Question

$H_2:{\mu }_1<{\mu }_2$

Short answer

Since the value of the test statistic is fall in the accepted region. Thus, the null hypothesis is not rejected.

Step-by-step solution

Given Information

The hypothesis is

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

and the level of significance is 0.1.

Explanation

Compute the sample mean as follow:

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

$=\frac{-17}{7}$

$=-1.714$

The value of the standard deviation is given below:

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

$=\sqrt{\frac{122-\frac{(-12{)}^2}{7}}{7-1}}$

$=4.1115$

The test statistics are,

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

Replace the above value in the given equation

$$\begin{gathered} z=\frac{-1.714}{\frac{4.1115}{\sqrt{7}}} \\ =-1.1 \end{gathered}$$

The value of the degree of freedom is,

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

The critical value for the level of significance of $0.1$is $-1.44.$