stat 2 interval/test examples

  • Introduction to Problem Solving Method

    • Mary Bell introduces a new method for teaching problem solving.

    • The goal is to improve students' performance compared to traditional methods.

  • Decision Tree Analysis

    • Determine whether the data involves one sample or two:

      • It is two samples (Mary Bell's students vs. control group).

    • Identify if it concerns means or proportions:

      • This problem involves means, as average scores are mentioned.

      • Mary Bell’s group average = 82, control group average = 75.

    • Choose between Z or T test:

      • Since sample standard deviations are involved, use T test.

  • Sample Sizes and Confidence Intervals

    • The sample sizes are small, so ensure distributions are normal (data not provided to check).

    • Define parameters of interest:

      • M1 = average score with new method, M2 = average score with old method.

    • No need for null and alternative hypotheses when constructing intervals.

  • Estimation of Scores

    • Calculate score difference:

      • Mary Bell's students performed 7 points better on average (82 - 75 = 7).

    • Use this value as the midpoint for constructing the confidence interval.

  • Constructing the Confidence Interval

    • Calculation yields confidence interval of (-0.44, 14.44).

    • Interpretation:

      • Best case: students using Mary Bell’s method could score 14.44 points better.

      • Worst case: could score 0.44 points worse.

  • Understanding the Results

    • State your confidence level:

      • "We are 95% confident that this interval captures the true difference in population means."

    • Note that the inclusion of zero means we cannot claim definitive superiority of Mary Bell’s method based solely on this analysis.

  • Examining a Political Context

    • Example: Analyzing support for Trump’s third term candidacy among Democrats and Republicans.

    • Sample sizes show 10% of Democrats support vs. 58.33% of Republicans.

    • Again, this requires a two-proportion analysis.

  • Steps in Two-Proportion Test

    • Define populations and parameters:

      • p_D = proportion of Democrats, p_R = proportion of Republicans.

    • Calculate confidence interval based on these proportions.

    • For Republicans supporting Trump's run: (0.583 - 0.10) gives a center of difference around 48%.

    • Finished interval shows that Republicans support Trump's candidacy significantly more than Democrats.

  • Conclusions from Two-Proportions Analysis

    • Effective summary sentence for confidence intervals:

      • "I am 95% confident that this interval captures the true difference in proportions."

    • A reminder of how hard it is to properly phrase results in these contexts to avoid confusion.

  • Exploring Additional Samples

    • Discussion on two samples of wealthy vs. poor divorced men regarding marriage length.

    • Analyze and set up similar hypothesis tests and analyses.

    • A brief reflection on the need for clarity in approaching different types of statistical tests and their interpretations.

  • Final Thoughts on Statistical Tests

    • Each type of test (T-tests and Z-tests) has specific conditions that dictate how to use them effectively.

    • Importance placed on accurate notation and definitions during computational analysis, especially with proportions and estimates.