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 33 years is selected from this population. what is the probability that the mean gain for the sample was between 400 and 700? assume σ = 1541. the probability is (round to four decimal places as needed.)

Explanation:

Step1: Calculate standard error

Standard error formula: $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$.
Given $\sigma = 1541$, $n = 33$.
$\sigma_{\bar{x}}=\frac{1541}{\sqrt{33}}\approx\frac{1541}{5.7446}\approx268.25$

Step2: Calculate z - scores

Z - score formula: $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$
For $\bar{x}=400$:
$z_1=\frac{400 - 653}{268.25}=\frac{- 253}{268.25}\approx - 0.94$
For $\bar{x}=700$:
$z_2=\frac{700 - 653}{268.25}=\frac{47}{268.25}\approx0.17$

Step3: Find probabilities using standard normal distribution table

$P(-0.94\lt Z\lt0.17)=P(Z\lt0.17)-P(Z\lt - 0.94)$
From standard normal table: $P(Z\lt0.17) = 0.5675$, $P(Z\lt - 0.94)=0.1736$
$P(-0.94\lt Z\lt0.17)=0.5675 - 0.1736=0.3939$

Answer:

$0.3939$