Question

In a test of the effectiveness of garlic for lowering​ cholesterol, 42 subjects were treated with garlic in a processed tablet form. Cholesterol levels were measured before and after the treatment. The changes ​(beforeminus​after) in their levels of LDL cholesterol​ (in mg/dL) have a mean of 3.6 and a standard deviation of 16.7. 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 1

x̅_d = 3.6, s_d = 16.7, n = 42

99% Confidence interval :

At α = 0.01 and df = n-1 = 41, two tailed critical value, t-crit = T.INV.2T(0.01, 41) = 2.701

Lower Bound = x̅_d - t-crit*s_d/√n = 3.6 - 2.701 * 16.7/√42 = -3.3606

Upper Bound = x̅_d + t-crit*s_d/√n = 3.6 + 2.701 * 16.7/√42 = 10.5606

-3.36 < µ_d < 10.56

Answer 2

To construct a confidence interval for the mean net change in LDL cholesterol after the garlic treatment, we can use the following information:

Sample size (n): 42 Mean net change (x̄): 3.6 Standard deviation (s): 16.7 Confidence level: 99% (α = 0.01)

The formula for calculating the confidence interval for the mean is:

CI = x̄ ± Z * (s / √n)

Where: CI = Confidence Interval x̄ = Sample mean Z = Z-score for the desired confidence level s = Sample standard deviation n = Sample size

First, we need to find the Z-score corresponding to a 99% confidence level. We can look it up in the Z-table or use a statistical calculator. For a two-tailed test with α = 0.01, the Z-score is approximately 2.626.

Now, let's calculate the confidence interval:

CI = 3.6 ± 2.626 * (16.7 / √42) = 3.6 ± 8.774 ≈ (-5.174, 12.374)

The 99% confidence interval estimate for the mean net change in LDL cholesterol after the garlic treatment is approximately (-5.174, 12.374) mg/dL.

Interpretation: Since the confidence interval contains both positive and negative values, it suggests that the mean net change in LDL cholesterol after the garlic treatment could range from a decrease of 5.174 mg/dL to an increase of 12.374 mg/dL. The effectiveness of garlic in reducing LDL cholesterol cannot be definitively determined based on this confidence interval alone. The interval includes zero, indicating that the effect of garlic on LDL cholesterol levels may not be statistically significant. Further studies and analysis are required to draw more conclusive findings about the effectiveness of garlic in reducing LDL cholesterol.