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 $n/N>0.05$. Here, $N = 700$ (population size) and $n = 50$ (sample size). Since $\frac{n}{N}=\frac{50}{700}\approx0.0714>0.05$, we use the finite - correction factor. 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 $\sigma = 0.009$, $n = 50$, and $N = 700$.
$\sigma_{\bar{x}}=\frac{0.009}{\sqrt{50}}\sqrt{\frac{700 - 50}{700 - 1}}\approx\frac{0.009}{\sqrt{50}}\sqrt{\frac{650}{699}}\approx\frac{0.009}{7.0711}\times0.9677\approx0.0012$

Step2: Calculate the z - score

The formula for the z - score is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$, where $\bar{x}=2.809$, $\mu = 2.813$, and $\sigma_{\bar{x}}\approx0.0012$.
$z=\frac{2.809 - 2.813}{0.0012}=\frac{- 0.004}{0.0012}\approx - 3.33$

Step3: Find the probability

We want to find $P(\bar{X}<2.809)$, which is equivalent to $P(Z < - 3.33)$ using the standard normal distribution table. From the standard normal table, $P(Z < - 3.33)=0.0004$.

Answer:

$0.0004$