# t-test for the mean using a TI83 or TI84 calculator (p-value method)

Do people tend to spend more than 2 hours on a computer every day? Can you say that the mean age of a college freshman in your state is not 18 years old? These are the types of questions that can be answered using collected data and a t-test for the mean. In this guide, you will see how you can use a TI83 or TI84 calculator to perform this test using the p-value method.

We will use an example to see how this process works. For this example, assume that the requirements for a hypothesis test for the mean are met (randomly selected sample, independent observations, large population size).

## Example: performing a t-test on the calculator

Suppose that a marketing firm believes that people who are planning to purchase a new TV spend more than 7 days researching their purchase. They conduct a survey of 32 people who had recently purchased a TV and found that the mean time spent researching the purchase was 7.8 days with a standard deviation of 3.9 days. At a significance level of 0.05, does this survey provide evidence to support the firm’s belief?

### Step 1: Write the null $(H_0)$ and alternative $(H_a)$ hypotheses

The alternative hypothesis is a statement about what you are testing. Here, you are testing the firm’s belief that the mean time spent is more than 7 days. In hypothesis testing, we are trying to understand a population value using a sample, so your hypothesis should be in terms of the population parameter. In this case, that is the population mean, $\mu$.

$\text{mean time spent is more than 7 days} \rightarrow H_a: \mu > 7$

The null hypothesis is the equality* statement using the same value:

$H_0: \mu = 7$

Putting these together, the null and alternative hypotheses are:

$H_0: \mu = 7\\ H_a: \mu > 7$

(*In some books, they use the statement that is the opposite of $H_a$ for the null hypothesis. Here, that would be $H_0: \mu \leq 7$. Make sure you use the form preferred in the class you are taking!)

### Step 2: Calculate the p-value using your calculator and the correct test

A t-test is used here since we have a big enough sample, and the population standard deviation $(\sigma)$ is unknown. (We only have the standard deviation from the sample: $s = 3.9$.) Note that if we knew the population standard deviation, we would use a z-test instead.

1. Press [STAT] then go the the TESTS menu.

2. Select “2. T-test”. Make sure that you highlight Stats and press [ENTER] if your screen looks different from this.

3. Enter the values and select the correct tail for the test.

4. Highlight Calculate and press [ENTER].

### Step 3: Compare the p-value to the significance level alpha $(\alpha)$ and make your decision

From the last line of the calculator, $\text{p-value} \approx 0.1274$. Further, the last part of the problem stated “At a significance level of 0.05, does this survey provide evidence to support the firm’s belief?”. Therefore, $\alpha = 0.05$.

To make the decision, use the decision rule:

In this problem:

### Step 4: Interpret your decision in terms of the problem

If you fail to reject $H_0$, then you are saying there is not enough evidence for $H_a$. Remember for this problem:

$H_0: \mu = 7\\ H_a: \mu > 7$

So, we are saying that there is not enough evidence that the population mean is greater than 7. In context, we are saying:

This sample does not provide evidence that the mean time spent researching a new TV purchase is more than 7 days.

Although our sample mean was in fact larger than 7, it wasn’t quite enough to suggest that this is true for the entire population. Remember, in hypothesis testing, that is what we are trying to determine – is the sample enough to say that the hypothesis holds for the entire population?