teaching activities towards achievement standard 2.9 use statistical methods to make an inference....
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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?