Question

1. The Registrars office would like to estimate the average commute time and determine a 95% confidence interval for the average commute time of evening University of Pittsburgh students from their usual initial starting point to campus. A member of the staff randomly chooses a parking lot and selects the first 100 evening students who park in the chosen lot starting at 5:00 PM. The confidence interval is: a. Not meaningful because of the lack of random sampling. b. Meaningful because the sample size exceeds 30 and the central limit theorem ensures normality of the sampling distribution of the sample mean. c. Not meaningful because the sampling distribution of the sample mean is probably not normal. d. Meaningful because the sample is representative of the population.

Answer 1

b. Meaningful because the sample size exceeds 30 and the central limit theorem ensures normality of the sampling distribution of the sample mean.

Explanation:

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

In this question:

Sample of 100, which means that the central limit theorem applies, no matter the distribution of the population. So the correct answer is given by option b.