Chapter 27, Problem 13E

a)

To determine

To state the hypotheses about the slope.

Answer to Problem 13E

  $\begin{array}{l}{H}_0:{\beta }_1=0\\ H_a:{\beta }_1\ne 0\end{array}$

Explanation of Solution

Given:

  

The null and alternative hypotheses:

  $\begin{array}{l}{H}_0:{\beta }_1=0\\ H_a:{\beta }_1\ne 0\end{array}$

That means,

  $H_0:$ There is no association between Marriage age difference and Year.

  $H_a:$ There is an association between Marriage age difference and Year.

b)

To determine

To explain whether the condition for inference are satisfied.

Answer to Problem 13E

Yes, the conditions are satisfied.

Explanation of Solution

Given:

  

  

Yes, the conditions are satisfied.

Random residuals condition: Satisfied, because there is no obvious pattern in the residual plot.

Does the plot thicken? condition: Satisfied, because the vertical spread in the residual plot is not increasing or decreasing.

Nearly normal condition: Satisfied, because the histogram of the residual is roughly symmetric and unimodal.

c)

To determine

To perform the hypothesis test.

Answer to Problem 13E

There is sufficient evidence that the slope is non zero and there is association between Year and marriage age difference.

Explanation of Solution

Given:

  

Formula:

Test statistic:

  $t=\frac{b-{\beta }_0}{S{E}_b}$

The null and alternative hypotheses:

  $\begin{array}{l}{H}_0:{\beta }_1=0\\ H_a:{\beta }_1\ne 0\end{array}$

Test statistic:

  $t=\frac{-0.02996-0}{0.0043}=-6.9674$

The degrees of freedom = df = 26

Therefore, p-value would be,

P-value = 0 …Using excel formula, =TDIST(6.9674,26,2)

Decision: P-value < 0.05, reject H₀.

Conclusion: There is sufficient evidence that the slope is non zero and there is association between Year and marriage age difference.