Question 1193670: A hypothesis has been suggested that a principal benefit of physical activity is to prevent sudden death from heart attack. The following study was designed to test this hypothesis: 100 men who died from a first heart attack and 100 men who survived a first attack in the age group were identified and their wives were each given a detailed questionnaire concerning their husbands’ physical activity in the year preceding their heart attacks. The men were then classified as active or inactive. Suppose that 30 of the 100 who survived and 10 of the 100 who died were physically active.
A) Present the data in the 2x2 table by constructing the observed and expected frequencies
B) State Ho, perform a test of this hypothesis
C) What is your conclusion about the relationship between physical activity and mortality from a heart attack
Answer by proyaop(69)   (Show Source):
You can put this solution on YOUR website! **A. 2x2 Table**
| | Survived | Died | Row Total |
|---------|----------|------|----------|
| Active | 30 | 10 | 40 |
| Inactive| 70 | 90 | 160 |
| Column Total| 100 | 100 | 200 |
**Expected Frequencies:**
| | Survived | Died | Row Total |
|---------|----------|------|----------|
| Active | 20 | 20 | 40 |
| Inactive| 80 | 80 | 160 |
| Column Total| 100 | 100 | 200 |
**Calculation of Expected Frequencies (for Active/Survived cell):**
* (Row Total for Active * Column Total for Survived) / Grand Total
* (40 * 100) / 200 = 20
**B. Hypothesis Testing**
* **H0 (Null Hypothesis):** There is no association between physical activity and mortality from a heart attack.
* **H1 (Alternative Hypothesis):** There is an association between physical activity and mortality from a heart attack.
**Chi-Square Test Statistic:**
* χ² = Σ [(O - E)² / E]
* Where:
* O = Observed frequency
* E = Expected frequency
* Calculate for each cell:
* (30 - 20)² / 20 = 5
* (10 - 20)² / 20 = 5
* (70 - 80)² / 80 = 1.25
* (90 - 80)² / 80 = 1.25
* Sum the values: χ² = 5 + 5 + 1.25 + 1.25 = 12.5
**Degrees of Freedom (df):**
* (Number of rows - 1) * (Number of columns - 1) = (2 - 1) * (2 - 1) = 1
* **Critical Value:**
* Using a chi-square distribution table with df = 1 and α = 0.05 (common significance level), the critical value is 3.841.
**Decision:**
* Since the calculated chi-square statistic (12.5) is greater than the critical value (3.841), we reject the null hypothesis.
**C. Conclusion**
* There is sufficient evidence at the 0.05 level of significance to conclude that there is an association between physical activity and mortality from a heart attack.
* The data suggests that physically active individuals are less likely to die from a heart attack.
**Note:**
* This analysis provides a basic framework for conducting a chi-square test for independence.
* Further analysis and consideration of other factors could provide a more comprehensive understanding of the relationship between physical activity and heart health.
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