Question 1170091: In a test of the effectiveness of garlic for lowering cholesterol, 43 subjects were treated with garlic in a processed tablet form. Cholesterol levels were measured before and after the treatment. The changes (beforeafter) in their levels of LDL cholesterol (in mg/dL) have a mean of 5.7 and a standard deviation of 17.9. Construct a 99% confidence interval estimate of the mean net change in LDL cholesterol after the garlic treatment. What does the confidence interval suggest about the effectiveness of garlic in reducing LDL cholesterol?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step.
**1. Identify the Given Information:**
* Sample size (n) = 43
* Sample mean (x̄) = 5.7 mg/dL
* Sample standard deviation (s) = 17.9 mg/dL
* Confidence level = 99%
**2. Determine the Critical Value:**
* Since the sample size is relatively small (n < 30) and the population standard deviation is unknown, we'll use a t-distribution.
* Degrees of freedom (df) = n - 1 = 43 - 1 = 42
* For a 99% confidence interval, the significance level (α) is 1 - 0.99 = 0.01.
* Since it's a two-tailed test, α/2 = 0.005.
* Using a t-table or calculator, the critical t-value (tα/2) for df = 42 and α/2 = 0.005 is approximately 2.698.
**3. Calculate the Margin of Error:**
* Margin of error (E) = tα/2 * (s / √n)
* E = 2.698 * (17.9 / √43)
* E ≈ 2.698 * (17.9 / 6.557)
* E ≈ 2.698 * 2.730
* E ≈ 7.366
**4. Construct the Confidence Interval:**
* Confidence interval = x̄ ± E
* Confidence interval = 5.7 ± 7.366
* Lower bound = 5.7 - 7.366 ≈ -1.666
* Upper bound = 5.7 + 7.366 ≈ 13.066
**5. Interpret the Confidence Interval:**
* The 99% confidence interval for the mean net change in LDL cholesterol is approximately (-1.666, 13.066) mg/dL.
* This means we are 99% confident that the true mean change in LDL cholesterol after garlic treatment is within this interval.
* Since the confidence interval includes zero, it suggests that there is a possibility that the garlic treatment has no effect on LDL cholesterol levels.
* The positive upper bound indicates that it is also possible that the garlic treatment increases LDL cholesterol.
* Therefore, the confidence interval does not provide strong evidence to support the effectiveness of garlic in reducing LDL cholesterol.
**Conclusion:**
The 99% confidence interval (-1.666, 13.066) includes zero, indicating that the garlic treatment's effect on LDL cholesterol could be zero. This suggests that the study does not provide convincing evidence that garlic is effective in reducing LDL cholesterol levels.
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