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Chapter 9 - Hypothesis Testing 138

Chapter 9 - Hypothesis Testing with 2 Samples

1. Introduction to Hypothesis Testing

  • Definition of hypothesis testing: A statistical method used to make inferences about population parameters based on sample data.

2. Distinction between t-test and z-test

  • Opening Activity: How do we tell the difference between a t-test and a z-test?

    • Key Differences:

      • t-test: Used when the sample size is small (typically less than 30) or when the population standard deviation is unknown.

      • z-test: Used when the sample size is large (typically 30 or more) and the population standard deviation is known.

3. Testing Between 2 Proportions

3.1 Key Details

  • HATNC: Hypotheses About Two Population Proportions

  • Differences:

    • Hypotheses are not about a number but about 2 population proportions.

    • Assumptions must be true for both samples, with a twist concerning the hypothesized population proportion (pc).

3.2 Example: Theta Drug Company

  • The Theta Drug Company claims their new drug is more effective for women than men.

  • Study parameters:

    • Sample Size: 100 women, 200 men.

    • Results: 38% of women caught colds; 51% of men caught the flu.

  • Objective: Determine if the drug is statistically more effective for women at a significance level of 0.01.

4. Testing Between 2 Means

4.1 Example: Restaurant Manager Performance

  • Scenario: Mio, the restaurant owner, wants to test if her two managers perform at the same level using data on customer complaints.

  • Conducting a two-sample test to determine if the mean complaints differ between the two managers.

  • Key Note: Ensure that all conditions for performing the test are met.

4.2 Caution in Testing

  • Do not confuse:

    • Two-sample test with matched pairs test.

    • Ensure samples are independent and from entirely different populations.

4.3 Example: Effectiveness of Hypnotism

  • A study investigates hypnotism in reducing pain using matched pairs.

  • Data shows before and after scores; differences calculated.

  • Objective: Test at a 5% significance level if average sensory measurements are lower after treatment.

5. Confidence Intervals for the Difference in Parameters

5.1 Overview

  • Use confidence intervals to determine differences between two population parameters (proportions or means).

6. Confidence Intervals for the Difference in Proportions

6.1 Constructing Confidence Intervals

  • Sample Scenario:

    • Age 18-29: 316 users, 26% use Twitter.

    • Age 30-49: 532 users, 14% use Twitter.

  • Task: Construct and interpret a 90% confidence interval for the difference in proportions of Twitter users among these age groups.

7. Confidence Intervals for the Difference in Means

7.1 Example with Professional Athletes

  • Sample 1: 45 professional football players, mean height = 6.18 feet, SD = 0.37.

  • Sample 2: 40 professional basketball players, mean height = 6.45 feet, SD = 0.31.

  • Task: Find a 95% confidence interval for the difference in mean heights.

  • Assess whether football players are generally shorter than basketball players.

8. Matched Pairs t-Interval

  • Study of high school students to compare sitting vs. standing pulse rates.

  • Objective: To conclude if sitting pulse rates are significantly lower than standing rates.

9. Calculator Tricks

9.1 Using Pooled Options for t-Intervals

  • For t-intervals, pooled option is usually insignificant as standard deviations are often unknown.

  • Recommendation: 99% of the time, leave as "no" for pooled standard deviation.

9.2 Matched Pairs t-Interval

  • Matched pairs t-intervals analyzed from a single group of subjects measuring differences between treatments.