a.
Construct a linear regression model for the data.
Test whether there is enough evidence to conclude that
Test whether there is enough evidence to conclude that
a.
Answer to Problem 8E
A simple linear regression model for the data is:
No, there is not enough evidence to conclude that
No, there is not enough evidence to conclude that
Explanation of Solution
Given info:
The data represents the values of the variables amount of deflection in mm
Calculation:
Linear regression model:
A linear regression model is given as
A linear regression model is given as
Regression:
Software procedure:
Step by step procedure to obtain regression using MINITAB software is given as,
- Choose Stat > Regression > General Regression.
- In Response, enter the numeric column containing the response data Y.
- In Model, enter the numeric column containing the predictor variables X.
- Click OK.
Output obtained from MINITAB is given below:
The ‘Coefficient’ column of the regression analysis MINITAB output gives the slopes corresponding to the respective variables stored in the column ‘Term’.
A careful inspection of the output shows that the fitted model is:
Hence, the linear regression model for the data is:
Test for slope coefficient
The test hypotheses are given below:
Null hypothesis:
That is, intercept of the regression model is not significant.
Alternative hypothesis:
That is, intercept of the regression model is significant.
Level of significance:
Since, the level of significance is not given. The prior level of significance
The ‘P’ column of the regression analysis MINITAB output gives the P- value corresponding to the respective variables stored in the column ‘Term’.
The P- value corresponding to the coefficient
Decision criteria based on P-value approach:
If
If
Conclusion:
The P-value is 0.000 and
Here, P-value is less than the
That is
By the rejection rule, reject the null hypothesis.
Therefore, intercept of the regression model is significant.
Thus, there is not enough evidence to conclude that
Test for slope coefficient
The test hypotheses are given below:
Null hypothesis:
That is, slope coefficient of the predictor variable distance is not significant.
Alternative hypothesis:
That is, slope coefficient of the predictor variable distance is significant.
From the MINITAB output, the P- value corresponding to the coefficient
Conclusion:
The P-value is 0.001 and
Here, P-value is less than the
That is
By the rejection rule, reject the null hypothesis.
Therefore, the slope coefficient of the predictor variable distance is significant.
Thus, there is not enough evidence to conclude that
b.
Construct a quadratic regression model for the data.
Test whether there is enough evidence to conclude that
Test whether there is enough evidence to conclude that
Test whether there is enough evidence to conclude that
b.
Answer to Problem 8E
The quadratic regression model for the data is
No, there is not enough evidence to conclude that
No, there is not enough evidence to conclude that
Yes, there is enough evidence to conclude that
Explanation of Solution
Calculation:
Quadratic model:
The quadratic regression model would be of the form:
Regression:
Software procedure:
Step by step procedure to obtain quadratic regression model using MINITAB software is given as,
- Choose Stat > Regression > General Regression.
- In Response, enter the numeric column containing the response data Y.
- In Model, enter the numeric column containing the predictor variables X and X-square.
- Click OK.
Output obtained from MINITAB is given below:
The ‘Coefficient’ column of the
A careful inspection of the output shows that the fitted model is:
Hence, the quadratic regression model for the data is:
Test for slope coefficient
The test hypotheses are given below:
Null hypothesis:
That is, intercept of the regression model is not significant.
Alternative hypothesis:
That is, intercept of the regression model is significant.
Level of significance:
Since, the level of significance is not given. The prior level of significance
The ‘P’ column of the regression analysis MINITAB output gives the P- value corresponding to the respective variables stored in the column ‘Term’.
The P- value corresponding to the coefficient
Decision criteria based on P-value approach:
If
If
Conclusion:
The P-value is 0.000 and
Here, P-value is less than the
That is
By the rejection rule, fail to reject the null hypothesis.
Therefore, intercept of the regression model is significant.
Thus, there is not enough evidence to conclude that
Test for slope coefficient
The test hypotheses are given below:
Null hypothesis:
That is, coefficient of the predictor variable distance is not significant.
Alternative hypothesis:
That is, coefficient of the predictor variable distance is significant.
From the MINITAB output, the P- value corresponding to the coefficient
Conclusion:
The P-value is 0.000 and
Here, P-value is less than the
That is
By the rejection rule, fail to reject the null hypothesis.
Therefore, coefficient of the predictor variable distance is significant.
Thus, there is not enough evidence to conclude that
Test for slope coefficient
The test hypotheses are given below:
Null hypothesis:
That is, coefficient of the predictor variable distance square is not significant.
Alternative hypothesis:
That is, coefficient of the predictor variable distance square is significant.
From the MINITAB output, the P- value corresponding to the coefficient
Conclusion:
The P-value is 0.000 and
Here, P-value is less than the
That is
By the rejection rule, fail to reject the null hypothesis.
