Question

question 5, 8.1.22 hw score: 50%, 4 of 8 points part 2 of 6 o points: 0 of 1 the most famous geyser in the world, old faithful in yellowstone national park, has a mean time between eruptions of 85 minutes. if the interval of time between the eruptions is normally distributed with standard deviation 21.25 minutes, complete parts (a) through (f). click here to view the standard normal distribution table (page 1). click here to view the standard normal distribution table (page 2). (a) what is the probability that a randomly selected time interval between eruptions is longer than 94 minutes? the probability that a randomly selected time interval is longer than 94 minutes is approximately 0.3372 (round to four decimal places as needed.) (b) what is the probability that a random sample of 10 time intervals between eruptions has a mean longer than 94 minutes? the probability that the mean of a random sample of 10 time intervals is more than 94 minutes is approximately \\(\square\\) (round to four decimal places as needed.)

Explanation:

Step1: Identify Distribution for Sample Mean

The population is normal with mean $\mu = 85$ and standard deviation $\sigma = 21.25$. For a sample of size $n = 10$, the sampling distribution of the sample mean $\bar{X}$ is normal with mean $\mu_{\bar{X}}=\mu = 85$ and standard deviation $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}=\frac{21.25}{\sqrt{10}}$.

Step2: Calculate Z - Score

We want $P(\bar{X}>94)$. The z - score is calculated as $z=\frac{\bar{x}-\mu_{\bar{X}}}{\sigma_{\bar{X}}}=\frac{94 - 85}{\frac{21.25}{\sqrt{10}}}$. First, calculate the denominator: $\frac{21.25}{\sqrt{10}}\approx\frac{21.25}{3.1623}\approx6.719$. Then the numerator is $94 - 85 = 9$. So $z=\frac{9}{6.719}\approx1.34$.

Step3: Find Probability

We need $P(Z > 1.34)$. Since $P(Z>z)=1 - P(Z\leq z)$, and from the standard normal table, $P(Z\leq1.34) = 0.9099$. So $P(Z > 1.34)=1 - 0.9099 = 0.0901$.

Answer:

0.0901