow at night? We are interested in answering a question about a population. © Project Maths Development Team 2014 • The population. www.projectmaths.ie »» Write the word “population” on the board and encourage students to write an explanation of the term in their copybooks. »» Do students understand that, in statistics, the complete set of people/items is known as "the population"? »» Do students understand that when we say “Irish post-primary students” we mean the population of Irish post‑primary students? KEY: » next step • student answer/response 6 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Actions Assessing the Learning »» Can anybody suggest how we might go about answering this question? • We need to survey some people. »» Write the second stage of the datahandling cycle “Collect Data” on the board. Link it to the first stage by means of an arrow. »» Can students identify the second stage of the data-handling cycle? »» Add to the diagram of the datahandling cycle to highlight the two general approaches to gathering dataconducting a census and sampling. »» Do students understand that, in general, when gathering data you can survey the entire population or a subset of the population? • We need some data. Teacher Reflections • We could ask everybody here in the room. »» If we were to gather the data ourselves, how many students could we ask? • We could ask them all. • We could ask some students. • We could take a sample of students. • 100. • 1,000. • All the students in our school. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 7 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Student Activities: Possible and Teacher Input Expected Responses Teacher’s Supports Assessing the Learning and Actions »» Can you explain why • Asking everybody should you might choose one provide a more accurate answer. approach over the other? • Asking everybody is expensive and takes a long time. »» Add some of the important advantages and disadvantages of sampling vs. conducting a census to the flow chart. • It wouldn’t be possible to ask every post‑primary student. »» Do students recognise that there are advantages and disadvantages to both approaches to collecting data? Teacher Reflections »» Can students identify the advantages and disadvantages of each approach to collecting data? • Sampling is faster and cheaper. • If you sample you mightn’t get an accurate answer. • When you sample you have to be careful to make sure the sample is representative. »» Do students recognise that sampling is used in the majority of statistical investigations? »» For many reasons you have just discussed, when answering a question in statistics, we usually use data from a sample instead of from the entire population. »» Do students understand why sampling is used in the majority of statistical investigations? »» Do students understand that the use of sampling raises the question of how accurate the results of a statistical investigation are? © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 8 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Teacher’s Supports and Actions Student Activities: Possible and Expected Responses Assessing the Learning »» You also pointed out a major disadvantage to sampling – that of accuracy. »» Do students understand that I have created a population so that I can investigate how reliable a sample is in describing a population? »» Because of this, we are going to carry out a small investigation to see if it’s possible to use the result from a sample to answer a question about a population. »» For the investigation, I have created (simulated) my own population of students using coloured counters. »» Show students the container of counters. »» Distribute a key to each group of students explaining what each colour counter represents. »» There are 300 students (or 300 counters) in my population. Each colour counter represents a different location in which students keep their mobile phone at night. »» I have purposely set up my population to have a specific proportion of students who keep their mobile phone under their pillow at night. »» I’ve written this proportion on a piece of paper in this envelope. »» We are now going to see if, by choosing a sample from my population, I can find out what this proportion is. »» Show students the envelope with the population proportion sealed in it. Pin it to the board. »» On one side of the board, write the heading “Population”. Underneath it write “No. of students in population = 300” and “Proportion of students in population who keep their mobile phone under their pillow = _____”. »» Encourage each group of students to replicate what’s written on the board on their own miniature whiteboard. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response »» Do students understand that each unit of my population is represented by a coloured counter? »» Do students understand that different colours represent different locations in which students keep their mobile phone? »» Do students understand that we are going to use the simulated population to see if a sample can be used to determine the population proportion? 9 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Actions Assessing the Learning »» In turns, I would like each group to choose a simple random sample of 25 students (counters) from the container and calculate the proportion of the sample who keep their mobile phone under their pillow at night. • Students draw 25 counters from container and record results. »» Across from the heading “Population” write a second heading “Sample”. Underneath it write “Number of students in my sample = 25” and “Proportion of students in my sample who keep their mobile phone under their pillow at night = _____ “ »» Do students understand that they are interested in the proportion of counters which are yellow? • Students calculate the proportion of their sample which is yellow. »» Do students understand how to choose a simple random sample? »» Do students understand how to calculate a proportion? »» This is stage three of the data-handling cycle – analyse the data. »» Encourage each group of students to replicate what’s written on the board on their own miniature whiteboard. »» Add in the third stage of the data-handling cycle to the flow chart on the board. »» Circulate to make sure students are completing the task correctly. »» Encourage students to write their proportion in the appropriate space on their whiteboard. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 10 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input »» Group 1, could you tell me the proportion of students in your sample who keep their mobile phone under their pillow at night? »» Is there any other way in which this result could be written? Student Activities: Possible and Teacher’s Supports and Actions Expected Responses • Note: This is only one of the possible proportions calculated from the sample. • or 40% or 0.4. • As a fraction or as a decimal or as a percentage. »» I am now going to use the result from Group 1’s sample to make a statement about the population of 300 students. This is the final stage in the data-handling cycle – interpret the results. »» The proportion of students in the population who keep their mobile phone under their pillow at night is 0.4. • Yes. • Yes, I got the same result. »» Write Group 1’s result in the appropriate space on the board in the form in which they reported it. »» Do students recognise that a proportion may be written in different ways? »» Encourage students to convert Group 1’s proportion to different representations. »» Do students understand that fractions, decimals and percentages are equally valid ways of representing a proportion? »» Add the different ways in which this proportion could be written to the board in the appropriate location. »» On the flow-chart showing the data-handling cycle, add in the final step of “Interpret the results”. • Well I got a different answer. • We all got different answers. • Why are we using Group 1’s answer? www.projectmaths.ie »» Can students easily change between the different ways of representing a proportion? »» Do students recognise that each group got a different sample proportion? »» Do students recognise that this makes it difficult to make any firm conclusions about a population based on the results of a single sample? • No, our group got a different proportion. »» Are you happy with this statement? © Project Maths Development Team 2014 Assessing the Learning »» Add Group 1’s result to the appropriate location under the heading “Population”. »» Write each group’s sample proportion in the correct location under the heading “Sample”. KEY: » next step • student answer/response 11 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Actions Assessing the Learning »» The fact that we all get different proportions when we sample is known as "sampling variability". • Our samples were randomly chosen. »» On the side of the board write the key term “sampling variability”. »» Can students explain what