TEACHING ACTIVITIES TOWARDS ACHIEVEMENT STANDARD 91264 (2.9) INTERNAL 4 CREDITS

Use statistical methods to make an inference.

Michelle Dalrymple

Use statistical methods to make an inference.

Population

Sample

Sample statistics

Population

parameterWhat 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 91035 (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 standardsBackground of studentsAccess 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 inference2. 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) 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-S7-1http://seniorsecondary.tki.org.nz/Mathematics-and-statistics/Achievement-objectives/Achievement-objectives-by-level/AO-S7-2

http://nzstatsedn.wikispaces.com/Gisborne+2012

Reminder – PPDAC cycle…

PPDAC REVIEWSAMPLING VARIABILITYPOINT ESTIMATES OF POPULATION PARAMETERS

Lesson 3 & 4

Population

Sample

Sample statistics

Population

parameterWhat 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 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

Comparative question progression – ASIDE from Pip ArnoldLevel 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

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

SAMPLING VARIABILITY – THE EFFECT OF SAMPLE SIZE

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 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 ___

AnalysisSample 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=15 from ___ to ___ from ___ to ___ Median-median = ___IQR-median = ___

n=30 from ___ to ___ from ___ to ___ Median-median = ___IQR-median = ___

n=50 from ___ to ___ from ___ to ___ Median-median = ___IQR-median = ___

n=100 from ___ to ___ from ___ 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 ____.

ConclusionAs 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

We want to plant a reflex…

Movies – one sample - summaryBox plot with memory…

SAMPLING VARIABILITY – THE EFFECT OF SPREAD OF POPULATION

Lesson 7

The scenarioIntermediate SchoolYear 7 & 8

Middle SchoolYear 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

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

WHY?

DEVELOPING THE FORMULA FOR INFORMAL CONFIDENCE INTERVALS FOR THE POPULATION MEDIAN

Lesson 8

Incorporating sample size

So far…

Population

Sample

Sample statistics

Population

parameterWhat 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…

2.2 2.4 2.6 2.8 3.0 3.2median

Measures from Sample of Kiw ipop Dot Plot

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 0.6805 kg

WIDTH 0.138 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 0.0349 kg

WIDTH 0.6805 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

n1

DEVELOPING THE FORMULA FOR INFORMAL CONFIDENCE INTERVALS FOR THE POPULATION MEDIAN

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?

weight1.5 2.0 2.5 3.0 3.5 4.0

Kiw ipop Dot Plot

3 different samples n = 303 different medians3 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:

nIQR

Worksheet 2Deciding 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 median2. 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 INHandout

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) 0.1222EG 5) 1.0399EG 6) 1.0005EG 7) 1.3007EG 8) 2.2880EG 9) 1.3370EG 10)

1.4119

0.113

0.113/0.12689= 0.89

0.159

0.159/0.1075= 1.479

0.212

0.212/0.1479= 1.433

3. 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

0.113

0.113/0.12689= 0.89

0.159

0.159/0.1075= 1.479

0.212

0.212/0.1479= 1.433

3. 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

Prezi recap [if time]

I’d lost them…

PPDAC FOR SUMMARY &CHECKING HOW WELL OUR INTERVALS CAPTURE THE POPULATION MEDIAN

Lesson 9

Problem

What is the median weight of New Zealand kiwis?

FORMULA 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?

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8weight

Sample of Kiw ipop Box Plot

Population median = 2.529 kg

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

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100

-4

-3

-2

-1

0

1

2

3

4

5

PPDAC FOR COMPARISON(CLEAR COMPARISON)

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… http://www.censusatschool.org.nz

AnalysisSelect and use appropriate displays and measures.Construct the informal confidence intervals

AnalysisDiscuss 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 ReganIn 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