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# 8.3: Testing a Mean Hypothesis

Difficulty Level: At Grade Created by: CK-12

### Evaluating Hypotheses for Population Means using Large Samples

When testing a hypothesis for the mean of a normal distribution, we follow a series of four basic steps:

1. State the null and alternative hypotheses.
2. Choose an \begin{align*}\alpha\end{align*} level.
3. Set the criterion (critical values) for rejecting the null hypothesis.
4. Compute the test statistic.
5. Make a decision (reject or fail to reject the null hypothesis).
6. Interpret the result.

If we reject the null hypothesis, we are saying that the difference between the observed sample mean and the hypothesized population mean is too great to be attributed to chance. When we fail to reject the null hypothesis, we are saying that the difference between the observed sample mean and the hypothesized population mean is probable if the null hypothesis is true. Essentially, we are willing to attribute this difference to sampling error.

Example: The school nurse was wondering if the average height of \begin{align*}7^{\text{th}}\end{align*} graders has been increasing. Over the last 5 years, the average height of a \begin{align*}7^{\text{th}}\end{align*} grader was 145 cm, with a standard deviation of 20 cm. The school nurse takes a random sample of 200 students and finds that the average height this year is 147 cm. Conduct a single-tailed hypothesis test using a 0.05 significance level to evaluate the null and alternative hypotheses.

First, we develop our null and alternative hypotheses:

\begin{align*}H_0: \mu &= 145\\ H_a: \mu &> 145\end{align*}

Next, we choose \begin{align*}\alpha=0.05\end{align*}. The critical value for this one-tailed test is 1.64. Therefore, any test statistic greater than 1.64 will be in the rejection region.

Finally, we calculate the test statistic for the sample of \begin{align*}7^{\text{th}}\end{align*} graders as follows:

\begin{align*}z=\frac{147-145}{\frac{20}{\sqrt{200}}} \approx 1.414\end{align*}

Since the calculated \begin{align*}z\end{align*}-score of 1.414 is smaller than 1.64, it does not fall in the critical region. Thus, our decision is to fail to reject the null hypothesis and to conclude that the probability of obtaining a sample mean equal to 147 if the mean of the population is 145 is likely to have been due to chance.

Again, when testing a hypothesis for the mean of a distribution, we follow a series of six basic steps. Commit these steps to memory:

1. State the null and alternative hypotheses.
2. Choose an \begin{align*}\alpha\end{align*} level.
3. Set the criterion (critical values) for rejecting the null hypothesis.
4. Compute the test statistic.
5. Make a decision (reject or fail to reject the null hypothesis).
6. Interpret the result.

For a step-by-step example of testing a mean hypothesis (4.0), see MuchoMath, Z Test for the Mean (9:34).

## Review Questions

1. In hypothesis testing, when we work with large samples, we use the ___ distribution. When working with small samples (typically samples under 30), we use the ___ distribution.
2. True or False: When we fail to reject the null hypothesis, we are saying that the difference between the observed sample mean and the hypothesized population mean is probable if the null hypothesis is true.
3. The dean from UCLA is concerned that the students' grade point averages have changed dramatically in recent years. The graduating seniors’ mean GPA over the last five years is 2.75. The dean randomly samples 256 seniors from the last graduating class and finds that their mean GPA is 2.85, with a sample standard deviation of 0.65.
1. What would the null and alternative hypotheses be for this scenario?
2. What would the standard error be for this particular scenario?
3. Describe in your own words how you would set the critical regions and what they would be at an alpha level of 0.05.
4. Test the null hypothesis and explain your decision.
4. For each of the following pairs of scenarios, state which option is more likely to lead to the rejection of the null hypothesis.
1. A one-tailed or two-tailed test
2. A 0.05 or 0.01 level of significance
3. A sample size of \begin{align*}n = 144\end{align*} or \begin{align*}n = 444\end{align*}

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