Would it be unusual for the sample mean to be less than 193 ?

a) The probability that the sample mean cholesterol level is greater than 204 is P(z > 1.32) = 0.0566.
b) The probability that the sample mean cholesterol level is between 190 and 197 is P(-2.06 < z < -0.32) = 0.3789
c) The probability of a z-score less than -1.32 is 0.0934
Based on the given information, we know that the mean serum cholesterol level for U.S. adults is 198, with a standard deviation of 39.
Given a simple random sample of 106 adults.
(a) To find the probability that the sample mean cholesterol level is greater than 204, we can use the z-score formula:
z = (x - μ) / (σ/ √n)
where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Substituting the given values, we get:
z = (204 - 198) / (39 / √106)
z = 1.58
Using a standard normal distribution table or Excel function, we can find that the probability of a z-score greater than 1.58 is 0.0566.
Therefore, the probability that the sample mean cholesterol level is greater than 204 is:
P(z > 1.32) = 0.0566
(b) To find the probability that the sample mean cholesterol level is between 190 and 197, we can use the same formula and calculate two z-scores:
z1 = (190 - 198) / (39 / √(106))
z1 = -2.06
z2 = (197 - 198) / (39 / √(106))
z2 = -0.32
Using a standard normal distribution table or Excel function, we can find that the probability of a z-score between -2.06 and -0.32 is 0.3789.
Therefore, the probability that the sample mean cholesterol level is between 190 and 197 is:
P(-2.06 < z < -0.32) = 0.3789
(c) To determine whether it would be unusual for the sample mean to be less than 193, we can use the same formula and calculate the z-score:
z = (193 - 198) / (39 / √(106))
z = -1.32
Using a standard normal distribution table or Excel function, we can find that the probability of a z-score less than -1.32 is 0.0934. Since this probability is greater than 0.05, which is often used as a threshold for unusual events, it would not be unusual for the sample mean to be less than 193.
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