5.3 - Hypothesis Testing for One-Sample Mean

In the previous section, 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.

Hypothesis Test for One-Sample Mean

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

Null:

\(H_0\colon \mu=\mu_0\)

Conditions:

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

Test Statistic:

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.

Notice when working with continuous data we are going to use a t statistic as opposed to the z statistic. This is due to the fact that the sample size impacts the sampling distribution and needs to be taken into account. We do this by recognizing “degrees of freedom”. We will not go into too much detail about degrees of freedom in this course.

Let’s look at an example.

Example 5-1

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.

5.3.1- Steps in Conducting a Hypothesis Test for \(\mu\)

Step 1: Set up the hypotheses and check conditions

One Mean t-test Hypotheses

\( H_0\colon \mu=\mu_0 \)

\( H_a\colon \mu\ne \mu_0 \)

Conditions: The data comes from an approximately normal distribution or the sample size is at least 30
Step 2: Decide on the significance level, \(\alpha \)
Typically, 5%. If \(\alpha\) is not specified, use 5%
Step 3: Calculate the test statistic
One Mean t-test: \( t^*=\dfrac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}} \)
Step 4: Compute the appropriate p-value based on our alternative hypothesis
Typically we will let Minitab handle this for us. But if you are really interested, you can look p values up in probability tables found in the appendix of your textbook!
Step 5: Make a decision about the null hypothesis
If the p-value is less than the significance level, \(\alpha\), then reject \(H_0\) (and conclude \(H_a \)). If it is greater than the significance level, then do not reject \(H_0 \).
Step 6
State an overall conclusion.

Minitab^®

Conduct a One-Sample Mean t-Test

Note that these steps are very similar to those for one-mean confidence interval. The differences occur in steps 4 through 8.

To conduct the one sample mean t-test in Minitab...

Choose Stat > Basic Stat > 1 Sample t.
In the drop-down box use "One or more samples, each in a column" if you have the raw data, otherwise select "Summarized data" if you only have the sample statistics.
If using the raw data, enter the column of interest into the blank variable window below the drop down selection. If using summarized data, enter the sample size, sample mean, and sample standard deviation in their respective fields.
Choose the check box for "Perform hypothesis test" and enter the null hypothesis value.
Choose Options.
Enter the confidence level associated with alpha (e.g. 95% for alpha of 5%).
From the drop down list for "Alternative hypothesis" select the correct alternative.
Click OK and OK.