sampling variability is? »» Can you explain why we all get different proportions i.e. can you explain why sampling variability occurs? • We all chose different samples from the container. »» Encourage students to discuss with each other • We chose our samples what sampling variability randomly so you wouldn’t means and to write the expect the answers to be term and its description the same. into their journals. Teacher Reflections »» Do students understand why sampling variability occurs? »» Can students explain why sampling variability occurs? »» The aim of this activity was to see if we can use a single sample to determine the proportion of students in a population who keep their mobile phone under their pillow at night. »» Because I simulated the population, I know what the population proportion is: remember it’s sealed in the envelope on the board. »» Given what we’ve just discovered, how confident would you be that Group 1’s proportion is the same as the population proportion? © Project Maths Development Team 2014 • Not very confident. • I’d say it’s around the right answer. • Reasonable confident. • I don’t think it’s likely to be the same. www.projectmaths.ie KEY: » next step • student answer/response 12 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Actions Assessing the Learning »» Can you explain to me why you’re not very confident with Group 1’s result? • Well, it’s just one of the possible results we could get. »» Encourage students to discuss their ideas with each other. • Because of sampling variability. »» Encourage each group to share their thinking with the other groups in the classroom. »» Do students understand that Group 1’s result is only one of the possible answers we can get when we sample a population? • Different groups got different values to Group 1. Teacher Reflections • There’s nothing special about Group 1’s result. • Maybe our result is the correct one. »» Would you have more confidence in the result from your own group? • Not really. • All the results are as good as each other. • One of the results is probably correct. »» Encourage students to discuss their ideas with each other. »» Encourage each group to share their thinking with the other groups in the classroom. »» Do students understand that while some results are better than others we have no way of knowing which are better? • Some of the results are probably closer to the real value than others. • There’s no way to know which result is best. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 13 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses »» So we agree that we are • The proportion of students not very confident using a in the population who proportion from a single keep their mobile phone sample to make conclusions under their pillow at night about the population. is around 0.4. »» With this in mind and based on all the information we have up on the board, could you come up with a statement about the population proportion in which you’d have greater confidence? • The proportion of students in the population who keep their mobile phone under their pillow at night is around 0.32. • The proportion of students in the population who keep their mobile phone under their pillow at night is somewhere between 0.24 and 0.52. • The proportion of students in the population who keep their mobile phone under their pillow at night is the average of all our results. © Project Maths Development Team 2014 www.projectmaths.ie Teacher’s Supports and Actions Assessing the Learning »» Encourage students to come up with a statement about the population by discussing it in groups. »» Do students understand that they cannot assume that their sample proportion is the same as the population proportion? »» Encourage students to justify their statement. Teacher Reflections »» Do students understand that the chance of their sample proportion being equal to the population proportion is low? »» Do students understand that they can make a more definite statement about the population proportion (i.e. a statement in which they have more confidence) using a range or interval of values? KEY: » next step • student answer/response 14 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses »» Can you explain why you have more confidence in a statement which is based on a range of values? • It says that the population proportion is around 0.4 not that it’s exactly equal to 0.4. Note: If students answer “Sampling variability", this idea should be discussed with the class. • It says that the population proportion could be lots of values, not just one. Teacher’s Supports and Actions Assessing the Learning »» Can students explain why they have more confidence in the last statement compared to previous statements? Teacher Reflections • It takes into account the fact that different samples give different answers. • Although different groups got different answers, they’re all close to each other and the last statement takes this into account. • It recognises the existence of sampling variability. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 15 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible Teacher’s Supports and Actions and Expected Responses »» Our class has just come up with one of the most important ideas in statistics and that is: the emergence of a range of values when making a statement about a population based on a single sample. »» On the board cross off the statement about the population and replace it with the following “The proportion of students in the population who keep their mobile phone under their pillow at night is between 0.24 and 0.52”. »» By using an interval we can have more confidence that what we are saying about the population is true. »» Write the term “confidence interval” on the board. »» Using an interval takes into account the existence of sampling variability. »» If I were to widen my interval to make the following statement “The proportion of students in my population who keep their mobile phone under their pillow at night is between 0.1 and 0.7 – would you be more or less confident in this statement? »» Encourage students to come up with an explanation of what a confidence interval is and to write this into their journal. • Less confident (wrong). • More confident. »» Draw this confidence interval on the board and compare it to our original interval. • More confident because we’re including more possible values. Assessing the Learning »» Do students understand that by using an interval I can be more confident that the statement I am making about the population is true? Teacher Reflections »» Do students understand what the term “confidence interval” means? »» Can students verbalise what a "confidence interval" is? »» Do students understand that a narrower interval affects how confident we are in our statement about the population? • More confident because a wider interval means it’s more likely to be true. »» Explain your reasoning. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 16 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible Teacher’s Supports and Actions and Expected Responses »» If I were to make my • Less confident. interval narrower with the following • Less confident – we’re statement “The only looking at a small proportion of students number of the answers in my population who we could get. keep their mobile phone under their • Less confident – there’s pillow at night is a smaller chance that we between 0.35 and 0.45 are capturing the real – would you be more population proportion. or less confident in this statement? »» Add the narrower confidence interval to the diagrams on the board. Assessing the Learning »» Do students understand that a narrower interval affects how confident we are in our statement about the population? Teacher Reflections »» Explain your reasoning. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 17 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Teacher’s Supports and Actions Possible and Expected Responses »» So when we make a statement about a population based on a single sample, how wide should we make our interval? And how confident should we be that the interval captures the population proportion? »» Do students understand that while we understand the need for a confidence interval when making statements about a population we have yet to discuss how to construct this interval using a single sample? »» Luckily for us, statisticians have already decided this. Using the empirical rule, they developed a simple method for creating this interval based on the results of a single sample. »» If we take the proportion calculated from • Students complete the calculation of our single sample and subtract the confidence from it we get the lower end of the interval, for the interval. example on the board. »» If we take the proportion calculated from to our single sample and add it we get the upper end of the interval. »» When we do this we create an interval for which we can be 95% confident that what we are saying about the population is true. Assessing the Learning »» Write the expression on the board. Teacher Reflections »» Go through a sample calcultion on the board using any sample proportion »» Are students able to (preferably one which wasn’t calculated by complete the any of the groups in the classroom). confidenceinterval »» Encourage students to complete the calculation? calculation using their calculators. »» This is known as the 95% confidence interval. