CNNBC recently reported that the mean annual cost of auto insurance is 1025 dollars. Assume the standard deviation is 147 dollars. You will use a simple random sample of 108 auto insurance policies.

Find the probability that a single randomly selected policy has a mean value between 992.5 and 1058.9 dollars.
\[
P(992.5< X< 1058.9)=
\]
Find the probability that a random sample of size $n=108$ has a mean value between 992.5 and 1058.9 dollars.
\[
P(992.5< M< 1058.9)=
\]
Enter your answers as numbers accurate to 4 decimal places.

Step 1 ：We are given that the mean annual cost of auto insurance is \(\mu = 1025\) dollars and the standard deviation is \(\sigma = 147\) dollars. We are asked to find the probability that a single randomly selected policy has a mean value between \(x_1 = 992.5\) and \(x_2 = 1058.9\) dollars.

Step 2 ：We first calculate the z-scores for \(x_1\) and \(x_2\) using the formula \(z = \frac{x - \mu}{\sigma}\). This gives us \(z_1 = -0.2211\) and \(z_2 = 0.2306\).

Step 3 ：We then find the area under the curve between these two z-scores, which represents the probability that a single randomly selected policy has a mean value between \(x_1\) and \(x_2\). This gives us \(P(992.5 < X < 1058.9) = 0.1787\).

Step 4 ：We are also asked to find the probability that a random sample of size \(n = 108\) has a mean value between \(x_1\) and \(x_2\).

Step 5 ：We first calculate the standard error using the formula \(SE = \frac{\sigma}{\sqrt{n}}\), which gives us \(SE = 14.1451\).

Step 6 ：We then calculate the z-scores for \(x_1\) and \(x_2\) using this standard error. This gives us \(z_{1_{sample}} = -2.2976\) and \(z_{2_{sample}} = 2.3966\).

Step 7 ：We then find the area under the curve between these two z-scores, which represents the probability that a random sample of size 108 has a mean value between \(x_1\) and \(x_2\). This gives us \(P(992.5 < M < 1058.9) = 0.9809\).

Step 8 ：\(\boxed{\text{Final Answer: The probability that a single randomly selected policy has a mean value between 992.5 and 1058.9 dollars is approximately 0.1787. The probability that a random sample of size 108 has a mean value between 992.5 and 1058.9 dollars is approximately 0.9809.}}\)