Assume that the heights of women are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. If 75 women are randomly selected, find the probability that they have a mean height
Between 63 and 65 inches.
A)0.9811
B)0.3071
C)0.0188
D)0.2119

Question

Quizplus · Accepted Answer

We need to find the probability that the sample mean lies between 63 and 65 inches. We can use the central limit theorem since the sample size is large enough (n=75).
z-score for 63 inches: (63-63.6)/(2.5/sqrt(75)) = -1.92
z-score for 65 inches: (65-63.6)/(2.5/sqrt(75)) = 1.92
Using a standard normal distribution table, the area between -1.92 and 1.92 is 0.9811. Therefore, the probability that the sample mean height is between 63 and 65 inches is 0.9811. We know that the sample mean follows a normal distribution with a mean of 63.6 inches and a standard deviation of 2.5 inches/sqrt(75) = 0.2892 inches.

To find the probability that the sample mean is between 63 and 65 inches, we need to standardize these values:

Z1 = (63 - 63.6) / 0.2892 = -2.07
Z2 = (65 - 63.6) / 0.2892 = 2.07

Using a standard normal table or calculator, we can find the area between these two values:

P(-2.07 < Z < 2.07) = 0.9811

Therefore, the answer is A) 0.9811.

Assume That the Heights of Women Are Normally Distributed with a Mean