A population has a mean of 400 and a standard deviation of 25. A sample of 100 observations will be taken. The probability that the sample mean will be greater than 405 is
A) .9772.
B) .0228.
C) .4207.
D) .3805.

Question

Quizplus · Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean will be normally distributed if the sample size is large enough (n ≥ 30 is a common rule of thumb), regardless of the shape of the population distribution. Given a population mean (μ) of 400 and a standard deviation (σ) of 25, for a sample size (n) of 100, the standard error of the mean (σ_x̄) is σ/√n = 25/√100 = 2.5. To find the probability that the sample mean is greater than 405, we first calculate the Z-score for 405, which is (405 - 400) / 2.5 = 2. The Z-score tells us how many standard deviations away from the mean our value is. Looking up the Z-score of 2 in the standard normal distribution table, we find that the area to the left of Z=2 is approximately .9772, which means the area to the right (the probability that the sample mean is greater than 405) is 1 - .9772 = .0228.

A Population Has a Mean of 400 and a Standard