Question
Adult males are taller, on average, than adult females. Visiting two recent American Youth Soccer Organization (AYSO) under-12-years-old (U12) soccer matches on a Saturday, you do not observe an obvious difference in the height of boys and girls of that age. You suggest to your little sister that she collect data on height and gender of children in 4th to 6th grades as part of her science project. The accompanying table shows her findings.
| Boys |
Girls |
| Sample Average Y |
Sample Variance S |
Sample Size n |
Sample Average Y |
Sample Variance S |
Sample Size n |
| 57.8 |
3.9 |
55 |
58.4 |
4.2 |
57 |
a. Let your null hypothesis be that there is no difference in the height of females and males at this age level. Specify the alternative hypothesis.
b. What is the unbiased estimate of the difference in height between boys and girls? Provide a formula and check the unbiasedness. Calculate the value of this estimate for the given sample.
c. Derive the formula for the variance of the estimate from (b). Calculate the estimate of the variance for the given sample.
d. Create a statistic for testing the hypothesis in (a) using the Central Limit Theorem and the Law of Large Numbers.
e. Calculate the t-statistic for comparing the two means. Is the difference statistically significant at the 1% level? Which critical value did you use? Why would this number be smaller if you had assumed a one-sided alternative hypothesis? What is the intuition behind this?
f. Generate a 95% confidence interval for the difference in height.