Venciss Watch your cholesterol: The mean serum cholesterol level for U.S. adults was 199, with a standard deviabion of 39.8 (the units are milligrams per decaiter). A simple random sample of 107 aduls is chosen. Use O Cumulatise Normat Pitribution Thbie if needed. Round the answers to at least four decimal places. Part: 0/3 Fart 1or:3 (a) What is the probability that the sumple mean cholesterol level is greater than 2077 The grobus is that the sample mean cholesterol level is greater than 207 is 0.0188 Part: 1/3 Pat 2 of 3 (0) What as the probability that the ample mean cholesterdi level is between 187 and 193 ? The probabin that the sample mean chalesterd level is betieen 187 and 193 is $\square$.

Step 1 ：The problem is asking for the probability that the sample mean cholesterol level is between 187 and 193. To solve this, we can use the properties of the normal distribution.

Step 2 ：The sample mean follows a normal distribution with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size.

Step 3 ：We can standardize the values 187 and 193 to z-scores using the formula \(Z = \frac{X - \mu}{\sigma / \sqrt{n}}\), where \(X\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Step 4 ：For \(X = 187\), the z-score is \(Z_1 = \frac{187 - 199}{39.8 / \sqrt{107}} = -3.1188\).

Step 5 ：For \(X = 193\), the z-score is \(Z_2 = \frac{193 - 199}{39.8 / \sqrt{107}} = -1.5594\).

Step 6 ：We then use the cumulative distribution function (CDF) of the standard normal distribution to find the probabilities. The probability that the sample mean is between 187 and 193 is the difference between the CDF values at 193 and 187.

Step 7 ：The CDF value at \(Z_1 = -3.1188\) is \(P_1 = 0.0009\).

Step 8 ：The CDF value at \(Z_2 = -1.5594\) is \(P_2 = 0.0594\).

Step 9 ：The probability that the sample mean cholesterol level is between 187 and 193 is \(P = P_2 - P_1 = 0.0594 - 0.0009 = 0.0585\).

Step 10 ：Final Answer: The probability that the sample mean cholesterol level is between 187 and 193 is \(\boxed{0.0585}\).