1. Test statistic:
To find the test statistic, we use the formula:
[tex]\begin{gathered} Z=\frac{\bar{X_d}-\mu_d}{\frac{s_d}{\sqrt[]{n}}} \\ \text{where,} \\ \bar{X}_d=sample\text{ difference} \\ \mu_d=\text{population difference} \\ s_d=\text{standard deviation of the differences } \\ n=\text{ number of people in the survey.} \\ \\ \text{ We use Z statistic because the number of people are more than 30} \end{gathered}[/tex]
Solving for Z, we have:
[tex]\begin{gathered} \bar{X}-\mu_d=3.1\text{ (Average difference given in the question)} \\ \\ \therefore Z=\frac{3.1}{\frac{13.8}{\sqrt[]{40}}}=1.4207\approx1.421\text{ (To 3 decimal places} \end{gathered}[/tex]
Thus, the test statistic is 1.421
2. P-value:
To find the p-value, we check the Z-distribution table.
The value for the p-value is
[tex]2\times0.077658=0.15532\approx0.1553\text{ (To 4 decimal places)}[/tex]
(We multiply by 2 because it is a two-tailed test.
3. Comparison:
The alpha level is 0.001.
Thus, the p-value is greater than the alpha level
