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 14 three year old girls are chosen. find ( p(\bar{x}<37) ) for ( n = 14 ). enter your answer as an area under the curve with 4 decimal places. ( p(\bar{x}<37)= ) question 11 1 pts

Explanation:

Step1: Calculate the standard error

The formula for the standard error \( \sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}} \). Given \( \sigma = 3.2 \) and \( n = 14 \), then \( \sigma_{\bar{x}}=\frac{3.2}{\sqrt{14}}\approx\frac{3.2}{3.742}\approx0.855 \).

Step2: Calculate the z - score

The z - score formula is \( z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}} \). Here, \( \bar{x} = 37 \), \( \mu=38.7 \), and \( \sigma_{\bar{x}}\approx0.855 \). So \( z=\frac{37 - 38.7}{0.855}=\frac{- 1.7}{0.855}\approx - 1.99 \).

Step3: Find the probability

Using the standard normal distribution table (or a calculator with a normal - distribution function, e.g., in Excel: NORM.S.DIST(-1.99, TRUE)), we find \( P(Z\lt - 1.99)\).

Answer:

\(0.0233\)