Two-Sample Tests Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Construct a 95% confidence interval for ฮผAโˆ’ฮผB\mu_A - \mu_B given: xห‰A=50\bar{x}_A=50, xห‰B=45\bar{x}_B=45, sA=6s_A=6, sB=8s_B=8, nA=nB=25n_A=n_B=25.

Solution

  1. 1
    Point estimate: xห‰Aโˆ’xห‰B=50โˆ’45=5\bar{x}_A - \bar{x}_B = 50 - 45 = 5
  2. 2
    SE: 3625+6425=10025=4=2\sqrt{\frac{36}{25} + \frac{64}{25}} = \sqrt{\frac{100}{25}} = \sqrt{4} = 2
  3. 3
    zโˆ—=1.96z^* = 1.96 (using normal approximation with large n)
  4. 4
    95% CI: 5ยฑ1.96(2)=5ยฑ3.92=(1.08,8.92)5 \pm 1.96(2) = 5 \pm 3.92 = (1.08, 8.92)

Answer

95% CI for ฮผAโˆ’ฮผB\mu_A - \mu_B: (1.08,8.92)(1.08, 8.92). Group A has significantly higher mean (CI excludes 0).
The CI for the difference in means excludes 0, confirming statistical significance. The interval (1.08, 8.92) also gives practical information: Group A scores 1 to 9 points higher than Group B with 95% confidence.

About Two-Sample Tests

Hypothesis tests and confidence intervals for comparing parameters (means or proportions) of two independent populations. The two-sample t-test compares means; the two-proportion z-test compares proportions.

Learn more about Two-Sample Tests โ†’

More Two-Sample Tests Examples

Example 1 medium

Test whether two teaching methods differ in effectiveness. Method A ([formula], [formula], [formula]

Example 3 easy

When should a two-sample t-test be used instead of a z-test, and what is the key assumption about th

Example 4 hard

Two independent samples: Group 1 ([formula]), Group 2 ([formula]). Calculate the t-statistic and df

โ† All Two-Sample Tests Examples Two-Sample Tests Concept