# Statistics Questions

Exam 2PSTAT 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.

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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.

2005

2015

Change

Mean

204.00

162.27

-41.73

Standard deviation

31.70

22.29

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.

2

2

• Let SX

, SY2 , SD

denote the sample variances and X̄, Ȳ , D̄ 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?

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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.)

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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?

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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:

2

SXY

=

n

1 X

(xi − x̄)(yi − ȳ)

n − 1 i=1

2

2

2

Show that SD

= SX

+ SY2 − 2SXY

.

2

.

(b) Find the numeric value of SXY

(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?

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