A news report stated that the mean distance

Question

Going to work: A news report stated that the mean distance that commuters in the United States travel each way to work is 14 miles. Assume the standard deviation is 8 miles. A sample of 70 commuters is chosen.

(a) What is the probability that the sample mean commute distance is greater than 

12

miles? Round the answer to at least four decimal places.The probability that the sample mean commute distance is greater than 12	miles is		.

Answer 1

First, I want you to imagine what you're solving for; picture a bell curve with a line directly through the middle that equates to the population mean of 14. Now, imagine there's a line on the bell curve at x=12 and we want to figure out how much of that bell curve is located to the right of that line - because all those values meet the criteria for the question (having a sample mean greater than 12). What we're trying to do here is calculate what percentage of the total bell curve (parabola) is to the right of the x=12 line.

Since this is a question of "what's the chance the number is greater than...", that makes it a right-tailed test. We need to find the probability - which requires us to calculate the z-score. The formula here is z=(x(bar)-μ)/(s/√n) where

xbar= sample mean (in this case, 12)

μ=population mean (in this case, 14)

s=standard deviation (in this case, 8)

n=number in sample (in this case, 70)

When you plug in the numbers, you get a z-score of around -2.09. Looking at a z-score table to find the p-value, you'll either find 0.0183 (if the table only shows left-tail scores) or 0.9817 (if it has a right-tailed score section). The only difference here is that if they only give the left-tail score, you convert it to a right-tailed score by subtracting the p-value from 1.

That gives you the probability that the sample mean is greater than 12.

Answer 2

For a sample of 70, the σ_x = σ/sqrt(70) = 8/sqrt(70)

Find z = (x - μ)/σ_x where x is 12, μ is 14

Use a standard normal table to look up Φ(z) (usually the area to the left of the z value)

take 1-Φ(z) That is your answer.

There are free z-calculators online or use TI-84 2nd Vars (Distr) and select normalcdf (You can adjust to get the right-side area directly)

Please consider a tutor. Take care.