A doctor want to estimate the HDL cholesterol of all 20- to 29- year old females. how many subjects are needed to estimate the HDL cholesterol within 3 points with 99% confindence assuming s=10.5. Suppose the doctor would be content with 90% confidence. How does the decrease in confidence affect the sample size​ required?

margin of error = z( s/sqrt n)

me^2 = (z^2)(s^2)/n

n = (z^2)(s^2)/me^2

n =(2.33)^2 ( 10.5)^2 /3^2
n = 5.43(110.25)/9
n = 66.5 so you need 67 people

repeat the problem using 1.28 for z.

To estimate the required sample size, we can use the formula for the sample size in estimating the population mean:

n = (Z * s / E)²

Where:
- n is the required sample size
- Z is the Z-score corresponding to the desired level of confidence
- s is the standard deviation of the population
- E is the desired margin of error

First, let's calculate the required sample size to estimate the HDL cholesterol within 3 points with 99% confidence assuming s = 10.5.

n₁ = (Z₁ * s / E)²

Using the Z-score for 99% confidence, which is approximately 2.576, and the given values of s = 10.5 and E = 3:

n₁ = (2.576 * 10.5 / 3)²
n₁ = 9.0207²
n₁ ≈ 81.43

Therefore, at a 99% confidence level, the doctor would need a sample size of approximately 82 subjects to estimate the HDL cholesterol within 3 points.

Now, let's consider the case where the doctor is content with 90% confidence. We'll calculate the required sample size using the same formula and the desired margin of error:

n₂ = (Z₂ * s / E)²

Using the Z-score for 90% confidence, which is approximately 1.645, and the given values of s = 10.5 and E = 3:

n₂ = (1.645 * 10.5 / 3)²
n₂ = 5.4355²
n₂ ≈ 29.53

Therefore, at a 90% confidence level, the doctor would need a sample size of approximately 30 subjects to estimate the HDL cholesterol within 3 points.

The decrease in confidence from 99% to 90% reduces the required sample size from approximately 82 to 30. As the confidence level decreases, the Z-score decreases, resulting in a smaller sample size needed to achieve the desired level of precision.