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.