Student’s t Distribution Lecture 33 Section 10.2 Wed, Dec 1, 2004

What if  is Unknown? It is more realistic to assume that the value of  is unknown. (If we don’t know the value of , then we probably don’t know the value of ). In this case, we use s to estimate .

What if  is Unknown? Let us assume that the population is normal or nearly normal. Then the distribution ofx is normal. That is,x is N(, /n). However,x is not N(, s/n) unless the sample size is large enough, (n  30).

What if  is Unknown? In other words, is not standard normal, so we can’t use the tables. If it is not N(0, 1) , then what is it?

Student’s t Distribution It has a distribution called Student’s t distribution. The t distribution was discovered by W. S. Gosset in 1908. He used the pseudonym “Student” to avoid getting fired for doing statistics on the job!!!

The t Distribution The shape of the t distribution is very similar to the shape of the standard normal distribution. However, the t distribution has a (slightly) different shape for each possible sample size. They are all symmetric and unimodal. They are all centered at 0.

The t Distribution They are somewhat broader than Z, reflecting the additional uncertainty resulting from using s in place of . As n gets larger and larger, the shape of the t distribution approaches the standard normal.

Degrees of Freedom If the sample size is n, then t is said to have n – 1 degrees of freedom. We use df to denote “degrees of freedom.”

Standard Normal vs. t Distribution The distributions t(2), t(30), and N(0, 1). t(2) t(30) N(0, 1)

Table IV – t Percentiles Table IV gives certain percentiles of t for certain degrees of freedom. Specific percentiles for upper-tail areas: 0.40, 0.30, 0.20, 0.10, 0.05, 0.025, 0.01, 0.005. Specific degrees of freedom: 1, 2, 3, …, 30, 40, 60, 120.

Table IV – t Percentiles The table tells us, for example, that P(t > 1.812) = 0.05, when df = 10. Since the t distribution is symmetric, we can also use the table for lower tails by making the t values negative. So, what is P(t < –1.812), when df = 10?

Table IV – t Percentiles The table allows us to look up certain percentiles, but it will not allow us to find probabilities.

TI-83 – Student’s t Distribution The TI-83 will find probabilities for the t distribution Press DISTR. Select tcdf and press ENTER. tcdf( appears in the display. Enter the lower endpoint. Enter the upper endpoint.

TI-83 – Student’s t Distribution Enter the number of degrees of freedom (n – 1). Press ENTER. The result is the probability.

Example Enter tcdf(1.812, 99, 10). The result is 0.0500. Thus, P(t > 1.812) = 0.05 when there are 10 degrees of freedom (n = 11).

TI-83 – Student’s t Distribution The TI-83 allows us to find probabilities, but it will not find percentiles for the t distribution.

Hypothesis Testing with t We should use the t distribution if The population is normal (or nearly normal), and  is unknown, so we use s in its place, and The sample size is small (n < 30). Otherwise, we should not use t.

Hypothesis Testing with t The hypothesis testing procedure is the same except for two steps. Step 3: Find the value of the test statistic. The test statistic is now Step 4: Find the p-value. We must look it up in the t table, or use tcdf on the TI-83.

Example Re-do Example 10.1 (by hand) under the assumption that  is unknown.

TI-83 – Hypothesis Testing When  is Unknown Press STAT. Select TESTS. Select T-Test. A window appears requesting information. Choose Data or Stats.

TI-83 – Hypothesis Testing When  is Unknown Assuming we selected Stats, Enter 0. Enterx. Enter s. (Remember,  is unknown.) Enter n. Select the alternative hypothesis and press ENTER. Select Calculate and press ENTER.

TI-83 – Hypothesis Testing When  is Unknown A window appears with the following information. The title “T-Test” The alternative hypothesis. The value of the test statistic t. The p-value. The sample mean. The sample standard deviation. The sample size.

Example Re-do Example 10.1 on the TI-83 under the assumption that  is unknown.

Let’s Do It! Let’s do it! 10.3, p. 582 – Study Time. Let’s do it! 10.4, p. 583 – pH Levels.