Question

1. the vacation times website rates recreational vehicle campgrounds using integers from 0 to 15. last year they rated over 1,000 campsites. the ratings were normally distributed with mean 7.6 and standard deviation 1.7. a. how high would a campsites rating have to be for it to be considered in the top 10% of rated campsites? round to the nearest hundredth. b. find the z - score for a rating of 5. round to the nearest hundredth. c. find the percentile for a rating of 7.5. round to the nearest percent. d. a campsite had a z - score of 2.4. what was its rating?

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the mean, and $\sigma$ is the standard deviation. We are given $\mu = 7.6$, $\sigma=1.7$.

Step2: Solve part a

We want to be in the top 10% of the ratings. The z - score corresponding to the top 10% (or the 90th percentile) from the standard normal distribution table is approximately $z = 1.28$. Using the z - score formula $z=\frac{x - \mu}{\sigma}$, we can solve for $x$. Rearranging the formula gives $x=\mu+z\sigma$. Substituting $\mu = 7.6$, $z = 1.28$, and $\sigma = 1.7$, we get $x=7.6+1.28\times1.7=7.6 + 2.176=9.776\approx9.78$.

Step3: Solve part b

Given $x = 5$, $\mu = 7.6$, and $\sigma = 1.7$. Using the z - score formula $z=\frac{x-\mu}{\sigma}$, we substitute the values: $z=\frac{5 - 7.6}{1.7}=\frac{- 2.6}{1.7}\approx - 1.53$.

Step4: Solve part c

We first find the z - score for $x = 7.5$. Using $z=\frac{x-\mu}{\sigma}$ with $\mu = 7.6$ and $\sigma = 1.7$, we have $z=\frac{7.5 - 7.6}{1.7}=\frac{-0.1}{1.7}\approx - 0.06$. We then find the percentile using the standard - normal distribution table. The area to the left of $z=-0.06$ is approximately $0.4761$ or $47.61\%\approx48\%$.

Step5: Solve part d

Given $z = 2.4$, $\mu = 7.6$, and $\sigma = 1.7$. Using the z - score formula $z=\frac{x-\mu}{\sigma}$ and rearranging for $x$ gives $x=\mu+z\sigma$. Substituting the values, we get $x=7.6+2.4\times1.7=7.6 + 4.08 = 11.68\approx11.7$.

Answer:

a. $9.78$
b. $-1.53$
c. $48\%$
d. $11.7$