Therefore, the coefficient of the predictor variable distance square is significant.
Thus, there is not enough evidence to conclude that
c.
Construct a cubic regression model for the data.
Test whether there is enough evidence to conclude that
Test whether there is enough evidence to conclude that
Test whether there is enough evidence to conclude that
Test whether there is enough evidence to conclude that
c.
Answer to Problem 8E
The quadratic regression model for the data is
No, there is enough evidence to conclude that
No, there is enough evidence to conclude that
Yes, there is enough evidence to conclude that
Yes, there is enough evidence to conclude that
Explanation of Solution
Calculation:
Cubic model:
The cubic regression model would be of the form:
Regression:
Software procedure:
Step by step procedure to obtain cubic regression model using MINITAB software is given as,
- Choose Stat > Regression > General Regression.
- In Response, enter the numeric column containing the response data Y.
- In Model, enter the numeric column containing the predictor variables X, X-square and X-cube.
- Click OK.
Output obtained from MINITAB is given below:
The ‘Coefficient’ column of the regression analysis MINITAB output gives the slopes corresponding to the respective variables stored in the column ‘Predictor’.
A careful inspection of the output shows that the fitted model is:
Hence, the cubic regression model for the data is:
Test for slope coefficient
The test hypotheses are given below:
Null hypothesis:
That is, intercept of the regression model is not significant.
Alternative hypothesis:
That is, intercept of the regression model is significant.
Level of significance:
Since, the level of significance is not given. The prior level of significance
The ‘P’ column of the regression analysis MINITAB output gives the P- value corresponding to the respective variables stored in the column ‘Term’.
The P- value corresponding to the coefficient
Decision criteria based on P-value approach:
If
If
Conclusion:
The P-value is 0.000 and
Here, P-value is less than the
That is
By the rejection rule, reject the null hypothesis.
Therefore, intercept of the regression model is significant.
Thus, there is not enough evidence to conclude that
Test for slope coefficient
The test hypotheses are given below:
Null hypothesis:
That is, coefficient of the predictor variable distance is not significant.
Alternative hypothesis:
That is, coefficient of the predictor variable distance is significant.
From the MINITAB output, the P- value corresponding to the coefficient
Conclusion:
The P-value is 0.002 and
Here, P-value is less than the
That is
By the rejection rule, reject the null hypothesis.
Therefore, coefficient of the predictor variable distance is significant.
Thus, there is not enough evidence to conclude that
Test for slope coefficient
The test hypotheses are given below:
Null hypothesis:
That is, coefficient of the predictor variable distance square is not significant.
Alternative hypothesis:
That is, coefficient of the predictor variable distance square is significant.
From the MINITAB output, the P- value corresponding to the coefficient
Conclusion:
The P-value is 0.081 and
Here, P-value is greater than the
That is
By the rejection rule, fail to reject the null hypothesis.
Therefore, the coefficient of the predictor variable distance square is not significant.
Thus, there is enough evidence to conclude that
Test for slope coefficient
The test hypotheses are given below:
Null hypothesis:
That is, coefficient of the predictor variable distance cube is not significant.
Alternative hypothesis:
That is, coefficient of the predictor variable distance cube is significant.
From the MINITAB output, the P- value corresponding to the coefficient
Conclusion:
The P-value is 0.627 and
Here, P-value is greater than the
That is
By the rejection rule, fail to reject the null hypothesis.
Therefore, the coefficient of the predictor variable distance cube is not significant.
Thus, there is enough evidence to conclude that
d.
Find the best model among the three models obtained in part (a), part (b) and part (c).
d.
Answer to Problem 8E
The model obtained in part (b) is the best model compared to the other two models in part (a) and part (c).
Explanation of Solution
Calculation:
In the model obtained in part (b), all the coefficients of the model are significantly different from zero.
The coefficient of determination is higher for the model obtained in part (b), than for the model obtained in part (a).
That is,
There is not much difference in the coefficient of determination for the model obtained in part (b) and part (c).
That is, 0.9976% and 0.9978% are not much distinct.
Increasing the number of predictors in an analysis increases the complexity of analysis. An investigator usually does not wish to increase the complications of analysis for such a small increase in
Thus, the model obtained in part (b) is the best model compared to the other two models in part (a) and part (c).
e.
Estimate the amount of deflection at a distance of 1m using the most appropriate method.
e.
Answer to Problem 8E
The estimate of the amount of deflection at a distance of 1m is 0.77231mm.
Explanation of Solution
Calculation:
The quadratic model is the most appropriate regression model among the obtained three models.
The quadratic regression model is,
Here,
Estimate of amount of deflection:
Thus, the estimate of the amount of deflection at a distance of 1m is 0.77231mm.
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Chapter 8 Solutions
Statistics for Engineers and Scientists (Looseleaf)
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