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 less than 402 is closest to _______.
A)0.3413
B)0.6826
C)0.8413
D)0.1587
E)0.9875

Question

Quizplus · Accepted Answer

To find the probability of drawing a sample with a mean less than 402, we use the Central Limit Theorem, which 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, a standard deviation (σ) of 24, and a sample size (n) of 144, we first calculate the standard error of the mean (σM) as σ/√n = 24/√144 = 24/12 = 2. We then calculate the Z-score for a sample mean of 402, which is (X̄ - μ) / σM = (402 - 400) / 2 = 1. A Z-score of 1 corresponds to the area to the left of it in the standard normal distribution, which is approximately 0.8413. This means there's about an 84.13% chance of drawing a sample with a mean less than 402.

Suppose a Population Has a Mean of 400 and a Standard