Exam 2
PSTAT 120B
Instructions: the exam is open book and open note and has no strict time limit. You can use any
course materials, and you are allowed to cite results in the book, lecture notes, and past homeworks
and quizzes without repeating proofs or derivations. Your work should be your own – please do not
consult or work with anyone else.
1
Exam 2 PSTAT 120B, Fall 2021
Background
The CDC’s National Center for Health Statistics website publishes state-level data on heart disease
mortality by year. The CDC comments, “Although adjusted for differences in age-distribution
and population size, rankings by state do not take into account other state specific population
characteristics that may affect the level of mortality. When the number of deaths is small, rankings
by state may be unreliable due to instability in death rates.” For this exam, you’ll investigate the
change in heart disease mortality rates over a ten-year period using tools you’ve learned in the class
based on a sample of 20 randomly selected states.
Below, Figure 1 shows the distributions of mortality rates for 20 randomly selected U.S. states
in 2005 and 2015, as well as a stacked bar chart of the rates for individual states. In the bar chart,
the height of the blue bar shows the rate in 2015, and the height of the red bar shows the rate in
2005.
Figure 1: Top, histograms of age-adjusted heart disease mortality rates (deaths per 100,000) for
each of 20 randomly selected states in 2005 and in 2015; bottom, stacked bar chart of rates for each
state by year.
2
Exam 2 PSTAT 120B, Fall 2021
Next, Figure 2 shows a histogram and bar chart of the differences by state in mortality rates
between the two years for the selected states.
Figure 2: Left, bar chart of change in age-adjusted heart disease mortality rates (deaths per 100,000)
by state between 2005 and 2015; right, histogram of differences by state between the two years.
The differences are calculated as (2015 rate − 2005 rate).
Finally, the summary statistics for the age-adjusted mortality rates in each year separately
(Figure 1) and for the change in rates (Figure 2) are shown below in Table 1.
Mean Standard deviation
2005 204.00 31.70
2015 162.27 22.29
Change -41.73 14.20
Table 1: Summary statistics (mean and standard deviation) for age-adjusted heart disease mortality
rates in 2005 and 2015, and for the difference in rates for each of the two years.
Throughout the exam, please use the following notations.
• Let Xi denote the rate for the ith state in 2005.
• Let Yi denote the rate for the ith state in 2015.
• Let Di = Yi −Xi denote the difference in rates for the ith state.
• Let S2X , S
2
Y , S
2
D denote the sample variances and X¯, Y¯ , D¯ denote the sample means.
• Let θX , θY denote the nationwide mean mortality rates in 2005 and 2015.
• Let θD = θY − θX .
The question of interest throughout the exam will be: has heart disease mortality decreased
in the U.S. from 2005 to 2015?
3
Exam 2 PSTAT 120B, Fall 2021
1. Answer the following questions using the differences D1, . . . , D20 shown in Figure 2.
(a) Test the hypothesis that the mean U.S. mortality rate θD remained the same or increased
from 2005 to 2015 against the alternative that the mean mortality rate decreased. State
the hypotheses, test statistic, rejection region, and p-value. Interpret the conclusion of
the test at level α = 0.01.
(b) Calculate the power of the test you performed in the previous part at θD = −10. What
is the type II error rate for this alternative scenario?
(c) Compute a 99% confidence interval for the change in mean heart rate mortality. Interpret
the interval in context.
(d) What assumption is being made about the variation in heart disease mortality across
the U.S.? (Consider whether your conclusions are consistent with the possibility that
mortality varies systematically by state as suggested by the CDC comment.)
4
Exam 2 PSTAT 120B, Fall 2021
2. Answer the following questions using the separate rates X1, . . . , X20 and Y1, . . . , Y20 shown in
Figure 1.
(a) Carry out the calculations for a test of the hypotheses that the mean U.S. mortality rate
θD remained the same or increased from 2005 to 2015 against the alternative that the
mean mortality rate decreased. State the hypotheses, test statistic, rejection region, and
p-value. Do not interpret the conclusion of the test.
(b) Compute a 99% confidence interval for the change in mean heart rate mortality. Do not
interpret the interval.
(c) How do the test and interval differ from those computed based on the differences in
question 1? What is responsible for this difference mathematically?
5
Exam 2 PSTAT 120B, Fall 2021
3. Consider your answers to the previous two questions and answer the following.
(a) The sample covariance for two equally-sized samples is given by:
S2XY =
1
n− 1
n∑
i=1
(xi − x¯)(yi − y¯)
Show that S2D = S
2
X + S
2
Y − 2S2XY .
(b) Find the numeric value of S2XY .
(c) In light of the above:
i. What is the difference between doing a two-sample test and a one-sample test using
the differences? Specifically, what is ignored in the two-sample test?
ii. What does the two-sample test assume that the one-sample test does not?
6