Suppose a population has a mean of 400 and a standard deviation of 24.If a random sample of size 144 is drawn from the population, the probability of drawing a sample with a mean between 395.5 and 404.5 is closest to _______.
A)0.9756
B)0.0244
C)0.0278
D)0.9722
E)1.0000

Question

Quizplus · Accepted Answer

Using the Central Limit Theorem, we can calculate the probability of drawing a sample with a mean between 395.5 and 404.5. The standard error (SE) of the sample mean is the population standard deviation divided by the square root of the sample size, which is 24 / sqrt(144) = 24 / 12 = 2. To find the Z-scores for 395.5 and 404.5, we subtract the population mean (400) and divide by the SE. For 395.5, Z = (395.5 - 400) / 2 = -2.25. For 404.5, Z = (404.5 - 400) / 2 = 2.25. Looking these Z-scores up in a standard normal distribution table gives us the probabilities of 0.0122 and 0.9878, respectively. The probability of the sample mean being between 395.5 and 404.5 is the difference, 0.9878 - 0.0122 = 0.9756.

Suppose a Population Has a Mean of 400 and a Standard