5. Data from the Framingham Study allow us to compare the distributions of initial serum cholesterol levels for two populations of males: those who go on to develop coronary heart disease and those who do not. The mean serum cholesterol level of the population of men who do not develop heart disease is u = 219 mg/100 ml and the standard deviation is o = 41 mg/100 ml. Suppose, however, that you do not know the true population mean; instead, you hypothesize that µ is equal to 244 mg/100 ml. This is the mean initial serum cholesterol level of men who eventually develop the disease. Since it is believed that the mean serum cholesterol level for the men who do not develop heart disease cannot be higher than the mean level for men who do, a one-sided test conducted at the a = 0.05 level of significance is appropriate. (a) What is the probability of making a type I error? (b) If a sample of size 25 is selected from the population of men who do not go on to develop coronary heart disease, what is the probability of making a type II error? (c) What is the power of the test?
5. Data from the Framingham Study allow us to compare the distributions of initial serum cholesterol levels for two populations of males: those who go on to develop coronary heart disease and those who do not. The mean serum cholesterol level of the population of men who do not develop heart disease is u = 219 mg/100 ml and the standard deviation is o = 41 mg/100 ml. Suppose, however, that you do not know the true population mean; instead, you hypothesize that µ is equal to 244 mg/100 ml. This is the mean initial serum cholesterol level of men who eventually develop the disease. Since it is believed that the mean serum cholesterol level for the men who do not develop heart disease cannot be higher than the mean level for men who do, a one-sided test conducted at the a = 0.05 level of significance is appropriate. (a) What is the probability of making a type I error? (b) If a sample of size 25 is selected from the population of men who do not go on to develop coronary heart disease, what is the probability of making a type II error? (c) What is the power of the test?
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
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Chapter1: Starting With Matlab
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Problem 1P
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![Certainly! Here's a transcription suitable for an educational website:
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**Statistical Analysis Using Framingham Study Data**
The Framingham Study provides data to compare initial serum cholesterol levels between two groups of males: those who develop coronary heart disease and those who do not. The mean serum cholesterol level for men who do not develop heart disease is \( \mu = 219 \) mg/100 ml with a standard deviation \( \sigma = 41 \) mg/100 ml.
Suppose we hypothesize that \( \mu \) equals 244 mg/100 ml, the mean level for men who do develop the disease. Since the hypothesis suggests that the mean cholesterol level for men who do not develop heart disease cannot be higher than that of men who do, a one-sided test at a significance level of \( \alpha = 0.05 \) is appropriate.
**Questions:**
(a) **Type I Error Probability**: What is the probability of incorrectly rejecting the true null hypothesis?
(b) **Type II Error Probability**: If a sample size of 25 is selected from men who do not develop coronary heart disease, what is the probability of failing to reject the false null hypothesis?
(c) **Test Power**: What is the power of the test, or the probability of correctly rejecting a false null hypothesis?
(d) **Increasing Test Power**: How can the power of the test be increased?
(e) **Hypothesis Testing**: Test the null hypothesis:
\[ H_0: \mu \ge 244 \text{ mg/100 ml} \]
against the alternative:
\[ H_A: \mu < 244 \text{ mg/100 ml} \]
at \( \alpha = 0.05 \). If the true population mean is as low as 219 mg/100 ml and you want only a 5% chance of failing to reject \( H_0 \), what sample size is required?
(f) **Sample Size Adjustment**: How would the sample size change if willing to risk a 10% chance of failing to reject a false null hypothesis?
---
This educational material encourages exploration of hypothesis testing, error types, and statistical power in the context of real-world data.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e7bd425-bf78-4dbf-9277-f1a3cc92645c%2F647d9a4f-e2fb-4801-be3f-4bf6125988bf%2F7jy608o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here's a transcription suitable for an educational website:
---
**Statistical Analysis Using Framingham Study Data**
The Framingham Study provides data to compare initial serum cholesterol levels between two groups of males: those who develop coronary heart disease and those who do not. The mean serum cholesterol level for men who do not develop heart disease is \( \mu = 219 \) mg/100 ml with a standard deviation \( \sigma = 41 \) mg/100 ml.
Suppose we hypothesize that \( \mu \) equals 244 mg/100 ml, the mean level for men who do develop the disease. Since the hypothesis suggests that the mean cholesterol level for men who do not develop heart disease cannot be higher than that of men who do, a one-sided test at a significance level of \( \alpha = 0.05 \) is appropriate.
**Questions:**
(a) **Type I Error Probability**: What is the probability of incorrectly rejecting the true null hypothesis?
(b) **Type II Error Probability**: If a sample size of 25 is selected from men who do not develop coronary heart disease, what is the probability of failing to reject the false null hypothesis?
(c) **Test Power**: What is the power of the test, or the probability of correctly rejecting a false null hypothesis?
(d) **Increasing Test Power**: How can the power of the test be increased?
(e) **Hypothesis Testing**: Test the null hypothesis:
\[ H_0: \mu \ge 244 \text{ mg/100 ml} \]
against the alternative:
\[ H_A: \mu < 244 \text{ mg/100 ml} \]
at \( \alpha = 0.05 \). If the true population mean is as low as 219 mg/100 ml and you want only a 5% chance of failing to reject \( H_0 \), what sample size is required?
(f) **Sample Size Adjustment**: How would the sample size change if willing to risk a 10% chance of failing to reject a false null hypothesis?
---
This educational material encourages exploration of hypothesis testing, error types, and statistical power in the context of real-world data.
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