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 18 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible Teacher’s Supports and and Expected Responses Actions Assessing the Learning »» Now, using your own sample proportion, I would like you to create a 95% confidence interval and use this to make a statement about the population. • Students complete their »» Circulate around the own confidence – interval room to ensure students calculation. understand the task and are completing it correctly. »» Do students understand that they are using their own sample proportion to construct their own confidence interval? »» We introduced the idea of a confidence interval because we realised that, due to sampling variability, we cannot simply use the result from a single sample to make statements about the population. »» Encourage students to write a statement about the population using their confidence interval. »» Do students know how to construct a 95% confidence interval? »» Get each group to write their confidence interval on the board beside their sample proportion. »» Can students use their 95% interval to make a statement about the population? Teacher Reflections »» Open the envelope to reveal the proportion of students in the population who keep their mobile phone under their pillow at night. »» Let’s see if our confidence-interval approach has worked i.e. does it allow us to make correct statements about the population? »» Because I simulated the population we investigated I know what the proportion of students in the population who keep their mobile phone under their pillow at night is: it’s 0.3. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 19 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Actions Assessing the Learning »» If you had used your sample proportion only to make a statement about the population, would your statement have been correct? • No. »» Highlight each group's result on the board and the fact that none of these equal the population proportion. »» Do students see that using just the sample proportion to make a statement about the population almost certainly leads to an incorrect statement and that if it doesn’t it is merely due to chance? »» Go through each group’s confidence interval on the board and use it to make a statement about the population. »» Do students understand that by creating an interval around our sample proportion we are now able to make a statement about the population which is very likely (95%) to be true? • It would have been close. • It wouldn’t have been too bad. Note: Because the sample size is 25, it is not possible for a student to get a sample proportion of 0.3 since all the sample proportions must be a multiple of 0.04. »» If you use your confidence interval to make a statement about the population, is your statement correct? • Yes. • Yes the answer lies within my interval. • Not for mine. »» Encourage students to answer if each statement about the population is correct or incorrect. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response Teacher Reflections 20 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses »» Would you expect every group’s statement about the population to be true / would you expect every group’s confidence interval to contain the true population proportion? • Yes (incorrect). • No. • Only 95% of the time. • There’s a small chance some won’t. Teacher’s Supports and Actions Assessing the Learning »» Do students understand that by creating an interval in this way there is still a chance that the interval will not contain the true population proportion/the statement that they make about the population will not be true? Teacher Reflections »» Do students understand that you would expect your statement about the population to be true only 95% of the time? »» Do students understand that there is only a 95% chance of their statement/ confidence interval being correct? © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 21 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Student Activities: Possible Teacher’s Supports and Actions Teacher Input and Expected Responses »» Open the GeoGebra file “Confidence Interval. »» Do students ggb”. understand that the line »» Click the button “Show population represents the proportion”. population proportion of »» Enter one group’s sample proportion in 0.3? the input box labelled “Single Sample Proportion” and click “Return” to plot it. »» Do students understand »» Repeat the previous step to input and plot that the graph every group’s sample proportion. reinforces the notion of sampling variability? »» Let’s look at this idea using a graphical representation. »» The dashed line represents the proportion of students in our population who keep their mobile phone under their pillow at night i.e. the line represents what we were trying to find out using our single samples. »» Do any of the sample proportions match the population proportion? • No. • Some are close to it. • Most of them are close to it but not a perfect match. • Most are close to it but a few are far away. »» If we had made a statement about the population using our sample proportion would it have been correct? • No. • Some of us would have been close. • Some groups’ statements would have been way off. © Project Maths Development Team 2014 Assessing the Learning www.projectmaths.ie Teacher Reflections »» Do students understand that none of their sample proportions is equal to the population proportion? KEY: » next step • student answer/response 22 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible Teacher’s Supports and Actions and Expected Responses »» Does every group’s 95% confidence interval cover the same range of values? • No. »» Can you explain why this is? • Some do but most do not. • Because of sampling variability. »» On the GeoGebra file, click the button “Show 95% confidence interval”. »» Encourage students to explain why each group got a different confidence interval. • Because they were all created using a different sample. Assessing the Learning »» Do students understand that each group’s confidence interval is different? Teacher Reflections »» Can students explain why each group’s confidence interval is different? • Because each group’s sample proportion was different. »» Does each group’s confidence interval contain the population proportion we were looking for? • Yes. »» Can you explain how this can be, given that the confidence intervals are all different? • The intervals overlap. © Project Maths Development Team 2014 • Most of them do. • 95% of them do. »» Encourage students to point out any intervals which do not capture the population proportion. »» Do students understand that although each group’s confidence interval is different »» Encourage students to discuss how they all have a great different intervals can all capture the deal of overlap? population proportion. • Although the intervals are different they capture a lot of the same values. www.projectmaths.ie KEY: » next step • student answer/response 23 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Actions Assessing the Learning »» Would you expect each group’s confidence interval to contain the population proportion? • Yes. »» Ask students to explain to each other what it means when we say the interval we construct is a 95% confidence interval. »» Do students understand that because we have chosen to construct a 95% confidence interval, sometimes the interval will not include the population proportion? »» Encourage students to point (on the board) those statements about the population that are correct and those that are not. »» Do students understand that when the 95% confidence interval captures the population proportion the resulting statement about the population will be true? • No. • I’d expect most of them to. • It’s a 95% confidence interval so I’d expect 95% of the intervals to contain the population proportion. »» If each group makes a statement about the population using their confidence interval, will they all be correct? • Yes. • No. • Most of them will be. • I’d expect 95% of them to be correct in their statement. • There’s a 95% chance that each statement will be correct. © Project Maths Development Team 2014 www.projectmaths.ie Teacher Reflections »» Do students understand that using this confidenceinterval approach to making statements about a population based on a single sample will be correct 95% of the time? KEY: » next step • student answer/response 24 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Teacher’s Supports and Actions Possible and Expected Responses Assessing