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. find p(x < 37).
enter your answer as an area under the curve with 4 decimal places.
p(x < 37) = ______
(this problem is just like ch 6.1 so you are calculating a regular z - score and looking up the la.)

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 38.7\) (mean), \(\sigma=3.2\) (standard deviation), and \(x = 37\).

$$ z=\frac{37 - 38.7}{3.2}=\frac{-1.7}{3.2}=- 0.53125 $$

Step2: Find the area corresponding to the z - score

We look up the value of \(P(Z\lt - 0.53125)\) in the standard normal distribution table. Using a standard normal table or a calculator with a normal - distribution function (e.g., in Excel: =NORM.S.DIST(-0.53125,TRUE) or in R: pnorm(-0.53125)).
The area \(P(Z\lt - 0.53125)\approx0.2971\)

Answer:

\(0.2971\)