Watch your cholesterol: The mean serum cholesterol level for U.S. adults was 199, with a standard deviation of 39.8 (the units are milligrams per deciliter). A simple random sample of 107 adults is chosen. Use the TI-84 Plus calculator. Round the answers to at least four decimal places.
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Part 1 of 3
(a) What is the probability that the sample mean cholesterol level is greater than 207 ?
The probability that the sample mean cholesterol level is greater than 207 is $\square$.
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Step 1 ：Calculate the standard error (SE) of the sample mean using the formula \(SE = \frac{\sigma}{\sqrt{n}}\). Given \(\sigma = 39.8\) and \(n = 107\), we get \(SE = \frac{39.8}{\sqrt{107}} = 3.85\).

Step 2 ：Calculate the z-score using the formula \(Z = \frac{X - \mu}{SE}\). Given \(X = 207\), \(\mu = 199\), and \(SE = 3.85\), we get \(Z = \frac{207 - 199}{3.85} = 2.0779\).

Step 3 ：Look up the z-score in the standard normal distribution table or use a calculator to find the probability. The table or calculator gives the probability that the value is less than 207, so we need to subtract this from 1 to get the probability that the value is greater than 207. We get \(P(X > 207) = 1 - P(X < 207) = 1 - 0.9808 = 0.0192\).

Step 4 ：\(\boxed{P(X > 207) = 0.0192}\)