STAT 501 Homework 7 Solution (Penn State) Latest Verified Review
2023 Practice Questions and Answers for Exam Preparation, 100%
Correct with Explanations, Highly Recommended, Download to
Score A+
1. (6x2 = 12 points) State whether each of the following statements is true or false. For
the statements that are false, explain why they are false.
(a) A confidence interval for the mean response will be valid only if all LINE conditions
are satisfied.
FALSE
The mean response confidence interval will be valid even if normality is not satisfied,
provided the sample size n is large enough
(b) Increasing the sample size, n, while keeping the confidence level fixed ensures that
the widths of both a mean response confidence interval and a new response
prediction interval will be decreased.
TRUE
As we increase sample size n, the width of confidence interval decreases
(c) A clear linear trend in a scatterplot with residuals on the vertical axis and fitted
values on the horizontal axis suggests that the multiple linear regression linearity
assumption is likely satisfied.
FALSE
If the multiple linear regression linearity assumption is satisfied, then a scatterplot with residuals
on the vertical axis and fitted values on the horizontal axis should appear as a random scatter of
points with no linear trend.
(d) Since a fitted value for a multiple linear regression model is simply a linear
combination of the predictors, once a residual scatterplot with fitted values on the
horizontal axis has been assessed, there is no need to also consider residual
scatterplots with each predictor on the horizontal axis.
FALSE
Residual patterns that violate the model assumptions can show up in a residual scatterplot with a
predictor on the horizontal axis even if the residual plot with fitted values on the horizontal axis
1
, looks fine
(e) A small p-value associated with the Anderson-Darling test for error normality
indicates that the regression errors are normally distributed.
FALSE
A large p-value indicates normality. It results in failure to reject this null hypothesis and
assumption that the errors have a normal distribution.
(f) A large p-value associated with the modified Levene test for constant error variance
indicates that the regression errors have constant variance.
TRUE
A large p-value results in failure to reject this null hypothesis and assumption that the
variance is constant
2. (3 + 5 + 5 + 6 + 4 + 4 = 27 points) Use the “InfctRskL7” dataset. This is part of a larger
data set and these observations come from n = 111 hospitals in the USA. The Y-variable
is Infct = percentage of patients who get an infection while in the hospital. The four
predictor variables are:
Stay = average length of patient stay in each hospital,
Age= age of the patient,
Culture= culture,
Xray = a measure of how often X-rays are given in each hospital.
(a) Fit the multiple linear regression model:
E(Y )= β0 + β 1 Stay+ β 2 Age+ β3 Culture+ β 4 Xray .
Report the SSE (sum of squared errors) and degrees of freedom (df) for SSE.
SSE = 103.670
Df for SSE = 106
(b) Do a general linear F-test, at 5% significance level, to see whether Age and Xray
provide significant information about the response, Infct, beyond the information
provided by the other predictor variables. Write down H 0, Ha, the test statistic, the p-
value, and the conclusion.
[In Minitab, you can find the information for the F-statistic either by selecting
Sequential sums of squares in the regression options for the model in the previous part
or by fitting a reduced model without Age and Xray. For the p-value for the F-statistic:
select Calc > Probability Distributions > F. . . . Select Cumulative probability and leave
the Noncentrality parameter set to 0.0. Next, enter in the respective Numerator
2
2023 Practice Questions and Answers for Exam Preparation, 100%
Correct with Explanations, Highly Recommended, Download to
Score A+
1. (6x2 = 12 points) State whether each of the following statements is true or false. For
the statements that are false, explain why they are false.
(a) A confidence interval for the mean response will be valid only if all LINE conditions
are satisfied.
FALSE
The mean response confidence interval will be valid even if normality is not satisfied,
provided the sample size n is large enough
(b) Increasing the sample size, n, while keeping the confidence level fixed ensures that
the widths of both a mean response confidence interval and a new response
prediction interval will be decreased.
TRUE
As we increase sample size n, the width of confidence interval decreases
(c) A clear linear trend in a scatterplot with residuals on the vertical axis and fitted
values on the horizontal axis suggests that the multiple linear regression linearity
assumption is likely satisfied.
FALSE
If the multiple linear regression linearity assumption is satisfied, then a scatterplot with residuals
on the vertical axis and fitted values on the horizontal axis should appear as a random scatter of
points with no linear trend.
(d) Since a fitted value for a multiple linear regression model is simply a linear
combination of the predictors, once a residual scatterplot with fitted values on the
horizontal axis has been assessed, there is no need to also consider residual
scatterplots with each predictor on the horizontal axis.
FALSE
Residual patterns that violate the model assumptions can show up in a residual scatterplot with a
predictor on the horizontal axis even if the residual plot with fitted values on the horizontal axis
1
, looks fine
(e) A small p-value associated with the Anderson-Darling test for error normality
indicates that the regression errors are normally distributed.
FALSE
A large p-value indicates normality. It results in failure to reject this null hypothesis and
assumption that the errors have a normal distribution.
(f) A large p-value associated with the modified Levene test for constant error variance
indicates that the regression errors have constant variance.
TRUE
A large p-value results in failure to reject this null hypothesis and assumption that the
variance is constant
2. (3 + 5 + 5 + 6 + 4 + 4 = 27 points) Use the “InfctRskL7” dataset. This is part of a larger
data set and these observations come from n = 111 hospitals in the USA. The Y-variable
is Infct = percentage of patients who get an infection while in the hospital. The four
predictor variables are:
Stay = average length of patient stay in each hospital,
Age= age of the patient,
Culture= culture,
Xray = a measure of how often X-rays are given in each hospital.
(a) Fit the multiple linear regression model:
E(Y )= β0 + β 1 Stay+ β 2 Age+ β3 Culture+ β 4 Xray .
Report the SSE (sum of squared errors) and degrees of freedom (df) for SSE.
SSE = 103.670
Df for SSE = 106
(b) Do a general linear F-test, at 5% significance level, to see whether Age and Xray
provide significant information about the response, Infct, beyond the information
provided by the other predictor variables. Write down H 0, Ha, the test statistic, the p-
value, and the conclusion.
[In Minitab, you can find the information for the F-statistic either by selecting
Sequential sums of squares in the regression options for the model in the previous part
or by fitting a reduced model without Age and Xray. For the p-value for the F-statistic:
select Calc > Probability Distributions > F. . . . Select Cumulative probability and leave
the Noncentrality parameter set to 0.0. Next, enter in the respective Numerator
2
