Question

Asked Mar 5, 2019

The mean height of women in a country (ages 20 −29) is 63.9 inches. A random sample of 55 women in this age group is selected. What is the probability that the mean height for the sample is greater than 64 inches? Assume σ= 25.2 .

The probability that the mean height for the sample is greater than 64 inches is________ (Round to four decimal places as needed.)

Step 1

**Introduction:**

A random sample of size *n* = 55 women in the age of 20-29 years is selected.

Let *X* denotes the height of a randomly selected women.

Denote the population mean, that is, the mean height of all the women in the age of 20-29 years as μ and the standard deviation in height of all the women in the age of 20-29 years as σ.

Here, μ =63.9 inches and σ = 25.2 inch.

Denote the sample mean height of 55 randomly selected women as *X*-bar.

Step 2

**Sampling distribution of sample mean:**

The sample mean of a random variable *X* has a sampling distribution that has population mean or expected value same as that of *X* and population standard deviation, called the standard error, which is the population standard deviation of *X* divided by the square root of the sample size *n*. The parameters are as follows:

Step 3

**Central limit theorem:**

If *X*1, *X*2, …, *Xn* are independent and identically distributed (iid) random variables with finite mean, *E*(*Xi*) = *μ* and finite variance *V*(*Xi*) = *σ*2, then *Z* will converge to ...

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