Question

The AAA district manager for the northeastern states would like to predict the number of calls they will receive based on the daily low temperatures measured in degrees. The linear regression equation is: y-hat = 4.79 - 0.08x

a. Draw a scatter plot using 10 datapoints. List your datapoints in a table.

b. Find the correlation coefficient, r.

c. What does r-square = .32 mean?

d. Give and interpret the slope and y-intercept.

e. What conclusions can be made about the prediction if x = 110 degrees?

Answer 1

Solution

Part (a)

Unable to draw scatter plot since data points are not available. Answer 1

Part (b)

Vide Part (c), r² = 0.32.

So, correlation coefficient, r = √0.32

= 0.5657 Answer 2

Part (c)

r² represents the proportion of the variation in the response variable that is explained by the variation in the predictor variable and is called coefficient of determination. Contextually, 32% of the variation in the number of calls received is determined by the temperature. Since 32% is not a high percentage, temperature cannot be an effective predictor for the number of calls received. Answer 3

Part (d)

In the estimated regression of Y on X given by: Y_hat = β_0cap + β_1capX,

β_0cap represents the y-intercept mathematically and physically represents the expected value of the response (dependent) variable when the predictor (independent) variable is zero and β_1cap represents the slope of the regression line mathematically and physically represents the expected change (increase/decrease) in value of the response (dependent) variable when the predictor (independent) variable changes (increases/decreases) by one unit.

Contextually, in the given regression equation, y-hat = 4.79 - 0.08x,

- 0.08 implies that when temperature increases/decreases by 1°, the number of calls received would decrease/increase by just 0.08; to get a feel of it, for the number of calls to come down by 1, the temperature has to go up by 12.5°.

4.79 would mean that the number of calls would be just 4.79 when temperature is zero degrees. This may not have a physical implication. Answer 4

Part (e)

Substituting x = 100 in the regression equation, y-hat = 4.79 - 0.08x, the prediction for the number of calls would be 12.79 or 8. Answer 5

[This prediction is subject to the condition that the given set of data points have x values close to 100, because prediction is allowed only if at least 2 or 3 data points are close to 100.]   

DONE