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Temperature and wind The average temperature (in degrees Fahrenheit) and average wind speed (in miles per hour) were recorded for 365 consecutive days at Chicago鈥檚 O鈥橦are International Airport. Here is the computer output for regression of y = average wind speed on x = average temperature:

a. Calculate and interpret the residual for the day where the average temperature was 42掳F and the average wind speed was 2.2 mph.

b. Interpret the slope.

c. By how much do the actual average wind speeds typically vary from the values predicted by the least-squares regression line with x = average temperature?

d. What percent of the variability in average wind speed is accounted for by the least-squares regression line with x = average temperature?

Short Answer

Expert verified

Part (a) Residual is 7.972528

Part (b) The average wind speed reduces by 0.041077miles per hour per degree Fahrenheit.

Part (c) The average deviation between the anticipated average wind speed and the actual average wind speed was 3.655950mph using the equation of the least square regression line.

Part (d) The least square regression line using average temperature as an explanatory variable explains 4.7874 percent of the variation in average wind speed.

Step by step solution

01

Part (a) Step 1: Given information

The relationship of average temperature and the average wind speed in given in the question.

02

Part (a) Step 2: Concept

The formula used:Residual=yy

03

Part (a) Step 3: Calculation

The general equation of the least square regression line is:

y=b0+b1x

As a result, the estimate of the constant b0 is given in the computer output's row "Intercept" and column "Estimate" as:

b0=11.897762

In the computer output, the estimate of the slope b1 is given in the row "Avg temp" and the column "Estimate" as:

b1=0.041077

As a result, when we plug the values into the general equation, we get:

y=b0+b1xy=11.8977620.041077x

As a result, the average wind speed at 42degrees Fahrenheit is:

y=11.8977620.041077x=11.8977620.041077(42)=10.172528

Thus the residual will be calculated as:.

Residual=yy=2.210.172528=7.972528

This means that while using the regression line to make a prediction, we underestimated the average wind speed on the day with an average temperature of 40Fby7.972528 mph.

04

Part (b) Step 1: Calculation

The question specifies the link between average temperature and average wind speed. The regression line is as follows:

y=11.8977620.041077x

The slope is the coefficient of x in the least-squares regression equation, and it reflects the average rise or decrease of y per unit of x as we all know. Thus,

b0=10.041077

As a result, the average wind speed reduces by0.041077 miles per hour per degree Fahrenheit.

05

Part (c) Step 1: Calculation

The question specifies the link between average temperature and average wind speed. The regression line is as follows:

y=11.8977620.041077x

The standard error of the estimate s is calculated as follows in the computer output following "root mean square error":

s=3.655950

The standard error of the estimations, as we all know, is the average error of forecasts, and thus the average difference between actual and predicted values. As a result, using the equation of the least square regression line, the predicted average wind speed differed by 3.655950 mph on average from the actual average wind speed.

06

Part (d) Step 1: Explanation

The question specifies the link between average temperature and average wind speed. The regression line is as follows:

y=11.8977620.041077x

The coefficient of determination is given as follows in the computer output after "RSquare":

r2=0.0478742=4.7874%

The coefficient of determination, as we all know, represents the proportion of variance in the answers y variable that can be explained by a least square regression model using the explanatory x variable. As a result, we can state that the least square regression line employing average temperature as an explanatory variable explains 4.7874 percent of the variation in average wind speed.

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Most popular questions from this chapter

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