Question

We wish to see if, on average, traffic is moving at the posted speed limit of 65 miles per hour along a certain stretch of Interstate 70. On each of four randomly selected days, a randomly selected car is timed and the speed of the car is recorded. The observed speeds are 70, 65, 70, and 75 miles per hour. Assuming that speeds are Normally distributed with mean μ, we test whether, on average, traffic is moving at 65 miles per hour, by testing the hypotheses H0: μ = 65, Ha: μ ≠ 65.

Answer 1

First you need to set you significant level. Since Ha: μ ≠ 65, you're dealing with a two-tailed test. If you set the significant level at 0.05 (5%), then it would be 0.025 (2.5%) per tail.

65 is you mean. You need to find the difference between μ and the sample mean. That difference would be in ''units'' of miles per hour. You need to convert that difference into ''units'' of standard deviation.

\(units\ of\ standard\ deviation = \large{ \frac{\bar{x}-\mu}{\sigma}}\)

Since you're not given σ, you need to estimate it with the sample standard deviation (sd).

Hence, what I've written in Latex code is called a t-statistics and you'll need a t-table to figure out the proportion (percentage) to see how the hypothesis testing goes.

Take the difference --> Estimate sd --> convert difference into units of sd --> take a look at your degrees of freedom --> Find the proportion with a t-table or t-calculator (you can find the proportion for a tails with 0.025 and then multiply that value by 2, to account for the other tail).

Answer 2

I use this T-calculator:

http://surfstat.anu.edu.au/surfstat-home/tables/t.php

It's neat.