# 6a.3 - Set-Up for One-Sample Hypotheses

We will continue our discussion by considering two specific hypothesis tests: a test of one proportion, and a test of one mean. We will provide the general set up of the hypothesis and the test statistics for both tests. From there, we will branch off into specific discussions on each of these tests.

In order to make a judgment about the value of a parameter, the problem can be set up as a hypothesis testing problem. We usually set the hypothesis that one wants to conclude as the alternative hypothesis, also called the research hypothesis.

Since hypothesis tests are about a parameter value, the hypotheses use parameter notation - $$p$$ for proportion or $$\mu$$ for mean - in their arrangement. For tests of a proportion or a test of a mean, we would choose the appropriate alternative based on our research question.

Below are the possible hypotheses from which we would select only one of them based on the research question. The symbols $$p_0$$ and $$\mu_0$$ are used in these general statements and in practice, get replaced by the parameter value, or constant, being tested.

One Sample Proportion
Research Question Is the population proportion different from $$p_0$$? Is the population proportion greater than $$p_0$$? Is the population proportion less than $$p_0$$?
Null Hypothesis, $$H_{0}$$ $$p=p_0$$ $$p= p_0$$ $$p= p_0$$
Alternative Hypothesis, $$H_{a}$$ $$p\neq p_0$$ $$p> p_0$$ $$p< p_0$$
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
*$$p_{0}$$ is the hypothesized population proportion

One Sample Mean
Research Question Is the population mean different from $$\mu_{0}$$? Is the population mean greater than $$\mu_{0}$$? Is the population mean less than $$\mu_{0}$$?
Null Hypothesis, $$H_{0}$$ $$\mu=\mu_{0}$$ $$\mu=\mu_{0}$$ $$\mu=\mu_{0}$$
Alternative Hypothesis, $$H_{a}$$ $$\mu\neq \mu_{0}$$ $$\mu> \mu_{0}$$ $$\mu<\mu_{0}$$
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
*$$\mu_{0}$$ is the hypothesized population mean

The most important step in hypothesis testing is choosing the correct parameter of interest and correctly setting up the alternative hypothesis.

## Example 6-3 Null and Alternative Hypotheses Section

In each of the following scenarios, determine the parameter of interest and the null and alternative hypotheses.

1. When debating the State Appropriation for Penn State, the following question is asked: "Are the majority of students at Penn State from Pennsylvania?"
The response variable is 'State' and is a qualitative variable. Therefore, the parameter of interest would be $$p$$ the population proportion of students from PA. The hypotheses should be in terms of $$p$$. The value we are testing is the “majority” (50%) or $$p_0=0.5$$. The majority also implies greater than 50%. Thus, the hypothesis set up would be a right-tailed test.

$$H_0\colon p=0.5$$ vs. $$H_a\colon p>0.5$$

2. A consumer test agency wants to see the whether the mean lifetime of a brand of tires is less than 42,000 miles. The tire manufacturer advertises that the average lifetime is at least 42,000 miles.
The response variable here is 'lifetime' and is a quantitative variable. Therefore, set up the hypotheses in terms of $$\mu$$. Here the value of $$\mu_0$$ is 42,000. With the consumer test agency wanting to research that the mean lifetime is below 42,000, we would set up the hypotheses as a left-tailed test:

$$H_0\colon \mu=42000$$ vs. $$H_a\colon \mu<42000$$

3. The length of a certain lumber from a national home building store 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.
The response variable is the 'length of lumber' and is quantitative. Therefore, we set up the hypotheses in terms of $$\mu$$. Here the value of $$\mu_0$$ is 8.5. With the builder wanting to check if the mean length is different from 8.5, she would set up the hypotheses as a two-tailed test:

$$H_0\colon \mu=8.5$$ vs $$H_a\colon \mu\ne 8.5$$

4. A political news company believes the national approval rating for the current president has fallen below 40%.
The response variable here is 'approval rating' and is a qualitative variable. Therefore, we will set up the hypothesis in terms of $$p$$. In this case, the $$p_0$$ value is 0.4 and the hypotheses would be set up as a left-tailed test:

$$H_0\colon p=0.4$$ vs. $$H_a\colon p<0.4$$