the Learning »» We have only looked at a small number of the different samples that could have been chosen from the population of 300 students. • Very few. »» Can students predict how many confidence intervals will capture the population proportion? • None. • There’s a tiny chance that one of the sample »» Using ICT I can very quickly proportions take many more samples and will be equal to see if what we’ve learned so the population far still holds true. proportion. »» If I took 100 different • Most of them. samples how many of these would you expect to be • Nearly all of them. the same as the population proportion? • 95% of them. »» If I took 100 different samples and used the results • Hardly any. to construct 100 different • Less than 5% of 95% confidence intervals, them. how many of these intervals would you expect to capture the population proportion? »» On the GeoGebra file click the “Reset” button. »» Click on the button “Show population proportion” »» Drag the slider “No. of samples” to create additional samples of size 25. »» Encourage students to identify any sample proportions which are identical to the population proportion. »» Click on the button “Show 95% confidence interval”. »» Encourage students to identify any confidence interval which does not capture the population proportion. »» How many confidence intervals would you expect not to capture the population proportion? © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response »» Do students understand that when we use a 95% confidence interval we would expect our interval to capture the population proportion 95% of the time? »» Do students understand that by using this approach, we expect the statement we make about the population to be correct 95% of the time? 25 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses »» Let’s take a minute and review what we’ve just discovered. Teacher’s Supports and Actions Assessing the Learning »» Encourage students to discuss these questions with each other and to explain their thinking to each other. »» Do students understand that due to sampling variability it is highly unlikely that a result from a sample will match the result for the population? »» If I take a sample from a population will the results from the sample match the results for the entire population? • No. • It’s highly unlikely. »» Can you explain why this is so? • Because of sampling variability. • Because you’re only looking at a piece of the population. »» Given this, is it possible to make a correct statement about the population based on the results from a single sample? • Yes. • Yes, but not just by using the result from our sample. • We can’t say for definite what the result for the population »» Where needed, is but we can say that it’s likely to be in a certain range. use appropriate • Yes we can, by using a confidence interval. questioning to help • Yes, by using a 95% confidence interval. students formulate an answer to each • It’s very likely that the interval we create captures the question. population proportion. • There’s a 95% chance that our interval holds the correct »» Write a summary value for the population. answer to each • There’s a 95% chance that the statement we make about question on the board the population is correct. – based on the class’s work and encourage • Yes. students to take • Yes, but it’s very unlikely. note of the questions • Yes, but there’s only a 5% chance of that happening. and answers in their copybooks. »» When we create a 95% confidence interval, what does this mean? »» Is it possible that when we use a confidence interval, the statement we make about the population will not be true? © Project Maths Development Team 2014 »» Encourage each group to write an agreed answer to each question on their whiteboards. www.projectmaths.ie »» Circulate the room to see if students understand the questions. KEY: » next step »» Do students understand that because of this we use a confidence interval when making a statement about a population based on a sample? »» Do students understand that the interval we choose to create is a 95% confidence interval? »» Do students understand that a 95% confidence interval means that there is a 95% chance that the interval captures the correct answer? »» Do students understand that by using a 95% confidence interval, there is a 95% chance that the statement we make about the population is true? »» Do students understand that there is a small (5%) chance that the confidence interval does not capture the correct answer? »» Do students understand that sometimes (5% of the time) the statement we make about the population based on the results from a sample will not be true? • student answer/response 26 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses • N and n. • • P and p (incorrect). p and p^. »» Could we write the idea of a confidence interval using mathematical notation? • • • • Yes, using an inequality. 0.28 ≤ p ≤0.28 + 0.28 - 0.2≤ p ≤0.28 + 0.2 0.08 ≤ p ≤ 0.48 »» We’ve just written down the confidence interval for one specific sample proportion from a sample of size 25. Could we write down the general form of a confidence interval for any sample proportion from any sized sample? • Yes. • No. • That sounds hard. »» What symbols might we use to represent the population proportion and the sample proportion? © Project Maths Development Team 2014 Assessing the Learning »» Do students »» On the diagram on the understand that it whiteboard showing the makes sense to use population and sample short-hand notation information, re-write each to represent the statement using the correct various quantities? notation. »» The statements we made on the board take up a lot of space and take a long time to write down. In maths we like to represent variables using symbols or letters, for this very reason. We will do the same here. »» What notation might we use the represent the number of units in our population and in our sample? Teacher’s Supports and Actions • p^ - www.projectmaths.ie ≤ p ≤ p^ + »» Use suitable questioning to help students write down the 95% confidence interval as an inequality. Teacher Reflections »» Can students write down the 95% confidence interval as an inequality? »» Use suitable questioning to »» Can students write down the general help students write down form of a 95% the general form of a 95% confidence interval confidence interval for a for a proportion? proportion. »» Encourage students to record this in their journals. KEY: » next step • student answer/response 27 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses »» To construct a 95% confidence interval we add and subtract the same quantity from our sample proportion. What is this quantity? • 0.2. • • • »» This quantity is known as the margin of error. Could you explain why it is so called? Teacher’s Supports and Actions Assessing the Learning on the board and »» Circle label it as the margin of error. »» Encourage students to write this term into their journals. »» Do students understand that the margin of error for a proportion is ? Teacher Reflections »» Do students understand why this is known as the margin of error? • No. • It takes into account the fact that our sample proportion may not be exactly right. • It allows for some error in relating our sample proportion to the population proportion. • It allows for the fact that the population proportion is probably different to the sample proportion. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 28 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Assessing the Learning Actions »» Can students apply their understanding of sampling and confidence intervals to complete Section A: Student Activity 1? »» We will now apply our newfound knowledge of sampling to another statistical investigation. »» Distribute copies of Section A: Student Activity 1 to all students. »» In groups, I would like you to complete Section A: Student Activity 1. »» Circulate around the room to ensure that students understand what they are supposed to do and are on task. »» Is the proportion of the sample who admits to regularly speeding 0.42? • Yes. »» How did you confirm this? • Yes, »» Did you think the newspaper headline was fair? Explain your reasoning. • Yes, the population proportion mightn’t be exactly 42% but it’s likely to be close. • No, the proportion of our sample is 42%. We don’t know what the proportion is for all lorry drivers. • No, if a different sample had been taken it would have given a different result. • No, this is the result for a single sample, not the entire population. • There is a 95% chance that the population proportion lies between 0.42 and 0.42 + . © Project Maths Development Team 2014 =0.42. www.projectmaths.ie »» Encourage students to discuss the questions with each other and to explain the reasoning behind their answers. »» Encourage students to use the relevant notation when writing down their answers. Teacher Reflections »» Can students calculate the sample proportion from the information given? »» Do students