Statistical Inference – Quiz 3

Quiz Question Explanations

Question 1

The predicted distribution of sampling means is derived from the Central Limit Theorem, and it shows that if the population mean is 9.6 and the standard deviation is 3.4 and the sample size is 25, then there is a 9.0% chance of any given sample mean being greater than 10.5. Since the area under the t-distribution curve sums to 1, this is equivalent to saying that 9% of the area under the curve is above 10.5. Moreover, since this t-distribution is symmetric, that also implies that there is a 9.0% chance of any given sample mean being less than 9.6-0.9 = 8.7.

Question 2

If the sample size increased from 25 to 50, then the predicted distribution of sample means according to the Central Limit Theorem would be more tightly clustered around the population mean, and the probability of any given sample mean being a certain distance from the population mean would fall. In other words, the area under the curve above 10.5 would shrink as well, and instead be shifted closer to the population mean. Finally, with a larger sample the Central Limit Theorem would provide a better approximation of the true distribution of sampling means as well.

Question 3

If there is a 9.0% chance of any given sample mean being greater than 10.5 given this population mean, SD, and sample size, the chance of any given sample mean being more than 15 (and farther from the population mean) must be less than 9.0%. However, the probability is not 0 (assuming that there are at least some students in the population who scored 15 or more on the test).