Use statistical methods to make an inference. Michelle Dalrymple

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

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 (1.10) Multivariate Data

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.

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

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

Sequence of learning experiences: Based on work by Lindsay Smith and Pip Arnold 1. Introduction to making an inference 2. Sampling methods 3. Using a sample to make a point estimate & sampling variability 4. Sampling variability: effect of sample size

Sequence of learning experiences: Based on work by Lindsay Smith and Pip Arnold 5. Sampling variability: effect of spread of population 6. Developing the formula for informal confidence interval for the population median 7. PPDAC for summary & checking how well our intervals capture the population median 8. PPDAC for comparison (clear difference) 9. PPDAC for comparison (not a clear difference) Handout

Original resources available on… Lindsay Smith (University of Auckland) teachers-day-years-12-and-13/ teachers-day-years-12-and-13/ Pip Arnold (Cognition) statistics/Achievement-objectives/Achievement- objectives-by-level/AO-S7-1 statistics/Achievement-objectives/Achievement- objectives-by-level/AO-S7-2

Reminder – PPDAC cycle…

Lesson 3 & 4

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

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?

Reminders… Question typesGood 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

Comparative question progression – ASIDE from Pip Arnold Level 6 Level 7  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 Level 8 – under development still…  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

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 Population information

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

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….

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

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

Lesson 5 & 6

Using technology…  Sampling kiwis  Collecting the medians from repeated sampling 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!

Showing this with technology  One sample One sample  Collecting medians Collecting medians

Your collection of medians...

Analysis  For each sample size: A. I notice that the sample median weights of kiwis for samples of size ___ vary from ___ to ___ B. I notice that the bulk of the sample median weights of kiwis for samples of size ___ ranged from ___ to ___ C. 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 ___

Analysis 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 IQR for the sample medians is ___ n=15from ___ to ___ Median-median = ___ IQR-median = ___ n=30from ___ to ___ Median-median = ___ IQR-median = ___ n=50from ___ to ___ Median-median = ___ IQR-median = ___ n=100from ___ to ___ Median-median = ___ IQR-median = ___

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 ____.

Showing this with technology  One sample One sample  Collecting medians Collecting medians

Question?  What do you think will happen to variation in the IQR of the samples as we increase the sample size?  12kiwipopsample2IQR 12kiwipopsample2IQR

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?

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

Developing a reflex…  Chris Wild movie - n = 30 Chris Wild movie - n = 30

We want to plant a reflex…

Movies – one sample - summary Box plot with memory…

Lesson 7

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 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 to make an estimate of the population median

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

Lesson 8 Incorporating sample size

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

Samples of size ___ were reliable enough

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

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 However in real life …

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

Your turn… More kiwis… Handout

Add your SAMPLE MEDIANS TO THE SHEET Now…

Add your IQR (box) TO THE SHEET Student worksheet

Complete Q3 – Q5 on the worksheet Student worksheet

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 WIDTH kg WIDTH kg 1/5

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 WIDTH kg WIDTH kg 1/20

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

Lesson 8 How wide should our interval be?

Kiwi kapers 3

Developing an informal confidence interval for the population median…  For our informal confidence interval for the population median we want to use  Sample median  Sample IQR/  n  We need to see how big to make this interval so we’re pretty sure the interval includes the population median  We want it to work about 90% of the time

 Remember we’re in TEACHING WORLD  We’re going to explore how wide our intervals should be when we can work backwards from a given population.

 Informal confidence intervals… sample median  k x sample IQR/  n  What would be the ideal number ( k ) of sample IQR/  n to use all the time to be pretty sure the interval includes the population median? 3 different samples n = 30 3 different medians 3 different IQRs

That is…  We know what the population median actually is  We can look and see how far away from the population median this is:

Worksheet 2 Deciding how many sample IQR/n we need for the informal confidence interval (finding k ) For each example… 1. Mark the sample median on the big graph and draw a line to the population median 2. Find the distance the sample median is from the population median (2.529kg) 3. Divide by sample IQR/  n  This gives the number of sample IQR /  n that the sample median is away from the population median  THIS IS THE NUMBER WE ARE INTERESTED IN Handout

1. Mark the sample median on the big graph and draw a line to the population median 2. Find the distance the sample median is from the population median (2.529kg) 3. Divide by sample IQR/  n

EG 4) EG 5) EG 6) EG 7) EG 8) EG 9) EG 10) / = / = / = Divide by sample IQR/  n This gives the number of sample IQR/  n that the sample median is away from the population median

From our 10 samples it would appear ±1.5 x IQR/sqrt(n) would be most effective. That is… it should capture the population median most of the time / = / = / = Divide by sample IQR/  n This gives the number of sample IQR/  n that the sample median is away from the population median

The final formula for the informal confidence interval is : Final formula for informal Confidence interval

PreziPrezi recap [if time] I’d lost them…

Lesson 9

Problem  What is the median weight of New Zealand kiwis?

F ORMULA FOR INFORMAL CONFIDENCE INTERVAL FOR THE POPULATION MEDIAN

Plan & Data  Simple random samples of 30 kiwis  I sampled for you Handout

Analysis  Box plot  Summary statistics  I did this for you as well YOU NEED TO…  Use the formula to construct an informal confidence interval for the population median for each sample of 30 kiwis

Analysis

Conclusion Use your interval from SAMPLE A to complete the conclusion  From my sample, I am pretty sure that the median weight of New Zealand kiwis is between ____ and ____

Teaching and learning world  How many of our informal confidence intervals captured the population median?

Population median = kg

How many “lots” of IQR/sqrt(n) our samples are away from the population median

Lesson 11 & 12 Handout

Investigative question  I wonder if the median height of NZ kiwi females tends to be greater than the median height of NZ kiwi males Population parameter Variable of interest Groups/sub-populations Population

Data  I sampled for you  Have a look how… 

Analysis Select and use appropriate displays and measures. Construct the informal confidence intervals

Analysis Discuss sample distributions by comparing features of them. Compare - shape - overlap - shift - spread - middle 50% - unusual or interesting

Conclusion - inference  From my informal confidence intervals, I am pretty sure that the population median height of NZ kiwi females is between ____ and ____.  Similarly, I’m pretty sure that the population median height of NZ kiwi males is between ____ and ____. Teaching and learning world NOTE – Matt Regan In the conclusion… We used "sure" rather than "confident" as we should reserve the use of the term 'confident' to ideas about the confidence we have in our interval estimate (i.e., our confidence interval) which is different from the confidence we have about the 'pattern repeatability' and we don't want students to get muddled.

Conclusion  Based on these samples I would make the call that the population median height of NZ kiwi females is greater then the population median height of NZ kiwi males. That is, I would make the call that NZ kiwi females tend to be taller than NZ kiwi males back in the two populations.

Conclusion - justification  The informal confidence interval for the population median height of NZ kiwi females is (much) further up the scale than the informal confidence interval for the population median height of NZ kiwi males and these informal confidence intervals do not overlap.  I am quite sure that if I were to take another sample of NZ kiwi females and another sample of NZ kiwi males girls this non- overlapping pattern in confidence intervals for the population medians would persist, thus giving the same conclusion.

Conclusion Further thoughts…  What would happen if you took another sample and completed this process again?  What would happen if to the informal confidence intervals if you increased the sample size?

Use statistical methods to make an inference - ASSESSMENT