recognise that we cannot say the population proportion is equal to the sample proportion because of sampling variability? »» Do students understand that to make a fair statement about the population, they should use a confidence interval? »» Can students calculate the 95% confidence interval? »» Where needed, use suitable questioning to help students complete the activity. »» Can students calculate the margin of error? »» Are students comfortable using mathematical notation to represent various quantities? KEY: » next step • student answer/response 29 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses »» What is the 95% confidence interval for the population proportion? ≤ p ≤ 0.42 + • 0.42• 0.3753 ≤ p ≤ 0.4647 »» What is the margin of error for our sample population proportion? • • 0.4472 Teacher’s Supports and Assessing the Learning Actions Teacher Reflections »» What statement can you • I‘m 95% confident that the make about the population proportion of all truck drivers who based on your 95% regularly speed is between 0.3753 confidence interval. and 0.4647. • It is most likely that the proportion of all truck drivers who regularly speed is somewhere between 37.53% and 46.47% © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 30 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Actions »» Can students complete Section A: Student Activity 2? »» When we calculate the margin of error or the 95% confidence interval we use sample size to do so. • Students complete Section A: Student Activity 2. Note: This activity could also be given as a homework exercise which would then be reviewed and discussed at the start of the next lesson. »» Can you describe what happens to the margin of error as sample size increases? • The margin of error gets smaller. »» Can you describe what happens to the 95% confidence interval as the sample size increases? • It gets narrower. © Project Maths Development Team 2014 Teacher Reflections »» Can students calculate the margin of error for a given sample size? »» It is important to understand how sample size affects the margin of error or how sample size affects the 95% confidence interval. »» To this end, in groups, I would like you to complete Section A: Student Activity 2. Assessing the Learning • It decreases. »» Distribute Section A: Student Activity 2 to all students. »» Circulate around the room to ensure students are completing the task correctly. »» Where needed, help students complete the activity by using suitable questioning. »» Can students calculate the 95% confidence interval for a given sample size? »» Can students plot a graph to show the relationship between the margin of error and sample size? »» Can students describe the relationship between the margin of error and sample size? • The 95% confidence interval covers a smaller range of values. www.projectmaths.ie KEY: » next step • student answer/response 31 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Student Activities: Possible Teacher Input and Expected Responses Teacher’s Supports and Actions Assessing the Learning »» Can you explain why this relationship exists between margin of error (or the 95% confidence interval) and sample size? »» Select different groups of students to fill in each row of the table on the board. »» Can students describe the relationship between the 95% confidence interval and sample size? »» Can students explain why these relationships exist? »» Do students understand how margin of error (or the width of the 95% confidence interval) affects our ability to make useful statements about a population? »» Do students understand how sample size affects our ability to make useful statements about the population? »» Do students understand the disadvantages to using larger sample sizes? © Project Maths Development Team 2014 • If you used a small number of people in your sample they might not be reflective of the general population. • Because of randomness, a small sample could give a result which is very different from the population. • A larger sample is likely to produce a more reliable result. • A large sample is more likely to give a result which is the same as the population. • A small sample could, by chance, give you an answer far away from the population proportion so to capture the population proportion you would need a wide interval. www.projectmaths.ie »» Ask one group of students to sketch the graph showing the relationship between margin of error and sample size. KEY: » next step • student answer/response 32 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Actions »» If I were to decrease the sample size, what effect would that have on the margin of error (95% confidence interval)? • The margin of error will increase. »» On the board highlight the statement made about the population based on a sample size of 10 to demonstrate how limited the statement is. • The width of the 95% confidence interval will increase. Assessing the Learning Teacher Reflections • The 95% confidence interval will get wider. »» Can you identify an advantage of using a larger sample size when making statements about a population? • Your statement covers less possible answers. • Your statement is more definite. • Your statement is more useful. »» Can you identify any disadvantages of using a larger sample size? • No. • Not really. • It costs more money. • It takes more time. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 33 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Student Activities: Possible Teacher Input and Expected Responses Teacher’s Supports and Actions Assessing the Learning »» Distribute Section A: Student Activity 3 to all students. »» Move around the room to ensure all students understand the task. »» If students are having difficulties completing the task, use suitable questioning to guide them on their way. »» Encourage students to write a description of how to use a sample to make a statement about a population into their copybooks. »» Can all students complete Section A: Student Activity 3? »» We have learned a huge amount about how to use the results from a sample to make statements about a population. »» To review the learning, in groups, I would like you to complete Section A: Student Activity 3. • Students complete Section A: Student Activity 3. »» What information would you need to complete this task? • The size of the sample. • The number of students who said they intended to continue into third-level education. • Use it to calculate a sample »» What would you do with this information? proportion. • Use it to calculate a sample proportion and a 95% confidence interval. • Calculate a 95% confidence interval to take sampling variability into account. • Construct a 95% confidence interval and use it to make a statement about the population. © Project Maths Development Team 2014 www.projectmaths.ie »» Can students describe what information they would need to complete this task? »» Can students describe the process of using the results from a single sample to make a statement about a population? »» Can students describe the process of constructing a 95% confidence interval and how to use it to make a suitable statement about a population? »» Can students explain why it is important to use a confidence interval when making a statement about a population based on a single sample? »» Can students explain what they would need to do to make a more definite statement about the population using a single sample? »» Do students understand that a larger sample size enable a more definitive statement about the population? KEY: » next step • student answer/response 34 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses »» What type of statement would you make about the population? • The proportion of Junior Certificate students who intend going to third level is between x and y. Teacher’s Supports and Actions Assessing the Learning Teacher Reflections • I am 95% confident that the proportion of Junior Certificate students who intend entering third-level education is between a and b. • I am 95% confident that: p^ - ≤ p ≤ p^ + . »» What could you do to improve the quality of the statement you make about the population? • Use a larger sample. • Increase my sample size. • Reduce the margin of error. • Make the 95% confidence interval narrower. • We could increase n. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 35 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports Assessing the and Actions Learning Section B – Hypothesis Testing »» Write some examples of claims on the board. »» We are now going to look at another area of statistics which is important in everyday life and that is determining if claims made by companies, governments, or by anybody are accurate. »» Could you give me an example of a claim you’ve seen or heard in the media? »» Is it important to know if a claim is accurate or not? Explain. Teacher Reflections »» Do students understand what a claim is? »» Can students recall examples of claims that they have seen or heard in everyday life? • Taking a certain supplement will help you lose weight. • Different creams can get rid of wrinkles. • Some yoghurts help boost your immune system. • Smoking causes cancer. • Eating fatty foods causes heart disease. • Support for a political party is at a certain level. »» Can students explain the importance of checking a claim? • Not really. • Yes, so that we don’t waste our money. • Yes, especially if it’s to do with your health. »» We would like to decide a fair way to determine if a claim is true or not. »» We’re going to look at a particular type of claim that is a claim about a proportion. