Hypothesis Test for Two Means (Independent Populations)

Module 6: Hypothesis Test for 2 Means (Independent Populations)

Independent Populations

• Definition: Two populations are independent if they do not influence each other.

  • Example 1: Comparing weights of males and females in a company.
  • Example 2: Comparing grades of students in Edmonton vs. Calgary.

• Key Notations:

  • Population 1:
  • Sample Mean: $\bar{y}_1$
  • Population Mean: $\mu_1$
  • Population Std Dev: $\sigma_1$
  • Sample Size: $n_1$
  • Population 2:
  • Sample Mean: $\bar{y}_2$
  • Population Mean: $\mu_2$
  • Population Std Dev: $\sigma_2$
  • Sample Size: $n_2$

Comparing Two Populations

• The goal is to determine if sample means are significantly different rather than just comparing their means directly.
• The observed difference is assessed against the variability within the populations.
• Inferential statistics necessitates an analysis of the sampling distribution of the difference between means ($\bar{y}<em>1 - \bar{y}</em>2$).

Sampling Distribution

• Assuming Equal Variances:
  • $\bar{y}<em>1 - \bar{y}</em>2 = \mu<em>1 - \mu</em>2$
• Assuming Unequal Variances:
  • Pooled Standard Deviation:
    $S<em>{p} = \sqrt{\frac{(n</em>1 - 1)S<em>1^2 + (n</em>2 - 1)S<em>2^2}{n</em>1 + n_2 - 2}}$
  • Standard Error of the Difference:
  • $S<em>{\bar{y}</em>1 - \bar{y}<em>2} = S</em>p \sqrt{\frac{1}{n<em>1} + \frac{1}{n</em>2}}$
• If both populations are normally distributed or have large sample sizes, the sampling distribution approaches normality.

Testing for Equal or Unequal Variance

• Criteria for Equal Variability:
  • Standard deviation ratio < 2
  • Similar Interquartile Range (IQR) in box plots.

T-Tests

• Pooled T-Test (Equal Variance):
  • Test Statistic:
    $t<em>0 = \frac{\bar{y}</em>1 - \bar{y}<em>2 - d</em>0}{S<em>p \sqrt{\frac{1}{n</em>1} + \frac{1}{n_2}}}$
  • Distribution: $t$-distribution with $df = n<em>1 + n</em>2 - 2$
• Non-Pooled T-Test (Unequal Variance):
  • Test Statistic:
    $t<em>0 = \frac{\bar{y}</em>1 - \bar{y}<em>2 - d</em>0}{\sqrt{\frac{s<em>1^2}{n</em>1} + \frac{s<em>2^2}{n</em>2}}}$
  • Distribution: $t$-distribution with degrees of freedom calculated using the formula:
    $df = \min(n<em>1 - 1, n</em>2 - 1)$

P-Value Interpretation

• Decision Criteria:
  • If $p$-value ≤ $\alpha$, reject $H_0$.
  • If $p$-value > $\alpha$, do not reject $H_0$.

Confidence Intervals

• Pooled T-Test Confidence Interval:
  • $\bar{y}<em>1 - \bar{y}</em>2 \pm t^* S<em>p \sqrt{\frac{1}{n</em>1} + \frac{1}{n_2}}$
• Non-Pooled T-Test Confidence Interval:
  • $\bar{y}<em>1 - \bar{y}</em>2 \pm t^* \sqrt{\frac{s<em>1^2}{n</em>1} + \frac{s<em>2^2}{n</em>2}}$
• Where $t^*$ represents the critical value from the $t$-distribution.

Example Analysis: Vitamin Supplement

• Scenario: Testing a vitamin's effect on recovery time.
• Sample Information:
  • No Vitamin:
  • Sample Size: 45, Mean: 7.2, Standard Deviation: 3
  • Vitamin:
  • Sample Size: 45, Mean: 6.3, Standard Deviation: 1.6

Hypotheses

• Null ($H<em>0$): $\mu</em>1 - \mu_2 = 0$ (no effect)
• Alternative ($H<em>a$): $\mu</em>1 - \mu_2 > 0$ (effect exists)

Test Execution

• Test Statistic:
  • Using pooled variance (assuming equal variance since the ratio is < 2):
  • $t_0 = \frac{7.2 - 6.3 - 0}{2.404 \sqrt{\frac{1}{45} + \frac{1}{45}}} = 1.780$
• Degrees of Freedom: $df = 88$
• P-Value Calculation Outcomes:
  • Since $p > 0.025$, do not reject $H_0$.
  • Conclusion: Insufficient evidence to support vitamin's effectiveness.

Confidence Interval Calculation

• For difference in means:
  • $\bar{y}<em>1 - \bar{y}</em>2 \pm t^*\sqrt{\frac{s<em>p^2}{n</em>1} + \frac{s<em>p^2}{n</em>2}}$
• Result Interval: $(-0.11, 1.91)$ indicates no significant mean difference.

Caveats for Non-Pooled T-Test

• Use of non-pooled test indicated if standard deviation ratio > 2.

Conclusion

• Findings do not support the claim that the vitamin decreases average recovery time. Further investigation required.

Notes

• Remember: Equal variances should only be assumed if the informal standard deviation ratio is less than two.