Question

A university conducted a study of whether running is healthy for men and women over age 50. During the first eight years of the study, 1.9% of the 457 members of a 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)

Define the Random Variables X and P', in words. which of the following?

a.) 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.

b.) X is the proportion of all runners that die, and P' is the number of all runners that die.

c.) 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.

d.) X is the number of all runners that die, and P' is the proportion of all runners that die.

Part (b)

Which distribution should you use for this problem? (Round your answers to four decimal places.)

P' ~_ (__,__)

Explain your choice. which if the following?

The Student's t-distribution should be used because

npq≤ 10, which implies a small sample.

a.) The binomial distribution should be used because the two outcomes are "the runner died" and "the runner did not die."

b.) The standard normal distribution should be used because we are interested in proportions.

c.) The Student's t-distribution should be used because we do not know the standard deviation.

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.)

(ii) Sketch the graph.

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

Part (d)

Explain what a "97% confidence interval" means for this study. which of the following?

a.) There is a 97% chance that a runner over the age of 50 will die.

b.) 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.

c.) We are 97% confident that the number of deaths of runners over the age of 50 is less than 3 people.

d.) We are 97% confident that fewer than 3% of people over the age of 50 are runners.

Answer 1

We are given = 0.019 and n = 457

Part (a)

Define the Random Variables X and P', in words. which of the following?

c.) 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.

Part (b)

Which distribution should you use for this problem? (Round your answers to four decimal places.)

P' ~_ (__,__)

b.) The standard normal distribution should be used because we are interested in proportions.

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.

We are given confidence level = 0.97

Therefore α = 1 - 0.97 = 0.03 , 1 - (α/2) = 0.985

So we have to find z score corresponding to area 0.9850 on z score table

So z = 2.17

Confidence interval :

=

=

= 0.0051 and 0.0329

(i) State the confidence interval. (Round your answers to four decimal places.)

0.0051 and 0.0329

(ii) Sketch the graph.

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

0.0329

Part (d)

Explain what a "97% confidence interval" means for this study. which of the following?

b.) 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.

.