During the summer months,the prices of nonsmoking rooms with a king-sized bed in hotels in a certain area are roughly Normally distributed with a mean of $131.80 and a standard deviation of $29.12.A travel agent randomly selects prices of nonsmoking rooms with a king-sized bed from 15 hotels in the area.What is the probability that their average cost will be more than $150?
A)0.0077
B)0.1125
C)0.2660
D)0.3678

Question

Quizplus · Accepted Answer

To find the probability that the average cost of the rooms is more than $150, we use the formula for the standard error of the mean: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the standard deviation of the population and $n$ is the sample size. Here, $\sigma = 29.12$ and $n = 15$, so $\sigma_{\bar{x}} = \frac{29.12}{\sqrt{15}} \approx 7.52$.Next, we calculate the z-score to find how many standard deviations $150 is from the mean ($131.80): $z = \frac{X - \mu}{\sigma_{\bar{x}}} = \frac{150 - 131.80}{7.52} \approx 2.42$.Looking up the z-score of 2.42 in the standard normal distribution table or using a calculator, we find that the area to the left of this z-score is approximately 0.9923. Therefore, the probability of the average cost being more than $150 is 1 - 0.9923 = 0.0077.

During the Summer Months,the Prices of Nonsmoking Rooms with a King-Sized