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:

We can perform the hypothesis test one of two ways:

Rejection region approach.

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

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.

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.

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