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Teaching activities towards Achievement Standard 91264 (2.9) internal 4 credits

Michelle Dalrymple. Use statistical methods to make an inference. Teaching activities towards Achievement Standard 91264 (2.9) internal 4 credits. Use statistical methods to make an inference. Population. Population parameter. Sample. What we’re trying to estimate. Sample statistics.

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Teaching activities towards Achievement Standard 91264 (2.9) internal 4 credits

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  1. Michelle Dalrymple Use statistical methods to make an inference. Teaching activities towards Achievement Standard 91264 (2.9) internal 4 credits

  2. Use statistical methods to make an inference. Population Population parameter Sample What we’re trying to estimate Sample statistics

  3. Historical development • This standard replaces the old sampling standard with making an inference about a single population • Extends development of the curriculum material developed by Chris Wild and his team at Auckland University • Follows on from 91035 (1.10) Multivariate Data

  4. What is new/changed? • Use of exploratory data analysis. • Statistical inference comparing two populations (or two groups within one population). • Informal confidence intervals for population medians. • Sampling variability. • Using relevant (given) contextual knowledge.

  5. Approaches The approach you take will depend on • Type of course offered • Time allowed for the topic • Incorporating Stat Lit (reports) material, or material from other Statistics standards • Background of students • Access to ICT

  6. Key ideas… • Statistical literacy • Correct vocabulary • Sampling variability • Impact of sample size • Impact of spread of population • Informal confidence intervals • Level 7 guide • Making a call based on these intervals

  7. Sequence of learning experiences: Based on work by Lindsay Smith and Pip Arnold • Introduction to making an inference • Sampling methods • Using a sample to make a point estimate & sampling variability • Sampling variability: effect of sample size

  8. Sequence of learning experiences: Based on work by Lindsay Smith and Pip Arnold • Sampling variability: effect of spread of population • Developing the formula for informal confidence interval for the population median • PPDAC for summary & checking how well our intervals capture the population median • PPDAC for comparison (clear difference) • PPDAC for comparison (not a clear difference) Handout

  9. Original resources available on… Lindsay Smith (University of Auckland) http://www.censusatschool.org.nz/2011/statistics-teachers-day-years-12-and-13/ Pip Arnold (Cognition) http://seniorsecondary.tki.org.nz/Mathematics-and-statistics/Achievement-objectives/Achievement-objectives-by-level/AO-S7-1 http://seniorsecondary.tki.org.nz/Mathematics-and-statistics/Achievement-objectives/Achievement-objectives-by-level/AO-S7-2 http://nzstatsedn.wikispaces.com/Gisborne+2012

  10. Reminder – PPDAC cycle…

  11. Lesson 3 & 4 PPDAC reviewsampling variabilitypoint estimates of population parameters

  12. Population Population parameter Sample What we’re trying to estimate Sample statistics

  13. Problem • What sort of question is this? • How would we have worded this question last year? (Level 1) • What other sort of investigative questions are there? • What makes a good question? I wonder what the median weight of Stage 1 Statistics students at Auckland University is?

  14. Reminders… Question types Good questions • SUMMARY • Description of one variable • COMPARISON • Comparing two (or more) subsets of data across a common numeric variable • RELATIONSHIP • Looking at the interrelationship between two paired numeric variables • Can be answered with the data • Population of interest is clear • Variable(s) of interest is clear • Intent (summary, comparison, relationship) is clear • Someone is interested in the answer

  15. Comparative question progression – ASIDE from Pip Arnold Level 6 Level 7 Level 8 – under development still… •  I wonder if heights of NZ Yr 11 boys tend to be greater than heights of NZ year 11 girls • looking for a tendency, do the boxes overlap or not, if they do is it too much • I wonder if the median height of NZ year 11 boys tends to be greater than the median height of NZ year 11 girls • seeing if the informal confidence interval overlap or not • I wonder what the difference in heights  is between NZ year 11 boys and NZ year 11 girls. • finding an interval for the difference – if zero in the interval then probably not making the  call

  16. I wonder what the median weight of Stage 1 Statistics students at Auckland University is? • What do you think the typical weight will be? • Why? • Sketch the shape of the distribution of weights of Stage 1 Statistics students from Auckland University. • Population information

  17. Use sample median to provide a point estimate of the population parameter Conclusion • From my sample data I estimate that the median weight for all Stage 1 statistics students at Auckland University is….

