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 solve the problem, we will use the properties of the sampling distribution of the sample mean. Given the mean serum cholesterol level (\(\mu\)) is 200 mg/dL and the standard deviation (\(\sigma\)) is 40.6 mg/dL, we can calculate the standard error (SE) of the sample mean for a sample size (\(n\)) of 106 adults. ### Step 1: Calculate the Standard Error (SE) The standard error is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} \] Substituting the known values: \[ SE = \frac{40.6}{\sqrt{106}} \] ### Step 2: Calculate the Probability for Part (a) We need to find the probability that the sample mean cholesterol level is greater than 210 mg/dL. We will first calculate the z-score for 210 mg/dL using the formula: \[ z = \frac{\bar{x} - \mu}{SE} \] Where: - \(\bar{x} = 210\) - \(\mu = 200\) ### Step 3: Calculate the Probability for Part (b) Next, we will find the probability that the sample mean cholesterol level is between 189 and 197 mg/dL. We will calculate the z-scores for both values (189 and 197) using the same z-score formula. Now, let's perform the calculations for both parts. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(210-200\right)}{\left(\frac{40.6}{\sqrt{106}}\right)}\) - step1: Remove the parentheses: \(\frac{210-200}{\frac{40.6}{\sqrt{106}}}\) - step2: Divide the terms: \(\frac{210-200}{\frac{203}{5\sqrt{106}}}\) - step3: Subtract the numbers: \(\frac{10}{\frac{203}{5\sqrt{106}}}\) - step4: Multiply by the reciprocal: \(10\times \frac{5\sqrt{106}}{203}\) - step5: Multiply: \(\frac{10\times 5\sqrt{106}}{203}\) - step6: Multiply: \(\frac{50\sqrt{106}}{203}\) Calculate or simplify the expression \( (189-200)/(40.6/\sqrt(106)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(189-200\right)}{\left(\frac{40.6}{\sqrt{106}}\right)}\) - step1: Remove the parentheses: \(\frac{189-200}{\frac{40.6}{\sqrt{106}}}\) - step2: Divide the terms: \(\frac{189-200}{\frac{203}{5\sqrt{106}}}\) - step3: Subtract the numbers: \(\frac{-11}{\frac{203}{5\sqrt{106}}}\) - step4: Multiply by the reciprocal: \(-11\times \frac{5\sqrt{106}}{203}\) - step5: Multiply: \(-\frac{11\times 5\sqrt{106}}{203}\) - step6: Multiply: \(-\frac{55\sqrt{106}}{203}\) Calculate or simplify the expression \( 40.6/\sqrt(106) \). Calculate the value by following steps: - step0: Calculate: \(\frac{40.6}{\sqrt{106}}\) - step1: Divide the terms: \(\frac{203}{5\sqrt{106}}\) - step2: Multiply by the Conjugate: \(\frac{203\sqrt{106}}{5\sqrt{106}\times \sqrt{106}}\) - step3: Multiply the numbers: \(\frac{203\sqrt{106}}{530}\) Calculate or simplify the expression \( (197-200)/(40.6/\sqrt(106)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(197-200\right)}{\left(\frac{40.6}{\sqrt{106}}\right)}\) - step1: Remove the parentheses: \(\frac{197-200}{\frac{40.6}{\sqrt{106}}}\) - step2: Divide the terms: \(\frac{197-200}{\frac{203}{5\sqrt{106}}}\) - step3: Subtract the numbers: \(\frac{-3}{\frac{203}{5\sqrt{106}}}\) - step4: Multiply by the reciprocal: \(-3\times \frac{5\sqrt{106}}{203}\) - step5: Multiply: \(-\frac{3\times 5\sqrt{106}}{203}\) - step6: Multiply: \(-\frac{15\sqrt{106}}{203}\) It seems there was an issue with calculating the probabilities directly. Let's break down the calculations manually for clarity. ### Step 1: Calculate the Standard Error (SE) Using the formula: \[ SE = \frac{40.6}{\sqrt{106}} \approx 3.9434 \] ### Step 2: Calculate the z-score for 210 mg/dL Using the z-score formula: \[ z = \frac{210 - 200}{SE} = \frac{10}{3.9434} \approx 2.5359 \] ### Step 3: Calculate the Probability for Part (a) To find the probability that the sample mean is greater than 210 mg/dL, we need to find \(P(Z > 2.5359)\). This can be found using the standard normal distribution table or a calculator. Using a standard normal distribution table, we find: \[ P(Z > 2.5359) \approx 0.0056 \] ### Step 4: Calculate the z-scores for 189 mg/dL and 197 mg/dL For 189 mg/dL: \[ z = \frac{189 - 200}{SE} = \frac{-11}{3.9434} \approx -2.7900 \] For 197 mg/dL: \[ z = \frac{197 - 200}{SE} = \frac{-3}{3.9434} \approx -0.7608 \] ### Step 5: Calculate the Probability for Part (b) Now we need to find the probability that the sample mean is between 189 and 197 mg/dL, which is \(P(-2.7900 < Z < -0.7608)\). Using the standard normal distribution table: 1. Find \(P(Z < -2.7900)\) 2. Find \(P(Z < -0.7608)\) Let's calculate these probabilities. Using the standard normal distribution table or calculator: - \(P(Z < -2.7900) \approx 0.0026\) - \(P(Z < -0.7608) \approx 0.2236\) Now, we can find the probability that the sample mean is between 189 and 197 mg/dL: \[ P(-2.7900 < Z < -0.7608) = P(Z < -0.7608) - P(Z < -2.7900) \approx 0.2236 - 0.0026 = 0.2210 \] ### Final Answers - (a) The probability that the sample mean cholesterol level is greater than 210 is approximately \(0.0056\). - (b) The probability that the sample mean cholesterol level is between 189 and 197 is approximately \(0.2210\).