AP Statistics Topic 4.10: Carrying Out a Test for the Difference Between Two Population Means

01 · Start here

The concept in 60 seconds

The null says the $2$ population means are equal. The test asks how unusual your observed sample mean difference would be under that assumption.

Difference → standardized evidence → probability → conclusion. Divide the observed difference by its standard error to get $t$ . Use the appropriate $t$ distribution to find the $p$ -value. Compare that $p$ -value with the planned significance level $\alpha$ .

A small $p$ -value makes the observed result hard to explain by sampling variation under equal population means. A large $p$ -value means the evidence is not strong enough to reject equality at the chosen $\alpha$ ; it does not prove the means are equal.

The direction of the alternative comes from the original question. You cannot switch to the favorable tail after seeing the data.

02 · One situation, deeper understanding

Finish the delivery-time test

Continue the study from Topic 4.9. Independently selected SRSs compare service A’s $2{,}000$ deliveries with service B’s $1{,}800$ deliveries during a defined period. The question is whether A has a longer population mean delivery time. Use $\alpha =0.05$ .

Delivery sample summaries
Service	$n$	Sample mean	Sample $\operatorname{SD}$
A	$40$	$32 \,\mathrm{min}$	$8 \,\mathrm{min}$
B	$36$	$28 \,\mathrm{min}$	$6 \,\mathrm{min}$

Define $\mu_{A}$ and $\mu_{B}$ as the population mean delivery times for these services during that period. Test $H_{0}: \mu_{A}-\mu_{B} = 0$ against $H_{a}: \mu_{A}-\mu_{B} \gt 0$ .

The independent SRSs support random sampling and group independence. The $10\%$ checks are $40\le 200$ and $36\le 180$ . Both sample sizes are at least $30$ . We can use an unpooled two-sample $t$ -test.

$\operatorname{SE}=\sqrt{\frac{8^2}{40}+\frac{6^2}{36}}=\sqrt{2.6}\approx1.612\,\mathrm{min}$
$t=\frac{(32-28)-0}{\sqrt{2.6}}\approx2.481$

Technology gives $\mathrm{df}\approx 71.753$ . The greater-than alternative uses the right-tail probability: $p\approx 0.00773$ .

Locate the observed result under the null model

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Carrying Out a Test for the Difference Between Two Population Means

By the end of this lesson, you should be able to:

The concept in 60 seconds

Finish the delivery-time test

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The concept in 60 seconds

Finish the delivery-time test

Main Menu

Quick Links​

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