2. High density lipoprotein (HDL) in healthy males follows a normal distribution with a mean of 50 and a standard deviation of 8.
c)What is the 99th percentile of HDL in healthy males?
d)If we consider samples of 10 patients, what is the 90th percentile of the mean HDL in healthy males?
c) Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.99) and its Z score. Insert data into equation below.
Z = (score-mean)/SD
d) Using same table,
Z = (score-mean)/SEm
SEm = SD/√n
To find the answers to these questions, we need to use the concept of the standard normal distribution. Here are the steps to solve each problem:
c) To find the 99th percentile of HDL in healthy males, we can use the z-table or a calculator that can calculate normal distribution percentiles.
The z-score formula is given by:
z = (x - mean) / standard deviation
To find the 99th percentile, we need to find the corresponding z-score. Looking up in the z-table, we find that the z-score for the 99th percentile is approximately 2.33.
Now, using the formula above, we can calculate the HDL value:
2.33 = (x - 50) / 8
Solving for x:
x - 50 = 2.33 * 8
x - 50 = 18.64
x = 18.64 + 50
x ≈ 68.64
Therefore, the 99th percentile of HDL in healthy males is approximately 68.64.
d) To find the 90th percentile of the mean HDL in healthy males, we need to consider samples of 10 patients. Since we are dealing with a sample mean, we need to use the central limit theorem and calculate the standard error.
The formula for standard error is given by:
standard error = standard deviation / sqrt(sample size)
standard error = 8 / sqrt(10)
standard error ≈ 2.53
Now, we can use the z-table to find the z-score corresponding to the 90th percentile. Looking it up, the z-score for the 90th percentile is approximately 1.28.
Using the z-score formula once again, we can calculate the mean HDL value:
1.28 = (x - 50) / 2.53
Solving for x:
x - 50 = 1.28 * 2.53
x - 50 ≈ 3.22
x ≈ 53.22
Therefore, the 90th percentile of the mean HDL in healthy males, considering samples of 10 patients, is approximately 53.22.