Question

A study involves a population of 400 tall women. This population has a mean height of 179.832 cm and a standard deviation of 12.192 cm. If 50 of these women are randomly selected, find the probability that the mean for this sample group is above 182.88

Answer 1

Answer: 0.0386

Explanation:

Given: The population of 400 tall women has a mean height
`(\mu)` of 179.832 cm and a standard deviation
`(\sigma)` of 12.192 cm.

Let X be a random variable that represents the height of woman.

Sample size : n= 50

The probability that the mean for this sample group is above 182.88 will be :

`P(\overline{X}>182.88)\\\\=P(\frac{\overline{X}-\mu}{(\sigma)/(√(n))}>(182.88-179.832)/((12.192)/(√(50))))\\\\ =P(Z>1.7678)\ \ \ [Z=\frac{\overline{X}-\mu}{(\sigma)/(√(n))}]\\\\=1-P(Z<1.7678)\\\\=1-0.9614\ \ \ [\text{By p-value table}]\\\\= 0.0386`

Hence, Required probability = 0.0386