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 36 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Actions »» There is an example of such a claim in Section B: Student Activity 1. »» In groups, I would like you to complete this activity. • Students complete Section B: Student Activity 1. »» Distribute Section B: Student Activity 1 to all students. »» To test a claim what is the first thing we need? • Gather some evidence. • Survey some customers. • Get some data. »» In Question 2 of Section B: Student Activity 1 you are asked to make a statement about satisfaction levels of all of the airline’s customers. Is there another word used to describe “all of the airline’s customers”? • The population. • The population of customers. • No. »» If I made the following statement: “The proportion • No, 0.64 is the sample proportion, not the population of the population that is proportion. satisfied with the service • No, this is the proportion from provided by the airline a single sample. A different is 64%” would you be sample could give a different happy with it? Explain your result. reasoning. • No. Because of sampling variability we cannot say that. • No, the chance of that being true is tiny. © Project Maths Development Team 2014 www.projectmaths.ie »» Move around the room to make sure all students understand what they are supposed to do. »» Use suitable questioning strategies to help students who are having difficulty completing the task. »» Encourage students to discuss each question and to come up with an agreed answer. »» Encourage students to explain their reasoning to each other. Assessing the Learning Teacher Reflections »» Do students understand that to test a claim we need to gather data? »» Do students understand that when we refer to all customers we are talking about the population of customers? »» Do students understand that we cannot say that the proportion of customers in our sample that is satisfied is unlikely to equal the proportion of customers in the population that is satisfied? KEY: » next step • student answer/response 37 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Student Activities: Possible and Expected Teacher’s Supports and Actions Teacher Input Responses »» What statement did you make about satisfaction levels amongst the population of the airline’s customers? • It is very likely that between 0.6084 and 0.6716 of the population are satisfied with the service provided by the airline. • There’s a 95% chance that between 0.6084 and 0.6716 of the population are satisfied with the service provided by the airline. • I am 95% confident that between 0.6084 and 0.6716 of the population are satisfied with the service provided by the airline. • 0.6084 ≤ p ≤ 0.6716. »» Based on our evidence do you think the airline is correct to claim that 70% of their customers are satisfied with the service they provide? Explain your reasoning. • No. • Yes. Their claim is close to the result we got. • No. There is a more than a 95% chance that they are wrong. • No. There is a less than 5% chance that their claim is right. • They could be right but it’s very unlikely. • No. We know there is a 95% chance that the true population proportion lies between 0.6084 and 0.6716 so their claim of 0.7 is unlikely to be true. © Project Maths Development Team 2014 www.projectmaths.ie Assessing the Learning »» Sketch a proportion line on the »» Can students make a board and mark in the sample fair statement about proportion. the proportion of the population which »» On the same diagram, mark in is satisfied with the the airline’s claim. airline’s service? »» Do students recognise »» On the same diagram shade in the need for a the 95% confidence interval. confidence interval when making a statement about the population using a single sample? »» Can students construct a 95% confidence interval correctly? Teacher Reflections »» Do students recognise that the chance of the true population proportion lying outside the 95% confidence interval is very low? »» Do students understand that, based on the 95% confidence interval, it is extremely unlikely that the airline’s claim is correct? KEY: » next step • student answer/response 38 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses »» Is it possible that the true proportion of the population that is satisfied is not between 0.6084 and 0.6716? • Yes. • Yes, but it’s very unlikely. • Yes, but the chance of that is less than 5%. Teacher’s Supports and Actions Assessing the Learning Teacher Reflections »» Do students understand that if a claim lies outside the 95% confidence interval we reject the claim (because it only has a 5% chance or less of being true)? »» When we test a claim we use a 95% confidence interval to determine if the claim is fair or not. »» If a claim lies outside the 95% confidence interval constructed using our data, we reject the claim. »» Does this mean we are rejecting Go Fast Airline’s customer-satisfaction claim of 70%? Explain. © Project Maths Development Team 2014 »» On the board write “I • Yes. reject the airline’s claim”. • Yes, because their claim lies outside our 95% confidence interval. • Yes, because 0.7 lies outside the 95% confidence interval we constructed. www.projectmaths.ie »» Do students understand that because the airline’s claim of 70% lies outside our 95% confidence interval we reject their claim? KEY: » next step • student answer/response 39 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses »» What do you think our conclusion would have been had the company’s claim lay within our 95% confidence interval. • We would have accepted the company’s claim. Teacher’s Supports and Actions Assessing the Learning »» Do students understand that if the claim lies within the 95% confidence interval our conclusion will be different? • We’d conclude that the company’s claim is true. Teacher Reflections • We’d conclude that it is reasonable to say that the proportion of all customers that is satisfied is 70%. »» Let’s look at Section B: Student Activity 2 to see if your last statements make sense. • Students complete Section B: Student Activity 2. »» Distribute Section B: Student Activity 2 to all students. »» Move around the room to make sure that all students are on task and know what to do. »» I would like you to complete the activity, working in groups. »» Help students who are having difficulties with the task using suitable questioning. © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 40 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Actions »» To test a claim, what • Gather some evidence. is the first thing we • Survey some customers. • Get some data. need to do? »» What statement did you make about satisfaction levels amongst the population of the airline’s customers? • There’s a 95% chance that between 0.5584 and 0.7216 of the population are satisfied with the service provided by the airline. • I am 95% confident that the proportion of the population satisfied with the airline’s service is between 0.5584 and 0.7216. »» Based on your data, would you reject the airline’s claim of a 70% satisfaction rating? Explain. • No. • No, because the airline’s claim lies within my 95% confidence interval. • No, because given my 95% confidence interval runs from 0.5584 to 0.7216 it is possible that the true proportion of customers satisfied with the service is 70%. • No, I’d accept the claim. »» Based on your data, would you accept the airline’s claim of a 70% satisfaction rating? Explain. • Yes. • Yes, because the claim lies within my confidence interval. © Project Maths Development Team 2014 www.projectmaths.ie Assessing the Learning »» Can students identify that the first thing needed to test a claim is some data? »» On the board, below the proportion line used in Section B: Student Activity 1, draw another proportion line. »» On the new proportion line, mark in the airline’s claim. »» On the new proportion line mark in the sample proportion. »» On the new proportion line shade in the 95% confidence interval. »» Can students construct a 95% confidence interval using the data and use it to make a statement about the population? »» Write the conclusion “I accept the airline’s claim” on the board. »» Do students naturally use the word “accept” when making a conclusion about a claim which lies within the 95% confidence interval? Teacher Reflections »» Can students make a suitable conclusion regarding the airline’s claim? KEY: » next step • student answer/response 41 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Actions Assessing the Learning »» We need to be a little careful here with how we describe our conclusion regarding the company’s claim. Let’s understand why. Teacher Reflections »» Section B: Student Activity 1 and Section B: Student Activity 2 are similar to each other. Can you identify the similarities between the two activities? »» They’re both about the same airline. »» The claim is the same in both cases. »» The sample proportion we calculate is the same (0.64) for both. »» Even though the sample proportion is the same for both activities our conclusions are very different for each. How can this be? »» Because the confidence intervals are different sizes. »» The margin of error is not the same for each. »» Because the confidence interval is wider in Section B: Student Activity 2, the airline’s claim lies within in. »» Because the confidence interval is narrower in Section B: Student Activity 1, it fails to capture the airline’s claim and so we reject the claim. © Project Maths Development Team 2014 www.projectmaths.ie »» Point out that the confidence intervals from Section B: Student Activity 1 and Section B: Student Activity 2 have different widths. 