Student’s t Distribution Lecture 32 Section 10.2 Fri, Nov 10, 2006

What if  is Unknown? It is more realistic to assume that the value of  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  ). (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 . In this case, we use s to estimate .

Example See Example 10.1 on page 616 and assume that  is unknown. See Example 10.1 on page 616 and assume that  is unknown. Step 1: State the hypotheses. Step 1: State the hypotheses. H 0 :  = 15 mg. H 0 :  = 15 mg. H 1 :  < 15 mg. H 1 :  < 15 mg. Step 2: State the significance level. Step 2: State the significance level.  =  = Step 3: What is the test statistic? Step 3: What is the test statistic? We must digress. We must digress.

What if  is Unknown? Let us assume that the population is normal or nearly normal. Let us assume that the population is normal or nearly normal. Then the distribution of  x is normal. Then the distribution of  x is normal. That is, That is, However, for small n, However, for small n,

What if  is Unknown? Why not? Why not? And if it is not N(0, 1), then what is it? And if it is not N(0, 1), then what is it?

Student’s t Distribution It has a distribution called Student’s t distribution. It has a distribution called Student’s t distribution. The t distribution was discovered by W. S. Gosset in The t distribution was discovered by W. S. Gosset in See Distribution.html See Distribution.html Distribution.html Distribution.html

The t Distribution The shape of the t distribution is very similar to the shape of the standard normal distribution. The shape of the t distribution is very similar to the shape of the standard normal distribution. It is It is symmetric symmetric unimodal unimodal centered at 0. centered at 0. But it is wider than the standard normal. But it is wider than the standard normal. That is because of the additional variability introduced by using s instead of . That is because of the additional variability introduced by using s instead of .

The t Distribution Furthermore, the t distribution has a (slightly) different shape for each possible sample size. Furthermore, the t distribution has a (slightly) different shape for each possible sample size. As n gets larger and larger, s exhibits less and less variability, so the shape of the t distribution approaches the standard normal. As n gets larger and larger, s exhibits less and less variability, so the shape of the t distribution approaches the standard normal. In fact, if n  30, then the t distribution is approximately standard normal. In fact, if n  30, then the t distribution is approximately standard normal.

Degrees of Freedom If the sample size is n, then t is said to have n – 1 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.” We use df to denote “degrees of freedom.” We will use the notation t(df) to represent the t distribution with df degrees of freedom. We will use the notation t(df) to represent the t distribution with df degrees of freedom. For example, t(5) is the t distribution with 5 degrees of freedom (i.e., sample size 6). For example, t(5) is the t distribution with 5 degrees of freedom (i.e., sample size 6).

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

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

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

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

The Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno Give up TBA

The Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno Give up Is the population normal? yes no

The Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno Give up Is the population normal? yes no Is n  30? yesno

The Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno Give up Is the population normal? yes no Is n  30? yesno

The Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno Give up Is the population normal? yes no Is n  30? yesno

The Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno Give up Is the population normal? yes no Is n  30? yesno Is n  30? yesno

The Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno Give up Is the population normal? yes no Is n  30? yesno Is n  30? yesno

The Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno Give up Is the population normal? yes no Is n  30? yesno Is n  30? yesno Give up

The Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno Give up Is the population normal? yes no Is n  30? yesno Is n  30? yesno Give up

The Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno Give up Is the population normal? yes no Is n  30? yesno Is n  30? yesno Give up

The Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno Is the population normal? yes no Is n  30? yesno Is n  30? yesno Give up Give up

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

Table IV – t Percentiles What is the 95 th percentile of the t distribution when df = 10? What is the 95 th percentile of the t distribution when df = 10? The table tells us that The table tells us that P(t > 1.812) = 0.05, when df = 10. Thus, the 95 th percentile is Thus, the 95 th percentile is

Table IV – t Percentiles Since the t distribution is symmetric, we can also use the table for lower tails by making the t values negative. Since the t distribution is symmetric, we can also use the table for lower tails by making the t values negative. So, what is the 5 th percentile of t when df = 10? So, what is the 5 th percentile of t when df = 10? What is the 90 th percentile of t when df = 25? What is the 90 th percentile of t when df = 25?

Table IV – t Percentiles The table allows us to look up certain percentiles, but it will not allow us to find probabilities in general. The table allows us to look up certain percentiles, but it will not allow us to find probabilities in general. For example, what is P(t > 2.4) when df = 12? For example, what is P(t > 2.4) when df = 12?

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

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

Examples Enter tcdf(1.812, E99, 10). Enter tcdf(1.812, E99, 10). The result is The result is Enter tcdf(-E99, , 10). Enter tcdf(-E99, , 10). The result is The result is Thus, P(t > 1.812) = 0.05 when there are 10 degrees of freedom (n = 11). Thus, P(t > 1.812) = 0.05 when there are 10 degrees of freedom (n = 11). Find P(t > 2.4) when df = 12. Find P(t > 2.4) when df = 12.

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

Hypothesis Testing with t Continue with Example 10.1, p Continue with Example 10.1, p Step 3: Find the value of the test statistic. Step 3: Find the value of the test statistic. The test statistic is now The test statistic is now Step 4: Compute t from  x, , s, and n. Step 4: Compute t from  x, , s, and n.

Example Step 5: Find the p-value. Step 5: Find the p-value. We must look it up in the t table, or use tcdf on the TI-83. We must look it up in the t table, or use tcdf on the TI-83. Step 6: Make the decision regarding H 0 and H 1. Step 6: Make the decision regarding H 0 and H 1. Step 7: State the conclusion about cigarettes and carbon monoxide. Step 7: State the conclusion about cigarettes and carbon monoxide.

Example H 0 :  = 15 mg H 0 :  = 15 mg H 1 :  < 15 mg  =  = p-value = P(t < ) = tcdf(-E99, , 24) = p-value = P(t < ) = tcdf(-E99, , 24) = Reject H 0. Reject H 0. The carbon-monoxide content of cigarettes today is less than it was in the past. The carbon-monoxide content of cigarettes today is less than it was in the past.

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

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

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

Example Re-do Example 10.1, p. 616, on the TI-83 under the assumption that  is unknown. Re-do Example 10.1, p. 616, on the TI-83 under the assumption that  is unknown.

Practice Use the TI-83 to find the following probabilities for the t distribution. Use the TI-83 to find the following probabilities for the t distribution. P(T > 2.5), df = 14. P(T > 2.5), df = 14. P(-3 < T < 3), df = 7. P(-3 < T < 3), df = 7. P(T < 3.5), df = 5. P(T < 3.5), df = 5. Use the TI-83 to find the following t-distribution percentiles. Use the TI-83 to find the following t-distribution percentiles. 95 th percentile, df = th percentile, df = th percentile, df = th percentile, df = 21. Endpoints of the middle 80%, df = 12. Endpoints of the middle 80%, df = 12.