INFERENCE

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.”