t-test for One Sample

Some times you have to decide if a sample mean is different from a hypothesized population mean. You have calculated mean value and standard deviation for the group assuming you have measurement data. For example:

• You hear that the average person sleeps 8 hours a day. You think college students sleep less. You ask 10 college kids how long they sleep on an average day.
• You get the data and the mean of sleep time is 6.5 hours.
• Is this luck? Did you happen to pick a group of light sleepers by chance? Or do college kids really sleep less?
• What you are asking is whether college kids come from a population separate from the rest of society. "Society" supposedly being all the people who contributed to the mean of 8 hours a night that we hear about.

The problem is that if you calculate a sample mean and it is physically different from the one hypothesized, there are two possible reasons for the difference:

• Your sample comes from a different population and the sample mean represents a different population mean. When this happens, you reject the Null Hypothesis.

• The group comes from the same population and the mean varies by chance. You just happened to pick a sample group with a mean that misrepresents the population it came from. The group isn't really different. When this happens you fail to reject the Null Hypothesis - see the graphic on the next page.

The way to decide which is the case is with the one sample t-test. You will compare the sample mean to the population mean and get an estimate of the probability that the sample mean is different by chance. How does the t-test work?

First, you will calculate the mean and standard deviation for your group. A computer might actually do all of this for you.

Let me introduce some notation. The sample mean is usually noted as ` X . The population mean is noted as m. You are testing to see whether or not ` X is different from m.

• You will use the t-test formula to compute the t-value.
• The t-score is like a z-score and tells you if the sample mean is far away from the population mean (m). You will use the sampling distribution of the mean to do this.
• If the t-value is big enough (you look it up in a t-table), you will reject the Null Hypothesis and say you have a difference. If it is not big enough you will say that you have not found a difference and fail to reject the Null Hypothesis.
  

How does the t ratio do this?:

• The t-test sets up a sampling distribution of means with a mean that is specified by the null hypothesis (m).

• The t-test calculates if your sample mean is far in the tails of the sampling distribution and far away from the population mean (m).

• Thus, the difference may be too big to be chance. If your group was really from the null hypothesis distribution, then the difference should be close to zero.

• The t-score is made a relative score by dividing the difference between the sample mean and m by the standard error of the mean. See your book for the standard error formula.

• That's how we know if your sample mean is rare by chance as we can calculate the areas in the tales (see the z-score workshop).