Suppose the population of all colleges shows the annual parking fee per student is $110 with a standard deviation of $18. If a random sample of size 49 is drawn from the population, the probability of drawing a sample with a mean of more than $115 is ___.
A) 0.9738
B) 0.4738
C) 0.0262
D) 0.6103
E) 0.1103

Question

Quizplus · Accepted Answer

The sampling distribution of the mean follows the normal distribution with a mean of the population and a standard deviation of the population standard deviation divided by the square root of the sample size, i.e., $\sigma/\sqrt{n}$. In this case, $n=49$, $\mu$ (population mean parking fee) = $110 and $\sigma$ (population standard deviation) = $18. Then the sampling distribution of the mean, $\bar{x}$ is $\mathcal{N}(110,18/\sqrt{49}) = \mathcal{N}(110,2.57)$. So, we need to find $P(\bar{x} > 115)$ or $P\left(Z > \dfrac{115-110}{2.57}\right) = P(Z > 1.946)$, where $Z$ is a standard normal random variable. Using the standard normal distribution table, $P(Z > 1.946) = 1 - P(Z < 1.946)$. We can look up the table or use software like Excel or Python to get $P(Z < 1.946) = 0.974$, approximately. Therefore, $P(Z > 1.946) = 1-0.974 = 0.026$. Hence, the answer is C.

Suppose the Population of All Colleges Shows the Annual Parking