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# Regression Analysis - Using PHStat or Excel

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 A research analyst for an oil company wants to develop a model to predict miles per gallon based on highway speed. An experiment is designed in which a test car is driven at speeds ranging from 10 miles per hour to 75 miles per hour. The results are in the data set (SPEED.xls).
a) Set up a scatter diagram for speed and miles per gallon.
b) Apply simple regression analysis, and then interpret the meaning of the slope b1 in this problem.
c) Interpret the meaning of the regression coefficient b0 in this problem.
d) Determine the coefficient of determination, r2, and interpret its meaning.
e) How useful do you think this regression model is for predicting mileage?

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#### Solution Preview

 A research analyst for an oil company wants to develop a model to predict miles per gallon based on highway speed. An experiment is designed in which a test car is driven at speeds ranging from 10 miles per hour to 75 miles per hour. The results are in the data set (SPEED.xls).
a) Set up a scatter diagram for speed and miles per gallon.

b) Apply simple regression analysis, and then interpret the meaning of the slope b1 in this problem.
The estimated regression equation is given by,
MPG = 12.7464 + 0.0391 * Speed
Slope, b1 = 0.0391
For an increase in the speed by 10 miles per hour, the mileage increase by 3.91 miles per gallon.
Details
Coefficients Standard Error t Stat P-value
Intercept 12.7464 2.4988 5.1011 0.0000
Speed 0.0391 0.0531 0.7356 0.4686

c) Interpret the meaning of the regression coefficient b0 in this problem.
b0 = 12.7464
If the speed of the car is zero, the mileage of the car will be 12.7464 MPG.
d) Determine the coefficient of determination, r2, and interpret its meaning.
Coefficient of determination, r2 = 0.0204
Interpretation: 2.04% of the variation in MPG can be explained by the estimated regression model.
Details
Regression Statistics
Multiple R 0.1428
R Square 0.0204
Standard Error 5.6658
Observations 28

e) How useful do you think this regression model is for predicting mileage?
The significance of the regression model is tested using F-test.
F-statistic = 0.5411
P-value = 0.4686
Clearly the F-statistic is insignificant at a significance level 0.05, as the p-value is greater than 0.05.
Hence we can conclude that the regression model is not significant in predicting MPG.
Details
ANOVA
df SS MS F Significance F
Regression 1 17.3697 17.3697 0.5411 0.4686
Residual 26 834.6289 32.1011
Total 27 851.9986

The model adequacy is measured using the R2 value. Here R2 = 0.0204. Thus only 2.04% variability in MPG can be explained by the regression model.
Hence the regression model is not appropriate for predicting mileage.

 Refer to the data set given in .
a) We want to assume a quadratic relationship between speed and mileage. Is there any indication in your work in  where this assumption might work as the next model to the approach in ? Explain.
From the scatter plot it is clear that the assumption of linearity is not valid and hence a linear model is not appropriate.
b) At the 0.05 level of significance, determine whether the quadratic model is a better fit than the linear regression model.
The significance of the regression model is tested using F-test.
F-statistic = 141.4596
P-value = 0.0000
Clearly the F-statistic is significant at the significance level 0.05, as the p-value is less than 0.05.
Hence we can conclude that the quadratic model is a better fit than the linear regression model.
Details
ANOVA
df SS MS F Significance ...

#### Solution Summary

The solution provides step by step method for the calculation of regression analysis. Formula for the calculation and Interpretations of the results are also included.

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