Question

determine if the finite correction factor should be used. if so, use it in your calculations when you find the probability. in a sample of 700 gas stations, the mean price for regular gasoline at the pump was $2.813 per gallon and the standard deviation was $0.009 per gallon. a random sample of size 50 is drawn from this population. what is the probability that the mean price per gallon is less than $2.809? click here to view information about the finite correction factor. the probability that the mean price per gallon is less than $2.809 is . (round to four decimal places as needed.)

Explanation:

Step1: Check finite - correction factor condition

The finite - correction factor is used when $n/N>0.05$, where $n$ is the sample size and $N$ is the population size. Here, $n = 50$ and $N=700$. Calculate $n/N=\frac{50}{700}\approx0.0714>0.05$, so the finite - correction factor should be used. The formula for the standard deviation of the sample mean with the finite - correction factor is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\sqrt{\frac{N - n}{N - 1}}$, where $\sigma$ is the population standard deviation, $n$ is the sample size, and $N$ is the population size. Given $\sigma = 0.009$, $n = 50$, and $N = 700$. First, calculate $\sqrt{\frac{N - n}{N - 1}}=\sqrt{\frac{700 - 50}{700 - 1}}=\sqrt{\frac{650}{699}}\approx\sqrt{0.9299}\approx0.9643$, and $\frac{\sigma}{\sqrt{n}}=\frac{0.009}{\sqrt{50}}\approx\frac{0.009}{7.0711}\approx0.0013$. Then $\sigma_{\bar{x}}=0.0013\times0.9643\approx0.00125$.

Step2: Calculate the z - score

The formula for the z - score is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean. Given $\mu = 2.813$, $\bar{x}=2.809$, and $\sigma_{\bar{x}}\approx0.00125$. Then $z=\frac{2.809 - 2.813}{0.00125}=\frac{- 0.004}{0.00125}=-3.2$.

Step3: Find the probability

We want to find $P(\bar{X}<2.809)$, which is equivalent to $P(Z < - 3.2)$ using the standard normal distribution. Looking up the value in the standard - normal table, $P(Z < - 3.2)=0.0013$.

Answer:

$0.0013$