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%
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.
To generate a 95% confidence interval for the difference in mean total cholesterol levels between treatments, we can follow these steps:
Step 1: Calculate the mean difference in total serum cholesterol levels between the new drug and placebo groups.
Mean difference = Mean of the new drug group - Mean of the placebo group
Mean difference = 185.0 - 204.3 = -19.3
Step 2: Calculate the standard error (SE) of the mean difference.
Standard error (SE) = Square root of [(SD1^2 / n1) + (SD2^2 / n2)]
where SD1 and SD2 are the standard deviations of the new drug and placebo groups respectively, and n1 and n2 are the sample sizes of the new drug and placebo groups respectively.
SE = Square root of [(24.5^2 / 75) + (21.8^2 / 75)]
SE ≈ 3.63
Step 3: Calculate the margin of error (ME) using the critical value for a 95% confidence interval (CI). The critical value can be found using a t-distribution table or calculator based on the degrees of freedom (df), which is equal to (n1 + n2 - 2).
For the df = (75 + 75 - 2) = 148, the critical value for a 95% CI is approximately 1.98.
ME = Critical value (t*) * SE
ME = 1.98 * 3.63
ME ≈ 7.19
Step 4: Calculate the upper and lower bounds of the confidence interval.
Lower bound = Mean difference - ME
Lower bound = -19.3 - 7.19
Lower bound ≈ -26.49
Upper bound = Mean difference + ME
Upper bound = -19.3 + 7.19
Upper bound ≈ -12.11
Step 5: Interpret the results.
The 95% confidence interval for the difference in mean total cholesterol levels between the new drug and placebo treatments is approximately -26.49 to -12.11. This means we are 95% confident that the true difference in mean total cholesterol levels falls within this range.
To generate a 95% confidence interval for the difference in mean total cholesterol levels between treatments, we need to calculate the standard error of the difference and then use it to calculate the interval.
1. Find the standard error of the difference:
The formula for the standard error of the difference between two means is:
SE_difference = sqrt((SD1^2 / n1) + (SD2^2 / n2))
where SD1 and SD2 are the standard deviations of the two groups, and n1 and n2 are the sample sizes of the two groups.
In this case, the standard deviations (SD) and sample sizes (n) are:
SD1 = 24.5, n1 = 75 (New Drug)
SD2 = 21.8, n2 = 75 (Placebo)
So the standard error of the difference is:
SE_difference = sqrt((24.5^2 / 75) + (21.8^2 / 75))
2. Calculate the margin of error:
The margin of error is the critical value multiplied by the standard error. For a 95% confidence interval, the critical value is approximately 1.96.
Margin of error = 1.96 * SE_difference
3. Calculate the lower and upper bounds of the confidence interval:
Lower bound = (Mean1 - Mean2) - Margin of error
Upper bound = (Mean1 - Mean2) + Margin of error
In this case:
Mean1 = 185.0 (New Drug)
Mean2 = 204.3 (Placebo)
Lower bound = (185.0 - 204.3) - Margin of error
Upper bound = (185.0 - 204.3) + Margin of error
Plug in the values and calculate the confidence interval.
Note: The mean values given in the table are not needed for calculating the confidence interval, as we are using the means of the two groups directly in the calculation.
Please note that other assumptions and considerations, such as the distribution of the data, may need to be considered for a more accurate assessment of the confidence interval.