Confidence Interval for the Difference Between Two Means
Construct a 95% confidence interval for the difference in mean blood pressure reduction between two medications using independent samples.
Problem Scenario
A clinical researcher compares two blood pressure medications. Drug A is administered to 20 patients, yielding a mean reduction of 12.4 mmHg with a standard deviation of 3.8 mmHg. Drug B is administered to 22 patients, yielding a mean reduction of 9.6 mmHg with a standard deviation of 4.2 mmHg. Construct a 95% confidence interval for the difference in mean blood pressure reduction (Drug A - Drug B). Assume equal variances.
Given Data
Requirements
- Calculate the pooled standard deviation and the standard error of the difference
- Construct the 95% confidence interval for mu_A - mu_B
- Interpret the interval in the context of the clinical trial
Solution
Step 1:
Calculate the point estimate: x-bar_1 - x-bar_2 = 12.4 - 9.6 = 2.8 mmHg. Drug A shows 2.8 mmHg more reduction on average.
Step 2:
Calculate the pooled variance: s_p^2 = [(n_1-1)s_1^2 + (n_2-1)s_2^2] / (n_1+n_2-2) = [(19)(14.44) + (21)(17.64)] / 40 = [274.36 + 370.44] / 40 = 644.80 / 40 = 16.12. So s_p = sqrt(16.12) = 4.015.
Step 3:
Calculate the standard error of the difference: SE = s_p * sqrt(1/n_1 + 1/n_2) = 4.015 * sqrt(1/20 + 1/22) = 4.015 * sqrt(0.05 + 0.04545) = 4.015 * sqrt(0.09545) = 4.015 * 0.3090 = 1.241.
Step 4:
Calculate the margin of error: ME = t* x SE = 2.021 x 1.241 = 2.508.
Step 5:
Construct the interval: (x-bar_1 - x-bar_2) +/- ME = 2.8 +/- 2.508 = (0.292, 5.308).
Step 6:
Interpretation: We are 95% confident that Drug A reduces blood pressure by 0.29 to 5.31 mmHg more than Drug B. Because the entire interval is above zero, we can conclude that Drug A is significantly more effective than Drug B at the 0.05 level.
Final Answer
The 95% confidence interval for (mu_A - mu_B) is (0.29, 5.31) mmHg. We are 95% confident that Drug A reduces blood pressure by between 0.29 and 5.31 mmHg more than Drug B. Since the interval does not contain 0, the difference is statistically significant.
Key Takeaways
- ✓If a confidence interval for the difference between two means does not contain zero, it indicates a statistically significant difference at the corresponding significance level.
- ✓The width of the interval depends on sample sizes, variability, and the confidence level. Larger samples produce narrower (more precise) intervals.
- ✓Clinical significance matters as much as statistical significance. A 0.29 mmHg advantage might not be clinically meaningful even if statistically significant.
Common Errors to Avoid
- ✗Using a z* critical value instead of t* when population standard deviations are unknown. With sample standard deviations, the t-distribution is appropriate.
- ✗Incorrectly pooling the standard deviations by simply averaging s_1 and s_2. The pooled estimate weights each sample variance by its degrees of freedom.
- ✗Interpreting the confidence interval as "there is a 95% probability that the true difference falls between 0.29 and 5.31." The true difference is fixed; the 95% refers to the long-run reliability of the procedure.
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Common questions about this problem type
If the interval includes 0, you cannot conclude that the two population means are different at the corresponding significance level. This is equivalent to failing to reject H_0 in a two-sample t-test.
Use the pooled approach when you can reasonably assume equal population variances (check with an F-test or Levene's test, or if the sample SDs are similar). If the equal variance assumption is doubtful, use the Welch approach, which adjusts the degrees of freedom and does not require pooling.