A random sample of size 36 is taken from a population with mean µ = 17 and standard deviation σ = 6. The probability that the sample mean is between 15 and 18 is ________.
A) 0.0228
B) 0.8186
C) 0.8413
D) 0.8641

Question

Quizplus · Accepted Answer

To find the probability that the sample mean is between 15 and 18, 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 in this case, so the condition is met). The mean of the sampling distribution is equal to the population mean (µ = 17), and the standard deviation of the sampling distribution (standard error) is equal to the population standard deviation (σ = 6) divided by the square root of the sample size (n = 36), which is 6/√36 = 1.Next, we calculate the Z-scores for 15 and 18:Z for 15 = (15 - 17) / 1 = -2Z for 18 = (18 - 17) / 1 = 1Using the Z-table, we find the probabilities corresponding to these Z-scores. The probability of Z being less than -2 is approximately 0.0228, and the probability of Z being less than 1 is approximately 0.8413. To find the probability that the sample mean is between 15 and 18, we subtract the two probabilities:P(15 < X̄ < 18) = P(Z < 1) - P(Z < -2) = 0.8413 - 0.0228 = 0.8185The answer closest to our calculation is 0.8186, which is option B.

A Random Sample of Size 36 Is Taken from a Population