Lesson 6b: Hypothesis Testing for One-Sample Mean

Overview Section

In the previous Lesson, we learned how to perform a hypothesis test for one proportion. The concepts of hypothesis testing remain constant for any hypothesis test. In these next few sections, we will present the hypothesis test for one mean. We start with our knowledge of the sampling distribution of the sample mean.

Recall that under certain conditions, the sampling distribution of the sample mean, \(\bar{x} \), is approximately normal with mean, \(\mu \), standard error \(\frac{\sigma}{\sqrt{n}} \), and estimated standard error \(\frac{s}{\sqrt{n}} \).

The conditions are:

  • The distribution of the population is Normal
  • The sample size is large \( n>30 \).

If at least one of conditions are satisfied, then...

\( t=\dfrac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}} \)

will follow a t-distribution with \(n-1 \) degrees of freedom.

We can use this information to make probability statements for \(\bar{x} \).

Let’s look at an example.


Length of Lumber Section

The mean length of the lumber is supposed to be 8.5 feet. A builder wants to check whether the shipment of lumber she receives has a mean length different from 8.5 feet. If the builder observes that the sample mean of 61 pieces of lumber is 8.3 feet with a sample standard deviation of 1.2 feet. What will she conclude? Is 8.3 very different from 8.5?


This depends on the standard deviation of \(\bar{x} \) .

\begin{align} t^*&=\dfrac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}\\&=\dfrac{8.3-8.5}{\frac{1.2}{\sqrt{61}}}\\&=-1.3 \end{align}

Thus, we are asking if \(-1.3\) is very far away from zero, since that corresponds to the case when \(\bar{x}\) is equal to \(\mu_0 \). If it is far away, then it is unlikely that the null hypothesis is true and one rejects it. Otherwise, one cannot reject the null hypothesis.

How do we determine whether to reject the null hypothesis? Section

It depends on the level of significance \(\alpha \) (step 2 of conducting a hypothesis test), and the probability the sample data would produce the observed result. In the next section, we set up the six steps for a hypothesis test for one mean.


Upon successful completion of this lesson, you should be able to:

  • Perform hypothesis testing for a population mean using the p-value approach and the rejection region approach.
  • Use confidence intervals to draw conclusions about two-sided tests.