Lecture Notes: Two-Sample Tests

Introduction to Two-Sample Tests

• Comparison of two different samples.
• Focus on independent and dependent samples.
• Discussing Z test, testing proportions, and T-test.

Five-Step Process of Hypothesis Testing

1. State the Null and Alternate Hypothesis
   • Null hypothesis: Assumed to be true (status quo).
   • Alternate hypothesis: What you aim to prove.
2. Select Level of Significance
   • Common values: 0.05 or 0.01 (anything <= 0.1 can be significant).
3. Identify the Test Statistic
   • Use Z if population standard deviation is known or for proportions.
   • Use T if population standard deviation is unknown.
4. Formulate the Decision Rule
   • Identify critical value from Z or T table for comparison.
5. Compute Test Statistic
   • Compare computed statistic to critical value to accept or reject the null hypothesis.
   • Rule: If computed value < critical value, do not reject null hypothesis.

Comparing Two Populations

Example: Residential Real Estate Values

• Comparing values of real estate sold by males vs. females.
• Test scores comparison between genders.

Z-test for Two Samples

• Formula:

  [ Z = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} ]

• Example scenario: Grocery store comparing self-checkout vs. standard checkout times.

• Steps to perform Z-test:

  1. State null (no difference in time) and alternate hypothesis (self-checkout is faster).
  2. Level of significance: 0.01.
  3. Use Z-test since population standard deviation is known.
  4. Determine critical Z value (1-tail, 0.01 significance) = 2.33.
  5. Compute Z value, compare with critical value, and decide.

Test of Proportions

• Proportions compare yes/no data (e.g., pass/fail, preferences).

• Steps for hypothesis test:

  1. State hypotheses (null: no difference, alternate: there is a difference).

  2. Use significance level of 0.05.

  3. Compute pooled proportion for Z test. Formula:

     [ P_{pooled} = \frac{x_1 + x_2}{n_1 + n_2} ]

• Example: Perfume preference comparison between younger and older women.

• Compute Z and compare with critical value to accept or reject the null hypothesis._

T-test for Two Samples

• Use T-test when population standard deviations are unknown.
• Example: Lawn mower assembly time comparison.
• Steps:
  1. State hypotheses (null: no difference, alternate: there is a difference).
  2. Use significance level of 0.1.
  3. Calculate degrees of freedom and critical value.
  4. Compute T statistic using samples.

Dependent Samples

• Dependent samples involve the same sample before and after (e.g., test scores before and after a course).
• Example: House appraisal comparison by two companies.
• Steps:
  1. State hypotheses (null: no difference, alternate: there is a difference).
  2. Significance level: 0.05.
  3. Calculate mean of differences, standard deviation of differences, and T statistic.
  4. Compare T statistic with critical value.

Conclusion

• Recap of the five-step process for hypothesis testing.
• Importance of using proper statistical tests based on data characteristics.