7.3.73 sample standard deviation

An ethanol railroad tariff is a fee charged for shipments of ethanol on public railroads. An agricultural association publishes tariff rates for​ railroad-car shipments of ethanol. Assuming that the standard deviation of such tariff rates is ​$ 1300​,determine the probability that the mean tariff rate of 350 randomly selected​ railroad-car shipments of ethanol will be within ​$110 of the mean tariff rate of all​ railroad-car shipments of ethanol. Interpret your answer in terms of sampling error.

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The probability is (Round to three decimal places as needed)

The sample size n = 350 since there are 350 tariff randomly selected.

Since the sample size is more than 30, the variable is normally distributed followed by the central limit theorem.

Population standard deviation \(\sigma=1300\)

n = 350  
sigma = 1300

First we need to find sample standard deviation by using the formula \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\)

s = sigma/sqrt(n)
s

## [1] 69.48792

Next, we need to find the test statistic by using the formula \(z=\frac{x-\mu}{\sigma}\)

Since our estimated points are within 110 of the mean, so $x = $ . It follows that \(z=\frac{\pm110}{\sigma_{\bar{x}}}\)

110 / s

## [1] 1.583009

-110/ s

## [1] -1.583009

We can get the area between two estimated points by using pnorm()

pnorm(110 / s) - pnorm(-110 / s) 

## [1] 0.8865806

Round to three decimal places

round(pnorm(110 / s) - pnorm(-110 / s) , 3)

## [1] 0.887

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Hope that helps!