To calculate the 95% confidence interval for the difference in mean total cholesterol levels between treatments, we'll first need to find the standard error of the difference and then use this to compute the margin of error.
1. Calculate the difference in mean total cholesterol levels between the two treatments:
Mean difference = Mean_new_drug - Mean_placebo
Mean difference = 185.0 - 204.3
Mean difference = -19.3
2. Calculate the standard error of the difference:
SE = sqrt((SD_new_drug^2 / n_new_drug) + (SD_placebo^2 / n_placebo))
SE = sqrt((24.5^2 / 75) + (21.8^2 / 75))
SE = sqrt((600.25 / 75) + (475.24 / 75))
SE ≈ sqrt(8.003 + 6.336)
SE ≈ sqrt(14.339)
SE ≈ 3.79
3. Calculate the margin of error at the 95% confidence level:
Margin of error = z-score * SE
For a 95% confidence level, the z-score is 1.96.
Margin of error = 1.96 * 3.79
Margin of error ≈ 7.43
4. Calculate the 95% confidence interval for the difference in mean total cholesterol levels between treatments:
Lower limit = Mean difference - Margin of error
Lower limit = -19.3 - 7.43
Lower limit ≈ -26.73
Upper limit = Mean difference + Margin of error
Upper limit = -19.3 + 7.43
Upper limit ≈ -11.87
The 95% confidence interval for the difference in mean total cholesterol levels between treatments is approximately (-26.73, -11.87). This means we can be 95% confident that the true difference in mean total cholesterol levels between the new drug and placebo is between -26.73 and -11.87 units.
8. The following data were collected in a clinical trial to compare a new drug to a placebo for its effectiveness in lowering total serum cholesterol. Generate a 95% confidence interval for the difference in mean total cholesterol levels between treatments.
New Drug
(n=75) Placebo
(n=75) Total Sample
(n=150)
Mean (SD) Total Serum Cholesterol 185.0 (24.5) 204.3 (21.8) 194.7 (23.2)
% Patients with Total Cholesterol < 200 78.0% 65.0% 71.5%
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