When we do not know the population standard deviation sigma, σ, and the sample size, n, is less than 30, we use the t-test to evaluate a claim. Consider the following problem:

A scientist thinks the mean waste recycled by adults in the US is now more than one pound per person per day. In a random sample of 12 US adults, the mean waste recycled per day per person is 1.9 pounds with a standard deviation of 0.3 pounds. At a 10% significance level, does the sample data support the claim?

In statistics, it is always a good idea to sketch the situation described in the problem:

The sample mean is far to the right of the assumed population mean, µ = 1.

To solve this problem, we start by stating the null and alternative hypotheses:

- The claim includes the phrase “more than” which indicates the math operator is
**> .** - Because the null must be a form of equality [ ≤, =, or ≥ ], the claim is the alternative.
- Ho: µ ≤ 1; Ha: µ > 1.

We can perform the hypothesis test one of two ways:

- Find the critical value of t and determine if the test statistic is in the rejection region.
- Calculate the p-value for the test statistic and compare it to alpha.

Rejection region approach.

- The math operator in the alternative, >, points to the right, so this is a right-tail test.
- To find the critical value of t, we need the degrees of freedom.
- N = 12
- df = n-1 = 11

- Using Excel,

- Using the StatCrunch command sequence,
**Stat > Calculators > T**. Enter the df and alpha. Select the “≥” operator to get the right tail. Click**Compute**. I like StatCrunch for this because it always creates the sketch which helps prevent “dumb” tail errors.

The critical value of t is 1.36 and the rejection region is t > 1.36, the area in red.

- Now we need to find the test statistic.

Although I can do this using the Compute function in StatCrunch, I like to use Excel for this because I can save the file and reuse on similar problems:

Since t = 10.39 falls in the rejection region to the right of t-critical = 1.36, we reject the null hypothesis.

Now let’s use the p-value approach.

- First, using Excel:

- Now using StatCrunch: Use the command sequence
**Stat > T Stats > One Sample > With Summary.**Enter the sample mean, sample standard deviation, and n. Note: do not convert the sample s to the standard error (sigma sub x-bar) as we did using Excel because StatCrunch has that step built-in.

StatCrunch gives includes the test statistic, 10.392, in the output along with the p-value <0.0001. Again, the decision is to reject the null and conclude:

There is sufficient evidence to support the claim that the mean amount of waste recycled per person per day by US adults is greater than 1 pound.

**Confidence Interval for sample mean**

Remember, if you are asked for the confidence interval around the sample mean you can get it quickly using the same StatCrunch tool. Click on the **Options** button in the upper left of the Output box, then **Edit,** and the dialog box will open. Then just click the radio button next to **Confidence interval for ****µ **and click** Compute!**

Getting the confidence interval using Excel is not difficult though you need to recall we use the critical value of t for alpha/2, which is the two-tail critical value even though we were running a one-tail hypothesis test. That trips up a lot of students who just use the t-critical for the rejection area.