Chapter PII, Problem 26RE

(a)

To determine

To explain which seems to be more useful in predicting January temperature, altitude or latitude.

Answer to Problem 26RE

Latitude seems to be more useful in predicting January temperature.

Explanation of Solution

It is given in the question that data from $55$ large U.S. cities show the mean January temperature, altitude and the latitude. And the correlation matrix for the variables is also given. Thus, it seems latitude to be more useful in predicting January temperature because there is a higher correlation between them.

(b)

To determine

To find out would be the correlation between temperature and latitude if the temperature were measured in degrees Celsius.

Answer to Problem 26RE

The correlation between the temperature and latitude will be $-0.848$ .

Explanation of Solution

It is given in the question that data from $55$ large U.S. cities show the mean January temperature, altitude and the latitude. And the correlation matrix for the variables is also given. Thus, the correlation between the temperature and latitude will be $-0.848$ and if the temperature were measured in degrees Celsius then it will not affect the correlation because units do not affect the correlation.

(c)

To determine

To find out would be the correlation if the temperature were measured in degrees Celsius and the altitude in meters.

Answer to Problem 26RE

The correlation between the temperature and altitude will be $-0.369$ .

Explanation of Solution

It is given in the question that data from $55$ large U.S. cities show the mean January temperature, altitude and the latitude. And the correlation matrix for the variables is also given. Thus, the correlation between the temperature and altitude will be $-0.369$ and if the temperature were measured in degrees Celsius and the altitude in meters then it will not affect the correlation because units do not affect the correlation.

(d)

To determine

To explain what would you predict about the January temperatures in a city whose altitude is two standard deviations higher than the average altitude.

Answer to Problem 26RE

It will be $0.738$ standard deviations lower than the average.

Explanation of Solution

It is given in the question that data from $55$ large U.S. cities show the mean January temperature, altitude and the latitude. And the correlation matrix for the variables is also given. Thus, we can predict that temperature is $0.738$ standard deviations lower than the average as it is two standard deviations so we will multiply the correlation by two and it is about the January temperatures in a city whose altitude is two standard deviations higher than the average altitude.