Chapter PVII, Problem 28RE

(a)

To determine

To explain do you think the assumptions for inference are met.

Answer to Problem 28RE

The assumptions for inference are met.

Explanation of Solution

In the question, the regression analysis with the scatterplot and residual plot for the education and income variables are given. Let us check the conditions as:

Independence Assumption: The sample is largely independent of each other.

Randomization Condition: The sample is randomly selected.

$10\%$ condition: The sample is less than $10\%$ of all population.

So, all the assumptions for inference are met.

(b)

To determine

To explain does there appear to be an association between educational and income levels in these cities.

Answer to Problem 28RE

There appear to be an association between educational and income levels in these cities.

Explanation of Solution

In the question, the regression analysis with the scatterplot and residual plot for the education and income variables are given. Let us define the hypothesis for the test as:

Null hypothesis: There does not appear to be an association between educational and income levels in these cities.

Alternative hypothesis: There appear to be an association between educational and income levels in these cities.

Thus, from the regression analysis we have,

  $\begin{array}{l}{R}^2=32.9\%\\ s=2991\\ df=55\\ P\le 0.0001\end{array}$

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we conclude that there appear to be an association between educational and income levels in these cities.

(c)

To determine

To explain would this association appear to be weaker, stronger or the same if the data were plotted for individual people rather than for cities in aggregate.

Answer to Problem 28RE

The association appear to be weaker.

Explanation of Solution

In the question, the regression analysis with the scatterplot and residual plot for the education and income variables are given. If the data were plotted for individual people rather than for cities in aggregate then the association appear to be weaker as the sample size will increase and the chances of error will also increase.

(d)

To determine

To create and interpret a $95\%$ confidence interval for the slope of the true line that describe the association between income and education.

Answer to Problem 28RE

We are $95\%$ confident that the slope of the true line that describe the association between income and education will lie between $1502.39\text{ and }3387.19$ .

Explanation of Solution

In the question, the regression analysis with the scatterplot and residual plot for the education and income variables are given. Thus, a $95\%$ confidence interval for the slope of the true line that describe the association between income and education is calculated from the regression analysis as:

  $\begin{array}{l}{b}_1\pm t*\times S{E}_1\\ =2444.79\pm 2.00\times 471.2\\ =(1502.39,3387.19)\end{array}$

Thus, we are $95\%$ confident that the slope of the true line that describe the association between income and education will lie between $1502.39\text{ and }3387.19$ .

(e)

To determine

To predict the median income for cities where residents spent an average of $11$ years in school and describe your estimate with a $90\%$ confidence interval and interpret the result.

Answer to Problem 28RE

The median income for cities is $32862.74$ dollars and we are $90\%$ confident that the association between income and education will lie between $1186.686\text{ and }3702.894$ .

Explanation of Solution

In the question, the regression analysis with the scatterplot and residual plot for the education and income variables are given. Thus, the regression equation can be calculated from the regression analysis as:

  $\begin{array}{l}\widehat{y}=b_0+b_{1}x\\ \Rightarrow \widehat{I}\mathrm{ncome}=5970.05+2444.79\times \text{Education}\end{array}$

So, the predicted median income for cities where residents spent an average of $11$ years in school is calculated as:

  $\begin{array}{l}\widehat{I}\mathrm{ncome}=5970.05+2444.79\times \text{Education}\\ =5970.05+2444.79\times 11\\ =32862.74\end{array}$

The median income for cities is $32862.74$ dollars. Now, $90\%$ confidence interval can be estimated as:

  $\begin{array}{l}{b}_1\pm t*\times S{E}_1\\ =2444.79\pm 2.67\times 471.2\\ =(1186.686,3702.894)\end{array}$

Thus, we are $90\%$ confident that the association between income and education will lie between $1186.686\text{ and }3702.894$ .