Stanford University conducted a study of whether running is healthy for men and women over age 50. During the first eight years of the study, 1.5% of the 451 members of the 50-Plus Fitness Association died. We are interested in the proportion of people over 50 who ran and died in the same eight-year period. NOTE: If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

Part (a)
In words, define the random variables X and P'.
X is the proportion of people who ran and died in the eight-year period, and P' is the number of people who ran and died in the eight-year period.
X is the number of all runners that die, and P' is the proportion of all runners that die.
X is the proportion of all runners that die, and P' is the number of all runners that die.
X is the number of people who ran and died in the eight-year period, and P' is the proportion of people who ran and died in the eight-year period.
Correct: Your answer is correct.
Correct! The random variable of interest, X, is the number of runners in the group who have died in the eight-year period, and P' represents the proportion of the runners in the group who have died in the eight-year period.
Part (b)
Which distribution should you use for this problem? (Round your answers to four decimal places.)
P' ~

Correct: Your answer is correct.

Correct: Your answer is correct.
,
Incorrect: Your answer is incorrect.

Explain your choice.
The binomial distribution should be used because the two outcomes are "the runner died" and "the runner did not die."
The Student's t-distribution should be used because we do not know the standard deviation.
The standard normal distribution should be used because we are interested in proportions.
The Student's t-distribution should be used because

npq
≤ 10,
which implies a small sample.
Correct: Your answer is correct.
Correct! We use the standard normal distribution to create confidence intervals about the population proportion.
Part (c)
Construct a 97% confidence interval for the population proportion of people over 50 who ran and died in the same eight-year period.
(i) State the confidence interval. (Round your answers to four decimal places.)

Incorrect: Your answer is incorrect.
,

(ii) Sketch the graph.

Correct: Your answer is correct.

Correct: Your answer is correct.

Correct: Your answer is correct.

(iii) Calculate the error bound. (Round your answer to four decimal places.)

Part (d)
Explain what a "97% confidence interval" means for this study.
We are 97% confident that the population proportion of runners over the age of 50 that will die in the next eight years is between the interval values.
We are 97% confident that the number of deaths of runners over the age of 50 is less than 3 people.
There is a 97% chance that a runner over the age of 50 will die.
We are 97% confident that fewer than 3% of people over the age of 50 are runners.