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 400 and 800? assume $sigma = 1540$. the probability is 0.5292 (round to four decimal places as needed.) interpret the result. select the correct choice and fill in the answer box to complete your choice. (round to two decimal places if needed.) a. about % of samples of 32 years will have an annual mean gain between 400 and 653 b. about % of samples of 46 years will have an annual mean gain between 400 and 800 c. about % of samples of 32 years will have an annual mean gain between 400 and 800 d. about % of samples of 32 years will have an annual mean gain between 653 and 800

Explanation:

Step1: Calculate z - scores

The formula for the z - score of the sample mean is $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$, where $\mu = 653$, $\sigma = 1540$, and $n = 32$.
For $\bar{x}=400$, $z_1=\frac{400 - 653}{\frac{1540}{\sqrt{32}}}=\frac{- 253}{\frac{1540}{5.657}}\approx\frac{-253}{272.23}\approx - 0.93$.
For $\bar{x}=800$, $z_2=\frac{800 - 653}{\frac{1540}{\sqrt{32}}}=\frac{147}{\frac{1540}{5.657}}\approx\frac{147}{272.23}\approx0.54$.

Step2: Find the probability

We want $P(400<\bar{X}<800)=P(-0.93 < Z < 0.54)$.
Using the standard normal distribution table, $P(-0.93From the standard - normal table, $P(Z < 0.54)=0.7054$ and $P(Z < - 0.93)=0.1762$.
So $P(-0.93 < Z < 0.54)=0.7054 - 0.1762=0.5292$.

The probability that the mean gain for the sample of 32 years was between 400 and 800 is 0.5292.

To interpret the result:
The probability 0.5292 means that about $52.92\%$ of samples of 32 years will have an annual mean gain between 400 and 800.

Answer:

C. About 52.92% of samples of 32 years will have an annual mean gain between 400 and 800.