# Mean hypothesis testing with t-distribution

### Mean hypothesis testing with t-distribution

#### Lessons

If $\sigma$ is not known, then we cannot use the test statistic:
$Z=\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$
We will instead use the test-statistic:
$Z=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}$
So at the very least we must know the sample standard deviation, $s$. Furthermore we will be using a t-distribution instead of our standard normal distribution to find our fail to reject region and our rejection region.
• Introduction
How do we test hypotheses about mean, when we don't know $\sigma$?

• 1.
Hypothesis Testing Mean Claims without Knowing $\sigma$
A gravel company has been known in the past to overload their trucks. The load capacity is 2500lbs of gravel for one of their standard trucks. A total of 41 trucks were sampled and had an average load of 2550lbs, with a standard deviation of 150lbs. With a significance level of $\alpha$=0.01 can it be said that this company overloads their trucks?

• 2.
"Redline motorcycles" is a company that fixes and tunes motorcycles. A sample of 75 of their motorcycles had an average of 135hp, and a standard deviation of 35hp. Test the following claims with a 99% confidence level:
a)
"The average motorcycle produced by Redline has more than 125hp"

b)
"The average motorcycle produced by Redline doesn't have 125 hp"

c)
Compare the two answer found in the previous two parts