In a study that was highly published doctors discovered that aspirin helps prevent heart attacks. The research project which was scheduled to last 5 years employed 22,000 american physicians (all male). Half took an aspirin tablet 3 times a week while the other half took a placebo on the same schedule. After 3 years researchers determined that 104 of those who took aspirin and 189 of those who took placebo had heart attacks. Do the results indicate that aspirin is effective in reducing the incidence of heart attacks? Provide the test statistic and the critical value.
To determine if the results indicate that aspirin is effective in reducing the incidence of heart attacks, we can perform a hypothesis test using the given data. Here, the null hypothesis (H0) would be that there is no difference in the incidence of heart attacks between those who took aspirin and those who took the placebo. The alternative hypothesis (H1) would be that aspirin is effective in reducing heart attacks.
To conduct the hypothesis test, we can use the chi-square test for independence. This test is appropriate when comparing categorical variables, such as the incidence of heart attacks between two groups (aspirin and placebo) in this case.
Let's calculate the test statistic and critical value step by step:
Step 1: Set up the hypotheses:
H0: Aspirin has no effect on reducing the incidence of heart attacks.
H1: Aspirin is effective in reducing the incidence of heart attacks.
Step 2: Choose the level of significance, usually denoted as α. Commonly used values are 0.05 (5%) and 0.01 (1%).
Step 3: Calculate the test statistic and critical value.
The test statistic for the chi-square test of independence is calculated using the formula:
χ² = Σ[(O - E)² / E]
Where:
O = Observed frequencies (heart attacks in each group)
E = Expected frequencies (based on the assumption of no effect)
In this case, we have:
Observed frequencies:
- Aspirin group (heart attacks): 104
- Placebo group (heart attacks): 189
Expected frequencies (assuming no effect):
- Aspirin group (heart attacks): (104 + 189) * (104 + 189) / (104 + 189 + 104 + 189) = 146.5
- Placebo group (heart attacks): (104 + 189) * (104 + 189) / (104 + 189 + 104 + 189) = 146.5
The degrees of freedom (df) for the chi-square test of independence is calculated using the formula:
df = (number of rows - 1) * (number of columns - 1)
In this case, we have 2 rows (aspirin and placebo) and 2 columns (heart attacks and no heart attacks), so df = (2 - 1) * (2 - 1) = 1.
Using a chi-square distribution table, you can find the critical value for a given level of significance and degrees of freedom. For example, if α = 0.05 and df = 1, the critical value is 3.841.
Step 4: Compare the test statistic with the critical value.
If the test statistic is greater than the critical value, we reject the null hypothesis in favor of the alternative hypothesis. If the test statistic is less than or equal to the critical value, we fail to reject the null hypothesis.
In this case, calculating the test statistic using the given observed and expected frequencies yields:
χ² = [(104 - 146.5)² / 146.5] + [(189 - 146.5)² / 146.5] = 13.83
Since the test statistic (13.83) exceeds the critical value (3.841) at the chosen significance level of 0.05, we reject the null hypothesis. Therefore, the results indicate that aspirin is effective in reducing the incidence of heart attacks.
Please note that it is always good to consult with a statistician or conduct a full statistical analysis of the data to ensure accuracy and consider any potential confounding factors.