Student’s t Distribution Lecture 35 Section 10.2 Tue, Apr 3, 2007
The t Distribution The shape of the t distribution is very similar to the shape of the standard normal distribution. It is symmetric unimodal centered at 0. But it is wider than the standard normal. 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. 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.
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 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).
Standard Normal vs. t Distribution 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). t(2)
Standard Normal vs. t Distribution 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). t(2) t(30) N(0, 1)
The Decision Tree Is known? yes no Is the population normal? TBA yes Is n 30? yes no Give up
The Decision Tree Is known? yes no Is the population normal? Is n 30? yes no Give up
The Decision Tree Is known? yes no Is the population normal? Is n 30? Is n 30? yes no yes no Give up
The Decision Tree Is known? yes no Is the population normal? Is n 30? Is n 30? yes no yes no Give up
The Decision Tree Is known? yes no Is the population normal? Is n 30? Is n 30? yes no yes no Give up
The Decision Tree Is known? yes no Is the population normal? Is n 30? Is n 30? Is n 30? yes no yes no yes no Give up
The Decision Tree Is known? yes no Is the population normal? Is n 30? Is n 30? Is n 30? yes no yes no yes no Give up
The Decision Tree Is known? yes no Is the population normal? Is n 30? Is n 30? Is n 30? yes no yes no yes no Give up Give up
When to Use Z-Test Is known? yes no Is the population normal? Is n 30? Is n 30? Is n 30? yes no yes no yes no Give up Give up
When to Use T-Test Is known? yes no Is the population normal? Is n 30? Is n 30? Is n 30? yes no yes no yes no Give up Give up
When to Use Either Z-Test or T-Test Is known? yes no Is the population normal? Is the population normal? yes no yes no Is n 30? Is n 30? Is n 30? yes no yes no yes no Give up Give up
When to Give Up Is known? yes no Is the population normal? