Quiz 2 | MATH 4993

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Q1. (Data file: quiz2q1.txt)
The percent survival of a certain type of animal semen after storage was measured at various combinations of
concentrations of three materials used to increase chance of survival.

(i) Fit a multiple linear regression model with an intercept and all three explanatory variables.


(ii) What is the estimate of 2? ________________________

(iii) Use the fitted model in (i) to predict the percent survival, given the respective values of 7.1, 10.2, and
8.11 (weight %) of 1, 2 and 3:

93% confidence interval of the mean
percent survival
93% prediction interval of the individual
percent survival
[ , ] [ , ]

(iv) Use the fitted model in (i) to test if the sum of the regression coefficients for 1 and 2 is equal to zero
at a significant level of 0.01.

p-value: ________________________________________

Conclusion: __________________________________________________________



Quiz 2 | MATH 4993

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Q2. (Data file: quiz2q2.txt)
A chemical engineer is studying a newly developed polymer to be used in removing toxic wastes from water.
Experiments are conducted at five different temperatures. The response noted is the percentage of impurities
removed by the treatment.

Construct a one-way ANOVA model.
Then, use a CONTRAST statement to test the following null hypotheses at 0.05 level of significance:

(i) 0: =

p-value: ________________________________________

Conclusion: __________________________________________________________


(ii) 0 :
+
2
=
+
2


p-value: ________________________________________

Conclusion: __________________________________________________________

Quiz 2 | MATH 4993

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Q3. (Data file: quiz2q3.txt)
Create the following data structure by yourselves in SAS.
There are 9 combinations. Each combination has two observations, sub = 1 and 2.

Then, use a two-way ANOVA model to study the effects of number of years of work experience (year) since
graduation and the geographic locations (region) in the US on the starting salaries (salary) of the graduates.

(i) Is there any suggestion of interaction effect? Please answer it at a 0.03 level of significance.

p-value:_________________________________________________________


Conclusion:______________________________________________________


(ii) Perform an analysis of variance using years 1 and 2 only. Based on an appropriate model, compare the
mean salaries in West versus (Midwest, Northeast) at a 0.04 level of significance.

F-value: ______________________________________________________

p-value: ______________________________________________________

Conclusion: ___________________________________________________

Quiz 2 | MATH 4993

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(iii) Assuming that there is no interaction between year and region in the model with years 1 and 2 only,
carry out a nonparametric test to test whether salaries among regions are different. What is the
conclusion from a parametric test? Is the conclusion different?

Use 0.04 level of significance.

[Non-parametric test]

p-value: ______________________________________________________

[Parametric test]

p-value: ______________________________________________________


Is the conclusion different?

Conclusion: ___________________________________________________


(iv) Consider the resulting model in (i) and use 4% level of significance to test the assumption of normality
based on residuals.














Conclusion: ___________________________________________________
Tests for Normality
Test p-value
Shapiro-Wilk test
Kolmogorov-Smirnov
Cramer-von Mises
Anderson-Darling
Quiz 2 | MATH 4993

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Q4. Use the data set in Q3.
Now, we consider year as a quantitative/ independent variable and region as a categorical
variable, and then fit an ANCOVA model.
(i) Write down fitted regression lines for each geographic location.

For Midwest,

̂ = + .


For Northeast,

̂ = + .


For West,

̂ = + .


(ii) Test whether three slopes are equal at 2% level of significance, and draw a specific conclusion.


p-value: ______________________________________________________

Conclusion: ___________________________________________________

(iii) Estimate the difference between the mean salary of Midwest and the average of mean salaries of
Northeast and West, −
ℎ+
2
, at year = 1.5. Test whether this difference is less than 2
at 2% level of significance, and draw a specific conclusion.

Estimate: _____________________________________________________________________


p-value: ______________________________________________________

Conclusion: ___________________________________________________

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