Answer:
Explanation:
We can use the central limit theorem to approximate the sampling distribution of the sample mean for large sample sizes. According to the central limit theorem, the sampling distribution of the sample mean for a sample size n will be approximately normal with mean μ and standard deviation σ/√n, where μ is the population mean and σ is the population standard deviation.
For the given problem, the mean cholesterol level in men is normally distributed with a mean of 180 mg/dL and a standard deviation of 40 mg/dL.
For a sample size of 16 men, the sample mean cholesterol level is approximately normally distributed with a mean of 180 mg/dL and a standard deviation of 40 mg/dL/√16 = 10 mg/dL. To find the probability that the sample mean exceeds 170 mg/dL, we standardize the sample mean as follows:
z = (170 - 180) / 10 = -1
Using a standard normal table or a calculator, the probability that a standard normal random variable is less than -1 is approximately 0.1587. Therefore, the probability that the mean cholesterol level of 16 randomly selected men exceeds 170 mg/dL is:
1 - 0.1587 = 0.8413 (rounded to 4 decimal places)
For a sample size of 25 men, the sample mean cholesterol level is approximately normally distributed with a mean of 180 mg/dL and a standard deviation of 40 mg/dL/√25 = 8 mg/dL. To find the probability that the sample mean exceeds 170 mg/dL, we standardize the sample mean as follows:
z = (170 - 180) / 8 = -1.25
Using a standard normal table or a calculator, the probability that a standard normal random variable is less than -1.25 is approximately 0.1056. Therefore, the probability that the mean cholesterol level of 25 randomly selected men exceeds 170 mg/dL is:
1 - 0.1056 = 0.8944 (rounded to 4 decimal places)
For a sample size of 64 men, the sample mean cholesterol level is approximately normally distributed with a mean of 180 mg/dL and a standard deviation of 40 mg/dL/√64 = 5 mg/dL. To find the probability that the sample mean exceeds 170 mg/dL, we standardize the sample mean as follows:
z = (170 - 180) / 5 = -2
Using a standard normal table or a calculator, the probability that a standard normal random variable is less than -2 is approximately 0.0228. Therefore, the probability that the mean cholesterol level of 64 randomly selected men exceeds 170 mg/dL is:
1 - 0.0228 = 0.9772 (rounded to 4 decimal places)