Teaching activities towards Achievement Standard 2.9

Use statistical methods to make an inference.

Lindsay Smith, University of Auckland Stats Day 2011

What is new/changed?

• Use of exploratory data analysis.• Statistical inference comparing two

populations.• Informal confidence intervals for population

medians.• Sampling variability.• Using relevant contextual knowledge (given).

Lindsay Smith, University of Auckland Stats Day 2011

Lindsay Smith, University of Auckland Stats Day 2011

Historical development

• Replaces making an inference about a single population

• Extends development of the curriculum material developed by Chris Wild and his team at Auckland University

Lindsay Smith, University of Auckland Stats Day 2011

Approaches

The approach you take will depend on• Course offered (with maths or without)• Time allowed for the topic• Incorporating Stat Lit (reports) material• Background of students• Access to ICT

Lindsay Smith, University of Auckland Stats Day 2011

Key ideas 1

Sampling Variability• Every sample contains sampling error due to the sampling

process• Other errors, non-sampling errors, may be present due to the

sampling method applied (bias)• Developing an understanding that confidence in the estimate

will vary depending on factors such as sample size, sampling method, the nature of the underlying population, sources of bias.

• Experiencing evidence for the central limit theorem by simulating samples and comparing the distribution of sample medians for samples of different sizes.

Lindsay Smith, University of Auckland Stats Day 2011

Sample statistics

Population parameter: median (or other statistic) of whole population (unknown)

populationsampleSample statistic:

median of sample (known)

Lindsay Smith, University of Auckland Stats Day 2011

Key ideas 2Using the Level 7 guideline for constructing

informal confidence intervals for the population medians

• Informal development of the formula

Lindsay Smith, University of Auckland Stats Day 2011

Key ideas 3

Statistical literacy• Using correct vocabulary: estimate, point

estimate, parameter, sample • Developing critical thinking with respect to the

media involving sampling to make an inference

• Applying the PPDAC cycle

Lindsay Smith, University of Auckland Stats Day 2011

http://www.nzherald.co.nz/

Introduction

Lindsay Smith, University of Auckland Stats Day 2011

Possible data sets

• Stats NZ: Surf (synthetic unit record files 2003)• Census at School: school survey data, Kiwi data, • http://seniorsecondary.tki.org.nz/Mathematics-and-s

tatistics/Achievement-objectives/AO-S7-1• Kiwi Kapers 1: explores the justification for using a

sample to make an inference and sampling variation• Kiwi Kapers 2: explores the effect of sample size so

that we can have confidence in our estimate• Sampling stuff: explores sampling methods to ensure

the sample is representative: stratified sampling

Lindsay Smith, University of Auckland Stats Day 2011

Using the Stage 1 data set

• Note sampling variability• Not the effect of increasing the sample size

Lindsay Smith, University of Auckland Stats Day 2011

Showing the interval for the sample medians

WEIGHTACTUAL40 50 60 70 80 90 100 110

Sample of Stage 1 Statistics Students Box Plot

WEIGHTACTUAL40 50 60 70 80 90 100 110

Sample of Stage 1 Statistics Students Box Plot

WEIGHTACTUAL40 50 60 70 80 90 100 110

Sample of Stage 1 Statistics Students Box Plot

WEIGHTACTUAL40 50 60 70 80 90 100 110

Sample of Stage 1 Statistics Students Box PlotWEIGHTACTUAL40 50 60 70 80 90 100 110

Sample of Stage 1 Statistics Students Box Plot

http://www.censusatschool.org.nz/2009/informal-inference/WPRH/

Lindsay Smith, University of Auckland Stats Day 2011

Observing sampling variability

http://www.censusatschool.org.nz/resources/data-analysis-tools/

InvestigationWhat is the weight of schoolbags carried by year 12 males?

http://www.censusatschool.org.nz/2009/informal-inference/WPRH/

Lindsay Smith, University of Auckland Stats Day 2011

Collections of medians

median

40 50 60 70 80 90 100 110

Measures from Sample size 15 Dot Plot

median

40 50 60 70 80 90 100 110

Medians from 200 samples of size 30 Dot Plot

median

40 50 60 70 80 90 100 110

Measures from Sample size 60 Dot Plot

Lindsay Smith, University of Auckland Stats Day 2011

What else might affect the uncertainty in estimating the population median?

• The spread of the population

• Comparing the heights of intermediate school (years 7 and 8) and the heights of junior high school students (years 7 to 10)

Lindsay Smith, University of Auckland Stats Day 2011

Sampling variability: effect of spread

height100 120 140 160 180 200

Intermediate Dot Plot

height120 140 160 180 200

Middle School Dot Plot

height100 120 140 160 180 200

Sample of Intermediate Box Plot

height120 140 160 180 200

Sample of Middle School Box Plot

height120 140 160 180 200

Sample of Intermediate Box Plot

height120 140 160 180 200

Sample of Middle School Box Plot

Lindsay Smith, University of Auckland Stats Day 2011

Estimating the spread of the population

• Best estimate: using the IQR of our sample• Using the quartiles of our sample as point

estimates for the quartiles of the population

Lindsay Smith, University of Auckland Stats Day 2011

Providing an interval estimate (a confidence interval) for the

population medianThere are two factors which affect the uncertainty of

estimating the parameter:1. Sample size2. Spread of population, estimated with sample IQR

• How confident do we want to be that our interval estimate contains the true population median?

Lindsay Smith, University of Auckland Stats Day 2011

Development of formula for confidence interval

population median = sample median ± measure of spread √sample size

To ensure we predict the population median 90% of the time

population median = sample median ± 1.5 measure of spread √sample size

population median = sample median ± 1.5 x IQR √n

Lindsay Smith, University of Auckland Stats Day 2011

Justification for the calculation

Based on simulations,• The interval includes the true population

median for 9 out of 10 samples - the population median is probably in the interval somewhere

• This leads to being able to make a claim about the populations when they do not overlap

• Sampling variation only produces a shift large enough to make a mistaken claim about once in 40 pairs of samples

Lindsay Smith, University of Auckland Stats Day 2011

Comparing two populations

• Sampling variation is always present and will cause a shift in the medians

• We are looking for sufficient evidence, a big enough shift in the intervals for the median to be able to make a claim that there is a difference back in the populations

Lindsay Smith, University of Auckland Stats Day 2011

Technical aside

When the calculated intervals do not overlap a confidence interval for the difference in the population medians ranges from the smaller distance between the intervals to the larger distance between the intervals.

Lindsay Smith, University of Auckland Stats Day 2011

Retinal image

• http://www.censusatschool.org.nz/2009/informal-inference/WPRH/

• Scroll down to two populations

Lindsay Smith, University of Auckland Stats Day 2011

Observing sampling variability when comparing two populations

Question to explore:

Do students who have a facebook account tend to have newer cellphones than those who do not?

http://www.censusatschool.org.nz/2010/data-viewer/

Lindsay Smith, University of Auckland Stats Day 2011

Applying the PPDAC cycle

• http://www.censusatschool.org.nz/2010/data-viewer/

Question to explore:

Is the average schoolbag weight carried by boys as they start secondary school more than the average weight carried by boys as they start intermediate school?