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MEM 512
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FINAL EXAM
(Duration: 3 hours)
__________________________
Thursday 23 December 2021
Student # _______________
Problem 1 (8 marks): Answer the following statistical inference questions. Don’t forget to write your
conclusion in words.
1. The weights, in pounds, of two-month-old babies in a sample of 15 are the following:
8.9
8.6
8.0
8.3
8.8
8.6
8.1
7.2
8.0
8.6
9.1
9.0
9.1
8.3
7.9
Assuming that the data were sampled from a normal population distribution, find a 95% confidence
interval for the mean weight of two-month-old babies.
2. The intelligence quotients (IQs) of 5 students from one area of a city are given as follows:
98
117
102
111
109
Assuming the IQ score is normally distributed, do the data suggest that the population mean IQ
exceeds 100? Carry out a test using a significance level of 0.05.
3. Out of a random sample of 1,500 people in a presidential election poll, 820 voters are found to
support a certain candidate. Find a 95% confidence interval for the support rate of this candidate.
4. A group of concerned citizens wants to show that less than half of the voters support the president’s
handling of a recent crisis. A random sample of 500 voters gives 228 in support. Does this provide
strong evidence for concluding that less than half of all voters support the president’s handling of the
crisis?
Problem 2 (8 marks): The Tastee Bakery Company supplies a bakery product to many supermarkets in a
metropolitan area. The company wishes to study the effect of the shelf display height employed by the
supermarkets on monthly sales (measured in cases of 10 units each) for this product. Shelf display
height, the factor to be studied, has three levels—bottom (B), middle (M), and top (T)—which are the
treatments. To compare these treatments, the bakery uses a completely randomized experimental
design. For each shelf height, six supermarkets (the experimental units) of equal sales potential are
randomly selected, and each supermarket displays the product using its assigned shelf height for a
month. At the end of the month, sales of the bakery product (the response variable) at the 18
participating stores are recorded, giving the data in the file “bake.csv”. Let μB, μM, and μT represent the
mean monthly sales when using the bottom, middle, and top shelf display heights, respectively.
1. Draw the box plots of the sales for each type of shelf (similar to the figure below). Which display
height seems to give the highest bakery product sales?
2. Run a one-way ANOVA of the bakery sales study data.
3. Test the null hypothesis that μB, μM, and μT are equal by setting α = 0.05. On the basis of this test, can
we conclude that the bottom, middle, and top shelf display heights have different effects on mean
monthly sales?
4. Consider the pairwise differences μM – μB, μT – μB, and μT – μM. Find a Tukey simultaneous 95 percent
confidence interval for each pairwise difference. Interpret the meaning of each interval in practical
terms. Which display height maximizes mean sales?
Problem 3 (9 marks): The Boston Housing Dataset in the file “Boston.csv” contains information collected
by the U.S Census Service concerning housing in the area of Boston Mass. There are 14 attributes in
each case of the dataset.
1. Read these data into a data frame called “boston”.
2. Run a regression to predict “medv” in terms of “lstat”. Write the regression equation.
3. Give an interpretation of the coefficients.
4. Find the coefficient of determination and give its interpretation.
5. Find the coefficient of correlation and give its interpretation.
6. Test whether there is a significant regression relationship between “medv” and “lstat”.
7. What would be the median value of a home if lstat = 12.65.
8. Draw a scatterplot of the data and the line of best fit on the same graph (similar to the figure below).
The Boston Housing Dataset contains information collected by the U.S Census Service concerning
housing in the area of Boston Mass. There are 14 attributes in each case of the dataset. They are:
crim
zn
indus
chas
nox
rm
age
dis
rad
tax
ptratio
black
lstat
medv
Per capita crime rate by town
Proportion of residential land zoned for lots over 25,000 sq.ft.
Proportion of non-retail business acres per town
Charles River dummy variable (1 if tract bounds river; 0 otherwise)
Nitric oxides concentration (parts per 10 million)
Average number of rooms per dwelling
Proportion of owner occupied units built prior to 1940
Weighted distances to five Boston employment centers
Index of accessibility to radial highways
Full value property tax rate per $10,000
Pupil teacher ratio by town
1000(Bk – 0.63)^2 where Bk is the proportion of blacks by town
% lower status of the population
Median value of owner occupied homes in $1000’s
Problem 4 (bonus 2 marks): An oil company wishes to improve the gasoline mileage obtained by cars
that use its regular unleaded gasoline. Company chemists suggest that an additive, ST-3000, be blended
with the gasoline. In order to study the effects of this additive, mileage tests are carried out in a
laboratory using test equipment that simulates driving under prescribed conditions. The amount of
additive ST-3000 blended with the gasoline is varied, and the gasoline mileage for each test run is
recorded. The file “mileage.csv” gives the results of the test runs. Here the dependent variable y is
gasoline mileage (in miles per gallon) and the independent variable x is the amount of additive ST-3000
used (measured as the number of units of additive added to each gallon of gasoline). One of the study’s
goals is to determine the number of units of additive that should be blended with the gasoline to
maximize gasoline mileage. The company would also like to predict the maximum mileage that can be
achieved using additive ST-3000.
1. Run a regression to relate mileage y to additive x by using the quadratic model y = β0 + β1x + β2×2 + ε.
Write the regression equation.
2. Plot the data and the best fit curve on the same graph (similar to the figure below).
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