The serum cholesterol levels in men aged 18 to 24 are normally distributed with a mean of 178.1 and a standard deviation of 40.7. A group of 9 men aged 18 to 24 randomly selected.
1. Find the probability that there are at least 2 men in the group whose serum cholesterol level is at most 230.
2. Suppose 90% of men's serum cholesterol level is less than c. What is the value of c?
To find the probabilities in this scenario, we can use the normal distribution and the z-score formula.
1. Probability of at least 2 men with serum cholesterol level at most 230:
To find this probability, we need to convert the value of 230 into a z-score, and then use the normal distribution table (or a calculator that provides the cumulative probability for the standard normal distribution) to find the desired probability.
The formula for z-score is:
z = (x - μ) / σ
Where:
x = the value of interest (230 in this case)
μ = the mean of the distribution (178.1)
σ = the standard deviation of the distribution (40.7)
Calculating the z-score:
z = (230 - 178.1) / 40.7
z = 1.2053 (rounded to 4 decimal places)
Now, we need to find the cumulative probability for this z-score. Using a normal distribution table or calculator, we can find the probability associated with a z-score of 1.2053. Let's assume this probability is P(z ≤ 1.2053).
Since we want to find the probability of at least 2 men with serum cholesterol level at most 230, we need to subtract the probability of 1 or 0 men with serum cholesterol level at most 230 from 1.
P(at least 2 men) = 1 - [P(0 men) + P(1 man)]
2. Finding the value of c for which 90% of men have a serum cholesterol level less than c:
To find the value of c, we need to find the z-score associated with a cumulative probability of 90%. Let's assume this z-score is z_90.
Using the normal distribution table or calculator, we can find the z-score that corresponds to a cumulative probability of 90%.
Once we have the z-score, we can use the z-score formula to find the value of c:
c = μ + (z_90 * σ)
Where:
c = the value of interest
μ = the mean of the distribution (178.1)
σ = the standard deviation of the distribution (40.7)
z_90 = the z-score associated with a cumulative probability of 90%