Question

$H_a:{\mu }_1\ne {\mu }_2$

Short answer

Since the value of test statistic is fall in the rejection region.

Thus, the nullhypothesis is rejected.

Step-by-step solution

Given Information

The hypothesis is

$$\begin{gathered} H_0:{\mu }_1={\mu }_2, \\ {H}_a:{\mu }_1\ne {\mu }_2 \end{gathered}$$

and the level of significance is $0.1.$

Explanation

We have to find the sample mean as follow:

$\overline{d}=\frac{\sum d}{n}$

$=\frac{18}{7}$

$=2.571$

Let's compute the standard deviation:

$s_d=\sqrt{\frac{\sum d_i^2-\frac{{\left(\sum d_i\right)}^2}{n}}{n-1}}$

$=\sqrt{\frac{76-\frac{(18{)}^2}{7}}{7-1}}$

$=2.2254$

The test statistics are,

$t=\frac{\overline{d}}{\frac{s_d}{\sqrt{n^n}}}$

Replace the given values in the equation above.

$z=\frac{2.571}{\frac{2204}{\sqrt{7}}}$

$=3.06$

Compute the degree of freedom as follow

$df=n-1$

$$\begin{gathered} =7-1 \\ =6 \end{gathered}$$

The critical value for a level of significance of $0.1$is $1.943.$