Question

a simple random sample of size ( n = 55 ) is obtained from a population with ( mu = 88 ) and ( sigma = 19 ).
enter your answer as an area under the curve with 4 decimal places.

( p(x<85)= )______

Explanation:

Step1: Calculate the z - score

The formula for the z - score of the sample mean \(\bar{x}\) is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). Given \(\mu = 88\), \(\sigma=19\), \(n = 55\), and \(\bar{x}=85\).

$$ z=\frac{85 - 88}{\frac{19}{\sqrt{55}}}=\frac{- 3}{\frac{19}{\sqrt{55}}}\approx\frac{-3}{2.56}\approx - 1.17 $$

Step2: Find the probability

We want to find \(P(\bar{X}<85)\), which is equivalent to \(P(Z < - 1.17)\) 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 Excel: NORM.S.DIST(-1.17,TRUE)).

From the standard normal table, \(P(Z < - 1.17)=0.1210\)

Answer:

\(0.1210\)