Watch your cholesterol: The mean serum cholesterol level for U.S. adults was 200, with a standard devlation of 40.6 (the units are milligrams per deciliter). A simple random sample of 106 adults is chosen. Use the T1-84 Plus calculator. Round the answers to at least four decimal places. Part 1 of 3 (a) What is the probability that the sample mean cholesterol level is greater than 210 ? The probability that the sample mean cholesterol level is greater than 210 is \( \square \) 0.0056 . Part: \( 1 / 3 \) Part 2 of 3 (b) What is the probability that the sample mean cholesterol level is between 189 and 197? The probability that the sample mean cholesterol level is between 189 and 197 is \( \square \) .

To find the probability that the sample mean cholesterol level is between 189 and 197, we will use the Central Limit Theorem. The standard deviation of the sample mean (also known as the standard error) is calculated as the population standard deviation divided by the square root of the sample size. In this case, the standard error is \( \sigma_{\bar{x}} = \frac{40.6}{\sqrt{106}} \approx 3.95 \). Next, we will convert the sample means of 189 and 197 to their corresponding z-scores: For 189: \( z_1 = \frac{189 - 200}{3.95} \approx -2.78 \) For 197: \( z_2 = \frac{197 - 200}{3.95} \approx -0.76 \) Now, we can use the T1-84 Plus calculator to find the probabilities corresponding to these z-scores. The probability for \( z_1 \) gives us \( P(Z < -2.78) \approx 0.0027 \) and for \( z_2 \) gives us \( P(Z < -0.76) \approx 0.2236 \). Finally, to find the probability that the sample mean is between 189 and 197, we calculate: \( P(-2.78 < Z < -0.76) = P(Z < -0.76) - P(Z < -2.78) \). This results in: \( 0.2236 - 0.0027 = 0.2209 \). Thus, the probability that the sample mean cholesterol level is between 189 and 197 is approximately \( \square \) 0.2209.