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INFERENCE. What can you find out about a population by looking at a sample?. Getting started. You need a population to sample There should be a reason to sample The koala learning activity was developed by Anthony Harradine (2008). Are the koalas healthy? . Take a sample and make a dotplot. - PowerPoint PPT Presentation
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INFERENCEWhat can you find out about a population by looking at a sample?
Getting started
You need a population to sampleThere should be a reason to sample
The koala learning activity was developed by Anthony Harradine (2008)
Are the koalas healthy?
Take a sample and make a dotplot
What do you notice?
What do you notice?
What do you notice?
The median of the population is likely to be within the range of
sample medians.
The median weight of the female koalas is likely to be between 4.7kg and 5.4kg.
Are the koalas healthy?
Making an inference
The actual population median is 5.1kg.Usually we only see one sample.We make an inference that the population median is the same as the sample median (even though we know that it is probably not exactly the same).This is called a point estimate.
Making an interval estimate
At NZC level 7, the idea of the interval is developed further. Taking samples of different sizes and collecting the medians, you can demonstrate that there is less variation in the medians of large samples than the medians of small samples.
Lindsay Smith, University of Auckland Stats Day 2011
Collections of medians
median40 50 60 70 80 90 100 110
Measures from Sample size 15 Dot Plot
median40 50 60 70 80 90 100 110
Medians from 200 samples of size 30 Dot Plot
median40 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 sampleUsing 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
“ NCEA level 2 is not an endpoint. It is a platform.”