t Tests

One-Sample t Test

A scenario for a one-sample t test can be formed as follows. For example, a student looking into gardening finds a source online that claims its tomato seeds will yield at minimum 30 tomatoes per plant, on average. Questioning such a claim, the student purchases the seeds to test the validity of the company’s statement. Because the students’ backyard has limited space, only 12 seeds could be planted in separate locations with similar conditions. After a couple of months, the tomatoes on the vines begin to ripen and the student records the data. The sample mean x comes out to 28.5 tomatoes with a sample SD (s) of 2.19 tomatoes. The student states the null hypothesis as the average tomato yield per plant to be μ = 30, and the alternate hypothesis as µ < 30. This way, if the student rejects the null, there is sufficient information that suggests a contradiction to the company’s claim. The student also sets the level of significance at α = .05. The student obtains the test statistic (t) through the equation $t=\frac{\overline{x}-{\mu }_0}{s/\sqrt{n}}$, where µ0 is the numeric value that the mean is being compared to (in this case, it is 30), and n = 12 is the sample size, which comes out to t = −2.37. Using the t table, the student calculates the critical value (t0.05) at 11 degrees of freedom (n − 1) which is −1.796. The critical value is a predetermined area under the density curve to the right of it. The degrees of freedom is the size of the sample minus the number of estimated distribution parameters. In this instance, the population SD is the only parameter estimated from the data, thus the number of degrees of freedom is reduced by one. Further, because the test statistic falls in the rejection region (t ≤ t0.05), the student rejects the null and concludes that there is sufficient evidence against the company’s claim.

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