Chapter 10.2, Problem 65E

a.

To determine

To draw: The conclusion by conducting the appropriate significance test.

Answer to Problem 65E

Yes, data set is providing sufficient evidences.

Explanation of Solution

Given:

  $\begin{array}{l}{\overline{x}}_1=68.7,{\overline{x}}_2=54.1\\ s_1=13.3,s_2=11.96\\ n_1=15,n_2=14\end{array}$

Formula used:

Test statistic formula is:

  $t=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$

Calculation:

The null and alternative hypotheses for the provided case are:

  $\begin{array}{l}{H}_0:{\mu }_1-{\mu }_2=10\\ H_1:{\mu }_1-{\mu }_2>10\end{array}$

The test statistic is computed as:

  $\begin{array}{c}t=\frac{{\overline{x}}_1-{\overline{x}}_2-\left({\mu }_1-{\mu }_2\right)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\\ =\frac{68.7-54.1-\left(10\right)}{\sqrt{\frac{{13.3}^2}{15}+\frac{{11.93}^2}{14}}}\\ =0.982\end{array}$

The degree of freedom is calculated as:

  $df=\min \left(n_1-1,n_2-1\right)=\min \left(15-1,14-1\right)=3$

The p-value:

  $P-value=0.828$

In this case,

  $P-value=0.828>0.05$

The null hypothesis could not be rejected which is not showing the sufficient evidences for the claim at significance level of 5%.

b.

To determine

To find: The error that is committed using the result of above part.

Answer to Problem 65E

Type II error.

Explanation of Solution

From above part, the null hypothesis has not been rejected. Thus, there is a possibility of committing the Type II error.