4. An article reported that in a sample of 244 men, 73 had elevated total cholesterol levels (more than 200 milligrams per deciliter). In a sample of 232 women, 44 had elevated cholesterol levels. Can we conclude at a = 0.05 that the proportion of people with elevated cholesterol levels differs between men and women? Please show all 4 steps of the classical approach clearly and interpret your conclusion.
4. An article reported that in a sample of 244 men, 73 had elevated total cholesterol levels (more than 200 milligrams per deciliter). In a sample of 232 women, 44 had elevated cholesterol levels. Can we conclude at a = 0.05 that the proportion of people with elevated cholesterol levels differs between men and women? Please show all 4 steps of the classical approach clearly and interpret your conclusion.
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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![**Problem 4:**
An article reported that in a sample of 244 men, 73 had elevated total cholesterol levels (more than 200 milligrams per deciliter). In a sample of 232 women, 44 had elevated cholesterol levels. Can we conclude at \( \alpha = 0.05 \) that the proportion of people with elevated cholesterol levels differs between men and women? Please show all 4 steps of the *classical approach* clearly and interpret your conclusion.
### Explanation of the Classical Approach:
1. **State the Hypotheses:**
- Null Hypothesis (\( H_0 \)): The proportion of men with elevated cholesterol levels is equal to the proportion of women with elevated cholesterol levels.
- Alternative Hypothesis (\( H_a \)): The proportion of men with elevated cholesterol levels is different from the proportion of women with elevated cholesterol levels.
2. **Set the Significance Level (\( \alpha \))**:
- \(\alpha = 0.05\)
3. **Compute the Test Statistic:**
- Calculate the sample proportions:
- Proportion of men, \( \hat{p}_1 = \frac{73}{244} \)
- Proportion of women, \( \hat{p}_2 = \frac{44}{232} \)
- Use the following formula to calculate the test statistic for the difference between two proportions:
\[
z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}
\]
where \(\hat{p}\) is the pooled sample proportion calculated as:
\[
\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}
\]
and \( \hat{p}(1 - \hat{p}) \) is the combined proportion of successes and failures.
4. **Make a Decision:**
- Compare the computed z-value with the critical z-value from the standard normal distribution for a two-tailed test at \( \alpha = 0.05 \) (which is approximately \(\pm 1.96\)).
- If the absolute value of the computed z-value is greater than](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F322ab9b5-8b73-42ca-a79a-5c7ee3611144%2Fc888c701-810c-441f-a6a2-6eb6ff53006e%2Fbw95sx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 4:**
An article reported that in a sample of 244 men, 73 had elevated total cholesterol levels (more than 200 milligrams per deciliter). In a sample of 232 women, 44 had elevated cholesterol levels. Can we conclude at \( \alpha = 0.05 \) that the proportion of people with elevated cholesterol levels differs between men and women? Please show all 4 steps of the *classical approach* clearly and interpret your conclusion.
### Explanation of the Classical Approach:
1. **State the Hypotheses:**
- Null Hypothesis (\( H_0 \)): The proportion of men with elevated cholesterol levels is equal to the proportion of women with elevated cholesterol levels.
- Alternative Hypothesis (\( H_a \)): The proportion of men with elevated cholesterol levels is different from the proportion of women with elevated cholesterol levels.
2. **Set the Significance Level (\( \alpha \))**:
- \(\alpha = 0.05\)
3. **Compute the Test Statistic:**
- Calculate the sample proportions:
- Proportion of men, \( \hat{p}_1 = \frac{73}{244} \)
- Proportion of women, \( \hat{p}_2 = \frac{44}{232} \)
- Use the following formula to calculate the test statistic for the difference between two proportions:
\[
z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}
\]
where \(\hat{p}\) is the pooled sample proportion calculated as:
\[
\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}
\]
and \( \hat{p}(1 - \hat{p}) \) is the combined proportion of successes and failures.
4. **Make a Decision:**
- Compare the computed z-value with the critical z-value from the standard normal distribution for a two-tailed test at \( \alpha = 0.05 \) (which is approximately \(\pm 1.96\)).
- If the absolute value of the computed z-value is greater than
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