Question

It is known that the serum cholesterol level of females is normally distributed and its population variance is 45 mg/100 ml. Researchers decided to estimate the population mean serum cholesterol level. They observed a sample average serum cholesterol level of 220 mg/ml in 35 females. Researchers at another university are going to collect new data and they are assuming that the female serum cholesterol level is Normally distributed with population variance of 45. How many subjects do they have to observe in order to have a 95% confidence interval with a final length of at most 3? The number is either 21 subjects or 77 subjects. Please specifically explain why one of the choices is correct and specially explain why the other choice is wrong.

Answer 1

Answer: It is known that the serum cholesterol level of females is Normally distributed and its population variance is 45 mg/100mL. Researchers decided to estimate the population mean serum cholesterol level. They observed a sample average serum cholesterol level of 220 mg/ml in 35 females.

Solution:

Variance = 45

σ = √variance = √45 = 6.7082

At 95% confidence interval, α = 0.05

Zα/2 = 1.96

We know that margin of error is half the length of confidence interval.

Therefore,

Margin of error, E = 3/2 = 1.5

Therefore, the required sample size, n:

n = (Zα/2/E)^2 * σ2

n = (1.96/1.5)^2 * 6.7082

n = 76.8319

n = 77

Therefore, subjects they have to observe in order to have a 95% confidence interval with a final length of at most 3.