Question

find the indicated probability and interpret the result.
from 1975 through 2020, the mean annual gain of the dow jones industrial average was 653. a random sample of 32 years is selected from this population. what is the probability that the mean gain for the sample was between 500 and 700? assume σ = 1541.
the probability is
(round to four decimal places as needed.)

Explanation:

Step1: Calculate the standard error

The formula for the standard error of the mean is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$.
Given $\sigma = 1541$ and $n = 32$, then $\sigma_{\bar{x}}=\frac{1541}{\sqrt{32}}\approx272.37$.

Step2: Calculate the z - scores

The z - score formula is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$.
For $\bar{x}=500$, $z_1=\frac{500 - 653}{272.37}\approx - 0.56$.
For $\bar{x}=700$, $z_2=\frac{700 - 653}{272.37}\approx0.17$.

Step3: Find the probability

We want to find $P(500<\bar{x}<700)=P(-0.56 < z < 0.17)$.
Using the standard normal distribution table, $P(z < 0.17)=0.5675$ and $P(z<-0.56)=0.2877$.
Then $P(-0.56 < z < 0.17)=P(z < 0.17)-P(z<-0.56)=0.5675 - 0.2877=0.2798$.

Answer:

$0.2798$