  18. Conclusion • But they’re all different! • Who is right? • From my sample data I estimate that the median weight for all Stage 1 statistics students at Auckland University is….

  19. Everyone’s plots • How can we use our sample to predict what is going on back in the population? • The sample median is our best idea of the population median

  20. Sampling error • The process of taking a sample and using the median of the sample to predict the population median will never produce the exact value of the population median. • This is called sampling error • The difference between the sample median and the true value back in the population

  21. Lesson 5 & 6 Sampling variability – the effect of sample size

  22. Remember we’re in TEACHING WORLD - in the ‘real world’ we wouldn’t be able to take lots and lots of samples to see what happens! Using technology… • Sampling kiwis • Collecting the medians from repeated sampling

  23. Showing this with technology • One sample • Collecting medians

  24. Your collection of medians...

  25. Analysis • For each sample size: • I notice that the sample median weights of kiwis for samples of size ___vary from ___ to ___ • I notice that the bulk of the sample median weights of kiwis for samples of size ___ranged from ___ to ___ • I notice that the median for the sample median weight of kiwis for samples of size ___ is ___ and that the median for the sample IQR is ___

  26. Analysis

  27. Analysis • I notice that the variation of the median weights of kiwis ________ as the sample size _________. • For samples of size 15 the median weight ranged from ____ to ____, a difference of _____, • Whereas for samples of size 100 the median weight ranged from ____ to ____, a difference of ____.

  28. Conclusion As the sample size increases, the variation of the medians __________ • What is a sensible and reliable sample size to use to make inferences about the population?

  29. Conclusion Remember • Our best point estimate of the population parameter – the population median is our sample median • The estimates vary, even with n = 100 • It is better to provide a range of possible values for the parameter, based on our estimate, rather than stating one value

  30. Developing a reflex… • Chris Wild movie - n = 30

  31. We want to plant a reflex…

  32. Movies – one sample - summary Box plot with memory…

  33. Lesson 7 Sampling variability – the effect of spread of population

  34. The scenario Intermediate School Year 7 & 8 Middle School Year 7 – 10 • An intermediate school wants to purchase new furniture for their students, based on the median height of students in years 7 and 8. • A teacher takes a sample of 30 intermediate students from C@S to make an estimate of the population median • A middle school wants to purchase new furniture for their students, based on the median height of middle school students. • A teacher takes a sample of 30 middle school students from C@S to make an estimate of the population median

  35. Which teacher is likely to get a better estimate of the students heights? WHY?

  36. Incorporating sample size Lesson 8 Developing the formula for informal confidence intervals for the population median

  37. So far… Population Population parameter Sample What we’re trying to estimate Sample statistics Median weight of kiwis is somewhere between ___ and ___

  38. Samples of size ___ were reliable enough

  39. Distribution of sample medians… The median weight of kiwis was somewhere between ___ and ___ (90% ish of our sample medians)

  40. However in real life … We don’t get to take multiple samples so this process WON’T work We need to find an informal confidence interval for the population median based ON A SINGLE SAMPLE

  41. Our informal interval needs… To take into account both Sample size and spread

  42. More kiwis… Your turn… Handout

  43. Now… Add your SAMPLE MEDIANS TO THE SHEET

  44. Student worksheet Add your IQR (box) TO THE SHEET

  45. Student worksheet Complete Q3 – Q5 on the worksheet

  46. Q3: I notice that the width of the IQR for sample medians when the sample size is 30 is approximately of the width of the population IQR 1/5 WIDTH 0.138 kg WIDTH 0.6805 kg

  47. Q4: I notice that the width of the IQR for sample medians when the sample size is 400 is approximately __________ of the width of the population IQR 1/20 WIDTH 0.0349 kg WIDTH 0.6805 kg

  48. Q5: Relationship between the width of the IQR for sample medians of sample size n and the population IR and the sample size… • IQR for sample medians (sample size = n) is approximately of the population IQR • When n = 400 the IQR of the sample medians is approximately ________________ of population IQR • When n = 30 the IQR of the sample medians is approximately ________________ of population IQR

  49. How wide should our interval be? Lesson 8 Developing the formula for informal confidence intervals for the population median

  50. Kiwi kapers 3

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