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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, 1.7% of the 455 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.) O Part (a) Define the Random Variables X and P', in words. O Xis the proportion of all runners that die, and P'is the number of all runners that die. O Xis 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. O Xis the number of all runners that die, and P' is the proportion of all runners that die. O Xis 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. O Part (b) Which distribution should you use for this problem? (Round your answers to four decimal places.) P-O Explain your choice. O The Student's t-distribution should be used because V npg s 10, which implies a small sample. O The standard normal distribution should be used because we are interested in proportions. O The binomial distribution should be used because the two outcomes are "the runner died" and "the runner did not die." O The Student's t-distribution should be used because we do not know the standard deviation.

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, 1.7% of the 455 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.) O Part (a) Define the Random Variables X and P', in words. O Xis the proportion of all runners that die, and P'is the number of all runners that die. O Xis 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. O Xis the number of all runners that die, and P' is the proportion of all runners that die. O Xis 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. O Part (b) Which distribution should you use for this problem? (Round your answers to four decimal places.) P-O Explain your choice. O The Student's t-distribution should be used because V npg s 10, which implies a small sample. O The standard normal distribution should be used because we are interested in proportions. O The binomial distribution should be used because the two outcomes are "the runner died" and "the runner did not die." O The Student's t-distribution should be used because we do not know the standard deviation.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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**Study Overview:** A university conducted a study to determine if running is healthy for individuals over 50. The study spanned eight years, during which 1.7% of 455 fitness association members died. The focus is on the proportion of over-50 individuals who ran and subsequently died in this period. **Note:** If using a Student’s t-distribution, assume the underlying population is normally distributed, although this must typically be proven. --- **Part (a): Define the Random Variables \(X\) and \(P'\).** 1. \(X\) is the proportion of all runners that die, and \(P'\) is the number of all runners that die. 2. \(X\) 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. 3. \(X\) is the number of all runners that die, and \(P'\) is the proportion of all runners that die. 4. \(X\) 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): Distribution Selection** - **Task:** Choose the appropriate distribution for this problem and round answers to four decimal places. Represent as: \(P' = \left(\_\_\_\_\_, \_\_\_\_\_\right)\) **Explain Your Choice:** - The Student’s t-distribution is appropriate because \(\sqrt{npq} \leq 10\), suggesting a small sample size. - The standard normal distribution is used as we are dealing with proportions. - Binomial distribution applies since outcomes are either "the runner died" or "the runner did not die." - Student’s t-distribution applies as the standard deviation is unknown.
Transcribed Image Text:**Study Overview:** A university conducted a study to determine if running is healthy for individuals over 50. The study spanned eight years, during which 1.7% of 455 fitness association members died. The focus is on the proportion of over-50 individuals who ran and subsequently died in this period. **Note:** If using a Student’s t-distribution, assume the underlying population is normally distributed, although this must typically be proven. --- **Part (a): Define the Random Variables \(X\) and \(P'\).** 1. \(X\) is the proportion of all runners that die, and \(P'\) is the number of all runners that die. 2. \(X\) 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. 3. \(X\) is the number of all runners that die, and \(P'\) is the proportion of all runners that die. 4. \(X\) 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): Distribution Selection** - **Task:** Choose the appropriate distribution for this problem and round answers to four decimal places. Represent as: \(P' = \left(\_\_\_\_\_, \_\_\_\_\_\right)\) **Explain Your Choice:** - The Student’s t-distribution is appropriate because \(\sqrt{npq} \leq 10\), suggesting a small sample size. - The standard normal distribution is used as we are dealing with proportions. - Binomial distribution applies since outcomes are either "the runner died" or "the runner did not die." - Student’s t-distribution applies as the standard deviation is unknown.
**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.** - **Graph Description**: The graph is a normal distribution curve with two tails. The central part of the curve, marked as "C.L.," represents the confidence level. The areas in the tails of the curve are marked as \(\alpha/2\). - \(\alpha/2 =\) ______ - C.L. = ______ - \(\alpha/2 =\) ______ - \(P'\) is marked on the horizontal axis under the peak of the curve. (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. - ○ We are 97% confident that fewer than 3% of people over the age of 50 will die. - ○ There is a 97% chance that a runner over the age of 50 will die.
Transcribed Image Text:**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.** - **Graph Description**: The graph is a normal distribution curve with two tails. The central part of the curve, marked as "C.L.," represents the confidence level. The areas in the tails of the curve are marked as \(\alpha/2\). - \(\alpha/2 =\) ______ - C.L. = ______ - \(\alpha/2 =\) ______ - \(P'\) is marked on the horizontal axis under the peak of the curve. (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. - ○ We are 97% confident that fewer than 3% of people over the age of 50 will die. - ○ There is a 97% chance that a runner over the age of 50 will die.
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