Module 23: Hypothesis Test for a Difference in Two Population Means

Overview of Hypothesis Testing Steps

1. Determine the Hypotheses

   • Null Hypothesis (H₀): No effect or no difference. For two population means, ( H₀: \mu_1 - \mu_2 = 0 ) or equivalently ( \mu_1 = \mu_2 ).
   • Alternative Hypothesis (H₁): Can be any of the following:
     • ( \mu_1 - \mu_2 < 0 ) (or ( \mu_1 < \mu_2 ))
     • ( \mu_1 - \mu_2 > 0 ) (or ( \mu_1 > \mu_2 ))
     • ( \mu_1 - \mu_2 \neq 0 ) (or ( \mu_1 \neq \mu_2 ))

2. Collect the Data

   • Requirements:
     • Samples must be random and representative.
     • Two random samples must be independent.
     • The variable should be normally distributed in both populations.
     • If the sample size is more than 30, the difference in sample means can be modeled by the T distribution.
     • Use a T test when conditions are met.

3. Assess the Evidence

   • Calculate the T test statistic using the formula: ( T = \frac{{\text{{Sample Statistic}} - \text{{Hypothesized Population Parameter}}}}{{\text{{Estimated Standard Error}}}} ).
   • Use statistical software like StatCrunch for calculations.
   • Degrees of freedom for two-sample T test is complex; use software for exact value.

4. State a Conclusion

   • Compare the p-value to the level of significance (( \alpha )).
   • If ( p \leq \alpha ), reject H₀ in favor of H₁.
   • If ( p > \alpha ), fail to reject H₀.
   • State the conclusion in context.

Case Study: Impact of Group Settings on Meal Purchases

• Research Question: Do women purchase fewer calories when eating with men compared to women eating with women?

  • Populations:
    • Population 1: Women eating with women.
    • Population 2: Women eating with men.
  • Variable: Calories in the meal.
  • Hypotheses:
    • ( H₀: \mu_1 - \mu_2 = 0 )
    • ( H₁: \mu_1 - \mu_2 > 0 ) (expect more calories when women eat with other women)

• Data Collection:

  • Sample from Indiana University during Spring 2006.
  • Sample limitations: Primarily white undergraduate women aged 18 to 24.
  • Convenience sample, observations between February 13-22, 2006.

• Results:

  • Sample 1 (women with women): Mean = 850 calories, SD = 252, ( n = 45 ).
  • Sample 2 (women with men): Mean = 719 calories, SD = 322, ( n = 27 ).
  • T statistic calculated using StatCrunch.

• Conclusion:

  • Reject ( H₀ ) since p-value ( < \alpha ) (0.0385 < 0.05).
  • Conclusion in context: Women order more calories when eating with women than with men.

Discussion on Statistical Practice

• Limitation acknowledgment is important in studies.
• Findings should be replicated with larger, more diverse samples.
• Authors provide disclaimers about sample representativeness.

Key Takeaways

• Hypothesis Testing: Consistent steps with unique conditions for different tests.
• Data Collection and Analysis: Importance of random, representative samples.
• Conclusion Drawing: Contextualizing results within study's limitations.