Confidence Intervals for Differences in Population Means (College Board AP® Statistics): Study Guide

Two-sample t-interval for difference in population means

What is a confidence interval for the difference between two population means?

• A confidence interval for the difference between two population means is

  • a symmetric range of values centered about the sample means difference

  • designed to capture the actual value of the difference between two population means

• Different samples generate different confidence intervals

  • e.g. a sample means difference of 5 may have a confidence interval of (4.5, 5.5)

How do I calculate a confidence interval for the difference between two population means?

• The confidence interval for the difference between two population means is given by

  • $\mathrm{difference}\mathrm{between}\mathrm{sample}\mathrm{means}\pm \left(\mathrm{critical}\mathrm{value}\right)\left(\mathrm{standard}\mathrm{error}\mathrm{of}\mathrm{sample}\mathrm{means}\mathrm{of}\mathrm{two}\mathrm{populations}\right)$

• Where:

  • The difference between the two sample means is calculated from the samples or is given to you

  • The critical value is the relevant t-value

    • The critical value depends on the confidence level C%

  • The standard error is an estimate of how different the difference between two population means is likely to be from the difference between the two sample means, $\begin{array}{rcl}& & \sqrt{\frac{{s_A}^2}{n_A}+\frac{{s_B}^2}{n_B}}\end{array}$

What are conditions for a confidence interval for a difference in population means?

• When calculating a confidence interval, you must show that:

  • the two samples are independent of each other

  • items in the samples (or experiments) must satisfy the independence condition

    • by verifying that data is collected by random sampling

    • or random assignment (in an experiment)

    • and, if sampling without replacement, showing that the sample size is less than 10% of the population size

  • the populations are approximately normally distributed

    • The distribution of both populations needs to be approximately symmetric

    • There should be no outliers

What is the margin of error?

• The margin of error is the half-width of the confidence interval

  • $\mathrm{margin}\mathrm{of}\mathrm{error}=\left(\mathrm{critical}\mathrm{value}\right)\left(\mathrm{standard}\mathrm{error}\mathrm{of}\mathrm{the}\mathrm{difference}\mathrm{of}\mathrm{the}\mathrm{sample}\mathrm{means}\right)$

• The confidence interval is

  • $\mathrm{difference}\mathrm{in}\mathrm{sample}\mathrm{means}\pm \mathrm{margin}\mathrm{of}\mathrm{error}$

• The total width of a confidence interval is $2\times \mathrm{margin}\mathrm{of}\mathrm{error}$

• You may be given an interval and asked to calculate its margin of error

  • or another value, such as $n$

    • This involves forming and solving an equation

How do I interpret a confidence interval for a difference in population means?

• You must conclude calculations of a confidence interval by referring to the context

  • Start by saying 'we can be C% confident that the interval from [lower limit] to [upper limit]...'

    • using the limits from the confidence interval

  • then end with it capturing the difference between the population means in context

    • e.g. 'captures the actual difference between the population means of the time taken by students from each school to run 100 m'

• Confidence intervals for differences may have negative limits

  • This means that the difference, ${\mu }_1-{\mu }_2$, is negative

    • so ${\mu }_1<{\mu }_2$

How do I use confidence intervals to justify a claim about a population mean difference?

• If a population mean difference is claimed to be a specific value

  • check if that value lies in your confidence interval

• If it does, the sample data provides sufficient evidence that the population mean difference is that value

  • If it does not, the sample data does not provide sufficient evidence that the population mean difference is that value

• Look out for confidence intervals for differences that contains zero

  • This means ${\mu }_1-{\mu }_2=0$ so there is evidence to suggest ${\mu }_1={\mu }_2$