Topic:Inference for Two Populations (Determining an Appropriate Research Question)

(10.1/11.1) One Quantitative Variable, Comparing the Mean for Two Populations:

Example using hypothetical height data for men and women and compare the average heights of the two populations. Do men and women have significantly different average heights? 1. Data Collection: - Let's say we collected the following height data (in inches) for a sample of 20 men and 20 women: Men: 68, 70, 72, 69, 71, 67, 73, 68, 70, 71, 69, 72, 68, 70, 71, 69, 72, 68, 70, 71 Women: 64, 65, 63, 66, 64, 65, 63, 66, 64, 65, 63, 66, 64, 65, 63, 66, 64, 65, 63, 66 2. Graphical Summary: - We can compose a boxplot to visualize the distribution of heights for men and women. 3. Numerical Summary: - For men: Mean (x): 70.05 Standard Deviation (s): 1.60 Sample Size (n): 20 - For women: Mean (x): 64.45 Standard Deviation (s): 1.26 Sample Size (n): 20 4. Confidence Interval: - Let's calculate a 95% confidence interval for the difference in means using the formula: CI = (x1 - x2) t*(s1^2/n1 + s2^2/n2) - Assuming a two-tailed test and a t-critical value of 2.093 (for 95% confidence with 38 degrees of freedom): CI = (70.05 - 64.45) 2.093*(1.60^2/20 + 1.26^2/20) CI = 5.6 1.378 CI = (4.222, 6.978)

5. Hypothesis Testing: - Null Hypothesis (H0): The average height of men is equal to the average height of women. Alternative Hypothesis (H1): The average height of men is different from the average height of women. - Conducting a two-sample t-test, we find a t-statistic of 16.15 and a p-value < 0.001. 6. Conclusion: - Both the confidence interval and hypothesis test suggest that there is a significant difference in average heights between men and women in our sample. - Men appear to be taller on average compared to women based on the data collected. This example illustrates how you can compare the means of two populations, conduct hypothesis testing, and interpret the results to answer a research question about the difference in a quantitative variable.

Please help make (#2. Graphical Summary) - We can compose a boxplot to visualize the distribution of heights for men and women.