Answer: We can use the z-score formula to standardize the cholesterol levels and convert them to a standard normal distribution:
z = (x - mean) / standard deviation
where x is the cholesterol level, mean is the mean cholesterol level, and standard deviation is the standard deviation of cholesterol levels.
a) To find the probability that a randomly selected man has a cholesterol level less than 160 mg/dL, we calculate the z-score as:
z = (160 - 180) / 40 = -0.5
Using a standard normal table or calculator, we find that the probability of getting a z-score less than -0.5 is 0.3085 (or 0.309 in four decimal places).
Therefore, the probability that a randomly selected man has a cholesterol level less than 160 mg/dL is 0.3085 (or 0.309 in four decimal places).
b) To find the probability that a randomly selected man has a cholesterol level above 260 mg/dL, we calculate the z-score as:
z = (260 - 180) / 40 = 2
Using a standard normal table or calculator, we find that the probability of getting a z-score greater than 2 is 0.0228 (or 0.023 in four decimal places).
Therefore, the probability that a randomly selected man has a cholesterol level above 260 mg/dL is 0.0228 (or 0.023 in four decimal places).
c) To find the probability that a randomly selected man has a cholesterol level between 160 and 200 mg/dL, we calculate the z-scores as:
z1 = (160 - 180) / 40 = -0.5
z2 = (200 - 180) / 40 = 0.5
Using a standard normal table or calculator, we find the probability of getting a z-score less than -0.5 is 0.3085, and the probability of getting a z-score less than 0.5 is 0.6915. Therefore, the probability of getting a z-score between -0.5 and 0.5 is:
P(-0.5 < z < 0.5) = P(z < 0.5) - P(z < -0.5) = 0.6915 - 0.3085 = 0.3830 (or 0.383 in four decimal places).
Therefore, the probability that a randomly selected man has a cholesterol level between 160 and 200 mg/dL is 0.3830 (or 0.383 in four decimal places).
d) To find the probability that a randomly selected man has a cholesterol level below 100 or above 260 mg/dL, we need to calculate the probability of each event separately and add them together:
The z-score for a cholesterol level of 100 mg/dL is:
z1 = (100 - 180) / 40 = -2
Using a standard normal table or calculator, we find the probability of getting a z-score less than -2 is 0.0228 (or 0.023 in four decimal places).
The z-score for a cholesterol level of 260 mg/dL is:
z2 = (260 - 180) / 40 = 2
Using a standard normal table or calculator, we find the probability of getting a z-score greater than 2 is 0.0228 (or 0.023 in four decimal places).
Therefore, the probability that a randomly selected man has a cholesterol level below 100 or above 260 mg/dL is:
P(x < 100 or x > 260) = P(z < -2) + P(z
Explanation: