Chapter PII, Problem 27RE

(a)

To determine

To find out what percent of the variation in January temperatures can be explained by variation in latitude.

Answer to Problem 27RE

  $71.91\%$ of the variation in January temperatures can be explained by variation in latitude.

Explanation of Solution

It is given in the question the summary statistics for the data relating the latitude and average January temperature for large U.S. cities and the correlation between them is $-0.848$ . Thus, the percent of the variation in January temperatures can be explained by variation in latitude can be calculated as:

  $\begin{array}{l}{R}^2=r^2\\ \Rightarrow R^2={(}^{-}\\ \Rightarrow R^2=0.07191\\ =71.91\%\end{array}$

(b)

To determine

To explain what is indicated by the fact that the correlation is negative.

Answer to Problem 27RE

As latitude increases then the temperature decreases.

Explanation of Solution

It is given in the question the summary statistics for the data relating the latitude and average January temperature for large U.S. cities and the correlation between them is $-0.848$ . Thus, the fact that the correlation is negative indicates that the relationship between the variables in negative. So, we can say that as latitude increases then the temperature decreases.

(c)

To determine

To write the equation of the line of regression for predicting January temperature from latitude.

Answer to Problem 27RE

The regression line is .

Explanation of Solution

It is given in the question the summary statistics for the data relating the latitude and average January temperature for large U.S. cities and the correlation between them is $-0.848$ . Thus, the equation of the line of regression is calculated as:

  $\begin{array}{l}\text{Slope}=r\times \frac{{\sigma }_2}{{\sigma }_1}\\ =-0.848\times \frac{13.49}{5.42}\\ =-2.111\\ y-\text{intercept}={\mu }_2-\beta {\mu }_1\\ =26.44-(-2.111)39.02\\ =108.8\end{array}$

So, the regression line is:

  

(d)

To determine

To explain what the slope of the line means.

Explanation of Solution

It is given in the question the summary statistics for the data relating the latitude and average January temperature for large U.S. cities and the correlation between them is $-0.848$ . And the regression line is:

  

Thus, from this regression line, the slope of the line means that for every degree the latitude increases the temperature drops by ${2.111}^0\text{F}$ .

(e)

To determine

To explain do you think the y -intercept is meaningful.

Explanation of Solution

It is given in the question the summary statistics for the data relating the latitude and average January temperature for large U.S. cities and the correlation between them is $-0.848$ . And the regression line is:

  

Thus, from the regression line, the y -intercept means that the average temperature in a city at zero latitude would be ${108.8}^0$ but this is an extrapolation so, it is hard to tell if it is important.

(f)

To determine

To predict the mean January temperature there.

Answer to Problem 27RE

  ${24.36}^0\text{F}$ is the predicted mean January temperature in Denver.

Explanation of Solution

It is given in the question the summary statistics for the data relating the latitude and average January temperature for large U.S. cities and the correlation between them is $-0.848$ . And the regression line is:

  

Thus, the latitude of Denver is $40$ degree North. Thus, the January temperature is predicted as:

  

(g)

To determine

To explain what does it mean if the residual for a city is positive.

Explanation of Solution

It is given in the question the summary statistics for the data relating the latitude and average January temperature for large U.S. cities and the correlation between them is $-0.848$ . And the regression line is:

  

Thus, if the residual for a city is positive then it means that the actual temperature is higher than the temperature predicted by the model.