Bob is a survey researcher who knows from census data that the mean age of the Memphis population is 34.3 with a standard deviation of 1.2 years. He has just collected a sample of 500 Memphians for his latest study and sees that the mean age in his sample is 32.6.
What is the standard error of the mean in the theoretical sampling distribution? (i.e. If Bob kept collecting sample after sample of Memphis residents, what would be the standard deviation of those sample means).
The 95% confidence interval spans from what age to what age?
Does Bob’s survey generalize to the population of interest, or is there reason for concern? Explain why or why not?
+6248 \(\sigma = \text{With sample size of }N \text{ and population standard deviation }\sigma_p\\ \text{the standard error of the mean }= \dfrac{\sigma_p}{\sqrt{N}}\\ \text{A 95% confidence interval for the sample is that interval centered about}\\ \text{the sample mean that contains 95% of the probability mass}\\ \text{This turns out to be the interval }[\mu_s - 1.96 \sigma,~\mu_s + 1.96\sigma]\)
I'll let you plug and chug the numbers but it should be reasonably clear that Bob's sample does look fishy.
