Chapter 11.1, Problem 1E

To determine

To find:

The test statistic for a hypothesis test for two population means.

Answer to Problem 1E

Solution:

The test statistic is 1.16

Explanation of Solution

Given information:

${\overline{x}}_1=9.21,{\sigma }_1=2.01,n_1=45,{\overline{x}}_2=8.76,{\sigma }_2=1.77,n_2=51,H_0:{\mu }_1-{\mu }_2\ge 0$

Where: ${\overline{x}}_1$ and ${\overline{x}}_2$ are the two sample means, ${\sigma }_1$ and ${\sigma }_2$ are the two population standard deviations, $n_1$ and $n_2$ are the two sample sizes and $H_0:{\mu }_1-{\mu }_2\ge 0$ is the null hypothesis.

Steps:

The difference between the means of two populations i.e. ${\mu }_1-{\mu }_2$ is estimated by the difference between their respective sample means. This is used for comparing the two populations.

Test statistics is a random variable which is calculated from the sample data and used in hypothesis testing.

Test statistic calculates the degrees of acceptance between sample data and null hypothesis.

Formula used:

The test statistic for a hypothesis test for two population means is:

$z=\frac{({\overline{x}}_1-{\overline{x}}_2)-({\mu }_1-{\mu }_2)}{\sqrt{\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_2}}}$

Where: ${\overline{x}}_1$ and ${\overline{x}}_2$ are the two sample means, ${\mu }_1-{\mu }_2$ is the hypothetical value of the difference between the two population means ${\sigma }_1$ and ${\sigma }_2$ are the two population standard deviations, $n_1$ and $n_2$ are the two sample sizes.

Calculation:

Test statistic:

Substituting ${\overline{x}}_1=9.21,{\sigma }_1=2.01,n_1=45,{\overline{x}}_2=8.76,{\sigma }_2=1.77,n_2=51,{\mu }_1-{\mu }_2=0$ in the above equation,

$\begin{array}{l}z=\frac{(9.21-8.76)-0}{\sqrt{\frac{{(2.01)}^2}{45}+\frac{{(1.77)}^2}{51}}}\\ =\frac{0.45}{\sqrt{0.0898+0.0614}}\\ =\frac{0.45}{0.3888}\\ =1.1574\approx 1.16\end{array}$