WK10: Statistical Inference - Comparing Two Means: Test of Significance for Two Independent Populations

Test of Significance for Two Independent Populations

Steps

• Step 1: State the null and alternative hypotheses, and the level of significance $\alpha$.

  • Parameter of interest: Difference between means of two populations.
  • Null hypothesis: $H<em>0: \mu</em>1 - \mu_2 = 0$ (no difference in means).
  • Alternative hypothesis:
    • Two-sided: $H<em>a: \mu</em>1 - \mu_2 \neq 0$
    • One-sided: $H<em>a: \mu</em>1 - \mu<em>2 < 0$ or Ha: \mu1 - \mu2 > 0

• Step 2: Check conditions and calculate the test statistic.

  • Conditions:
    • Both samples from the populations of interest must be random.
    • Population distributions are either close to normal or sample sizes are large.
  • Test statistic:
    • General formula: $\frac{\text{point estimate} - \text{null value}}{\text{standard error}}$
    • For two independent samples: $t = \frac{\bar{x}<em>1 - \bar{x}</em>2 - (\mu<em>1 - \mu</em>2)}{\sqrt{\frac{s<em>1^2}{n</em>1} + \frac{s<em>2^2}{n</em>2}}}$
    • Under the null hypothesis ($\mu<em>1 - \mu</em>2 = 0$): $t = \frac{\bar{x}<em>1 - \bar{x}</em>2}{\sqrt{\frac{s<em>1^2}{n</em>1} + \frac{s<em>2^2}{n</em>2}}}$

• Step 3: Find the P-value.

  • The test statistic follows a T distribution.
  • Degrees of freedom: Use the conservative approach: $df = \min(n<em>1 - 1, n</em>2 - 1)$
  • Use Table C to find the P-value range based on the t statistic and degrees of freedom.

• Step 4: Make a statistical decision and draw a conclusion.

  • If P-value < \alpha, reject the null hypothesis.
    • The samples provide statistically significant evidence that the alternative hypothesis is true at level of significance $\alpha$.
  • If P-value > \alpha, fail to reject the null hypothesis.
    • The samples do not provide enough evidence to support the alternative hypothesis at level of significance $\alpha$.

Example: Exercise and Pulse Rate

• Study: Compare the mean resting pulse rate of adults who regularly exercise to those who do not.
• Question: Do these two populations differ in their mean resting pulse rates?
• Level of Significance: $\alpha = 0.05$
• Direction of Difference: Non-exercisers - Exercisers

Hypotheses

• Null hypothesis: $H<em>0: \mu</em>{\text{non-exercisers}} - \mu_{\text{exercisers}} = 0$
  • The mean resting pulse rate of adults who do not regularly exercise is the same as that of those who do.
• Alternative hypothesis: $H<em>a: \mu</em>{\text{non-exercisers}} - \mu_{\text{exercisers}} \neq 0$
  • The mean resting pulse rate of adults who do not regularly exercise is different from that of those who do.

Conditions

• The sum of sample sizes is greater than 40 (specifically, 60), so proceed with T procedures.

Test Statistic

• Given sample means and standard deviations, the calculated test statistic is $t = 3.96$
• Degrees of freedom: $df = \min(n<em>1 - 1, n</em>2 - 1) = 28$

P-value

• For $df = 28$, $t = 3.96$ is larger than the last tabulated value of 3.674.
• Therefore, the P-value is less than 0.001 (for a two-sided test).

Conclusion

• Since the P-value (< 0.001) is less than the level of significance ($\alpha = 0.05$), we reject the null hypothesis.
• The samples provide statistically significant evidence that, at a 5% level of significance, the mean resting pulse rate of adults who do not regularly exercise is different from that of adults who do regularly exercise.