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 »» Do students recognise that the confidence intervals in Section B: Student Activity 1 and in Section B: Student Activity 2 are different widths? 0.85 KEY: » next step • student answer/response 42 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses »» Why is the confidence interval wider in Section B: Student Activity 2 compared to Section B: Student Activity 1? • Because the sample size is smaller. • Because we didn’t sample as many customers. • Because the margin of error gets bigger as the number in your sample gets smaller. • Because the smaller n is, the wider your 95% confidence interval will be. Teacher’s Supports Assessing the Learning and Actions »» Do students understand what causes the difference in the confidence-interval widths? »» In Section B: Student Activity 2, the • Probably not. consumer agency had originally planned • No, because the 95% confidence interval to sample 1000 customers but only would have been much narrower. managed to sample 150. If they had • No, because we would have gotten a similar sampled 1000 customers do you think result to Section B: Student Activity 2. the 95% confidence interval would still have captured the airline’s claim of 0.7? Explain. »» With this in mind, for Section B: Student Activity 2, can you explain why the airline’s claim falls within the 95% confidence interval? Is it because the claim is now true? • • • • Yes. Not necessarily. Maybe. It’s because the confidence interval is now wider. • It’s because we had a much smaller sample. »» Do students understand that the reason we get two different conclusions for Section B: Student Activity 1 and Section B: Student Activity 2 is because of the different confidence-interval widths? »» When we test a claim and find that it lies within the 95% confidence interval, what could the reason be? • Because the claim is true. • Because a small sample was used making the 95% confidence interval very wide. • It could be for two reasons: The claim could be true or we mightn’t have used a big enough sample to find out that it’s false. »» Do students understand that a claim may lie inside our 95% confidence interval, not because it is true but because the confidence interval is really wide? »» Do students understand that a claim may lie inside our 95% confidence interval because we used a small sample to test it? © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 43 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses »» For this reason when a claim lies within the 95% confidence interval we don’t say we accept the claim, rather we say we fail to reject the claim. »» While each of these statements may seem to say the same thing, they do not. »» If we say “we accept the claim” what does this mean? • It means the claim is true. • It means we accept the claim to be true. »» If we say “we fail to reject the claim” what does this mean? • It means the claim could be true. • It means the claim could be true or we haven’t used a big enough sample to find out. »» When we are evaluating a claim using a sample what are the two possible conclusions we can make? • We can reject the claim or we can fail to reject the claim. © Project Maths Development Team 2014 www.projectmaths.ie Teacher’s Supports and Actions Assessing the Learning »» Draw a line through the statement “I accept the company’s claim” and replace it with “I fail to reject the company’s claim”. »» Underline the two possible conclusions when testing a claim “reject” and “fail to reject”. »» Do students understand that “accept a claim” and “fail to reject a claim” do not mean the same thing? Teacher Reflections »» Do students understand why we use the term “fail to reject” instead of “accept” when making a conclusion about a claim? »» Do students understand that when making a conclusion about a claim the two options are “reject the claim” or “fail to reject the claim”? KEY: » next step • student answer/response 44 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Student Activities: Possible and Expected Responses Teacher Input Teacher’s Supports and Actions Assessing the Learning »» In groups, I would like you to complete Question 1 of Section B: Student Activity 3. • Students complete Question 1 of Section B: Student Activity 3. »» Distribute Section B: Student Activity 3 to all students. »» Can students test the claim contained in the Activity Sheet? »» What was your conclusion about the Department of the Environment’s claim? • I rejected the claim (wrong). • I accepted the claim (incorrect language). • I failed to reject the claim. »» Move around the room, using suitable questioning to test students’ understanding of the activity. »» Do students use the correct language when making their conclusion on the claim? »» Can students justify their work? »» Draw a proportion line on the board and fill in the claim, the sample proportion and the 95% confidence interval. »» Why did you fail to reject the claim? • Because the claim lies within our 95% confidence interval. • Because the claim of 0.85 lies within our 95% confidence interval of 0.7984 and 0.8616. »» Can students justify the use of “fail to reject” when making their conclusion? »» Why is it incorrect to say we accept the claim? • Because the fact that the claim lies within my 95% confidence interval does not necessarily mean it’s true. • The existence of the claim within my 95% confidence interval might be due to a small sample and not because the claim is true. • If I took a larger sample, my margin of error would be smaller and the claim might lie outside my 95% confidence interval. »» Can students explain why using the word “accept” is not correct? © Project Maths Development Team 2014 www.projectmaths.ie KEY: » next step • student answer/response 45 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Actions Assessing the Learning »» As with all areas of mathematics, there is formal language to describe the process of testing a claim. Teacher Reflections »» I am going to describe the • Students answer Question formal language used and 2 of Section B: Student Activity 3. as I do so I would like you to answer Question 2 of Section B: Student Activity 3. »» On one side of the board write the term “Hypothesis Test”. »» The process of testing a claim is known as a hypothesis test. »» The claim being made is known as the null hypothesis. The shorthand notation for the null hypothesis is H0. »» What is the null hypothesis • The proportion of Irish in Section B: Student Activity households who pay the 3? local property tax is 0.85. • 85% of Irish households pay the local property tax. • © Project Maths Development Team 2014 p = 0.85. www.projectmaths.ie »» Do students understand that the null hypothesis means the claim being tested? »» Can students identify the null hypothesis? »» Do students understand that the alternative hypothesis is the counter claim? KEY: » next step • student answer/response 46 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Actions Assessing the Learning »» For every claim that’s made we can make a counter claim. The counter claim is known as the alternative hypothesis. The shorthand notation for the alternative hypothesis is HA or H1. Teacher Reflections »» What is the alternative hypothesis in Section B: Student Activity 3? • 85% of Irish households do not pay the local property tax (wrong). »» When testing a claim (or carrying out a hypothesis test) it is usual to start off by stating the null hypothesis and the alternative hypothesis. • The proportion of Irish households that pays the local property tax is not 0.85. © Project Maths Development Team 2014 • »» Can students identify the alternative hypothesis? p ≠ 0.85. www.projectmaths.ie KEY: » next step • student answer/response 