Question

the heights of children are normally distributed. for three year old girls, the mean height is 38.7 inches and the standard deviation is 3.2 inches. a random sample of 5 three year old girls are chosen. find p( x < 37) for n = 5. enter your answer as an area under the curve with 4 decimal places. p( x < 37) = ______

Explanation:

Step1: Calculate the standard error

The formula for the standard error of the mean is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$. Given $\sigma = 3.2$ and $n = 5$, we have $\sigma_{\bar{x}}=\frac{3.2}{\sqrt{5}}\approx1.431$.

Step2: Calculate the z - score

The z - score formula is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$. Here, $\bar{x}=37$, $\mu = 38.7$. So $z=\frac{37 - 38.7}{1.431}\approx - 1.19$.

Step3: Find the probability

We want to find $P(\bar{X}<37)$, which is equivalent to $P(Z < - 1.19)$ using the standard normal distribution. Looking up the value in the standard normal table or using a calculator with a normal - distribution function (e.g., in R: pnorm(-1.19) or in Excel: NORM.S.DIST(-1.19,TRUE)), we get $P(Z < - 1.19)=0.1170$.

Answer:

$0.1170$