47 Teaching & Learning Plan: Inferential Statistics for Proportions Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Supports and Actions Assessing the Learning »» We have learned a huge amount about testing a claim or about carrying out a hypothesis test. Teacher Reflections »» Let’s take some time to review what we’ve learned. »» Working in pairs, I would like you to complete Question 1 of Section B: Student Activity 4. • Students complete Question 1 of Section B: Student Activity 4. »» Distribute Section B: Student Activity 4 to all students. »» When you have completed Question 1 of Section B: Student Activity 4, I want you to swap your worksheet with the group next to you. »» Move around the room to ensure all students are fully engaged and that they understand what they are meant to do. »» When you’ve received the other group’s worksheet, I want you to answer Question 2 of Section B: Student Activity 4 using their instructions on how to carry out a hypothesis test. »» Emphasise that, having swapped worksheets with another group, students must use the other group’s instructions to answer Question 2 of Section B: Student Activity 4. »» When you’ve answered Question 2, I want you to correct the other group’s Question 1 and give them feedback on their description of a hypothesis test. »» Encourage each group to give constructive feedback to each other in relation to their description of a hypothesis test. © Project Maths Development Team 2014 www.projectmaths.ie »» Can students accurately describe the steps involved in carrying out a formal hypothesis test? »» Can students complete a formal hypothesis test, using appropriate language? »» Can students critically evaluate other students’ description of a hypothesis test? KEY: » next step • student answer/response 48 Teaching & Learning Plan: Inferential Statistics for Proportions Section A: Student Activity 1 The Road Safety Authority of Ireland (Údaras Um Shábháilteacht Ar Bhóithre) is interested in how many lorry drivers speed on a regular basis. To answer this question they choose a simple random sample of lorry drivers from the members list of the Irish Road Haulage Association and asked them if they regularly break the speed limit. Of the 500 drivers who replied, 210 stated that they regularly speed. 1. Using suitable calculations, confirm that the proportion of the sample that admit to regularly speeding is 0.42. 2. A national newspaper includes the following headline on its front page: “42% of all Irish lorry drivers admit to regularly speeding”. Is this a fair statement? Explain your reasoning. 3. Construct a 95% confidence interval for the proportion of Irish lorry drivers who admit to speeding. 4. Write down the margin of error for this sample. 5. Using your 95% confidence interval, make a statement about all lorry drivers which you consider fair. © Project Maths Development Team 2014 www.projectmaths.ie 49 Teaching & Learning Plan: Inferential Statistics for Proportions Section A: Student Activity 2 The government is interested in finding out what proportion of the Irish population is in favour of introducing water metering. To find this out, they commission RED C Research and Marketing Ltd. to carry out a survey on a sample of the population. RED C find that 0.37 of their sample are in favour of introducing water metering. 1. The table below shows some of sample sizes RED C may have used to calculate the proportion of 0.37. For each sample size presented: a Calculate the margin of error (correct to three decimal places). b Construct a 95% confidence interval for the population proportion. c Shade in the 95% confidence interval on the proportion line. d Make a statement about the proportion of the Irish population which is in favour of introducing water metering. Sample size Margin 95% The Proportion Line of error confid ence interval 10 Statement about the population I can be 95% confident that between __ and __ of the population are in favour of water metering. 50 100 200 500 700 1000 © Project Maths Development Team 2014 www.projectmaths.ie 50 Teaching & Learning Plan: Inferential Statistics for Proportions Section A: Student Activity 2 (continued) 2. Draw a graph of margin of error versus sample size. 3. By referring to your table and graph complete the following statements: a As the size of a sample increases, the margin of error _________. b As the size of a sample increases, the 95% confidence interval ________________. 4. By referring to the fourth column of the table, can you see any advantage to using a larger sample size? Explain your reasoning. © Project Maths Development Team 2014 www.projectmaths.ie 51 Teaching & Learning Plan: Inferential Statistics for Proportions Section A: Student Activity 3 For future planning, The Central Application Office (CAO) are interested in what proportion of current Junior Certificate students think it likely that they will continue to third-level education after leaving second-level education. Due to cost and time concerns they decide to use a sample of students to help them answer this question. Describe how the CAO should use a single sample of students to make a fair statement about all students. As part of your description you should consider the following: a What information would the CAO need? b What the CAO would do with this information and why would they do this? (Include a description of any calculations needed.) c The type of statement would the CAO make about the population. d What the CAO could do to improve the quality of the statement they make about the population? © Project Maths Development Team 2014 www.projectmaths.ie 52 Teaching & Learning Plan: Inferential Statistics for Proportions Section B: Student Activity 1 Go Fast Airlines advertise that 70% of their customers are satisfied with the service they provide. A consumer agency wants to determine if the company’s claim is true or not. 1. Describe how the consumer agency could go about determining if the claim is true or not? 2. The consumer agency survey 1,000 customers and find that 640 of these are satisfied with the service provided by the airline. Using this information make a fair statement about satisfaction levels amongst all of the airline’s customers. 3. Based on your answer to Question 2, do you think the airline’s claim of a 70% satisfaction rating is correct? Explain your reasoning. © Project Maths Development Team 2014 www.projectmaths.ie 53 Teaching & Learning Plan: Inferential Statistics for Proportions Section B: Student Activity 2 Go Fast Airlines advertise that 70% of their customers are satisfied with the service they provide. A consumer agency wants to determine if the company’s claim is true or not. 1. Describe how the consumer agency could go about determining if the claim is true or not? 2. The consumer agency plan to survey 1000 customers to help them test this claim, however due to time and money constraints they only manage to survey 150 customers. Of these, 96 say that they are satisfied with the service provided by the airline. Use this information to make a fair statement about satisfaction levels amongst the population of the airline’s customers. 3. Based on your answer to Question 2 what would you conclude about the airline’s claim. © Project Maths Development Team 2014 www.projectmaths.ie 54 Teaching & Learning Plan: Inferential Statistics for Proportions Section B: Student Activity 3 The Department of the Environment, Community and Local Government (Roinne Comhshaoil, Pobail agus Rialtais Áitiúil) claims that 85% of households now pay the local property tax (LPT). An opposition party commissions research to test the validity of this claim. This research finds that, out of a sample of 1,000 households surveyed, 830 confirm that they pay the LPT. 1. Based on the evidence presented, what would you conclude about The Department of the Environment’s claim? 2. Fill in each of the following terms for the above claim. Null Hypothesis Alternative Hypothesis © Project Maths Development Team 2014 www.projectmaths.ie 55 Teaching & Learning Plan: Inferential Statistics for Proportions Section B: Student Activity 4 1. In the space below, write a brief, step-by-step guide of how to carry out a hypothesis test. As part of this you should include explanations of why you carry out each step. Step 1 Step 2 Step 3 © Project Maths Development Team 2014 www.projectmaths.ie 56 Teaching & Learning Plan: Inferential Statistics for Proportions Section B: Student Activity 4 (continued) 2. Using the above instructions, answer the following question: Manchester United claims that 90% of fans were in favour of replacing David Moyes as manager. The Manchester United fanzine, www.rednews.com carries out a survey of 2,000 of its members, 1,700 of whom state that they were in favour of replacing David Moyes. Conduct a hypothesis test of Manchester United’s claim using the evidence presented. Step 1 Step 2 Step 3 © Project Maths Development Team 2014 www.projectmaths.ie 57

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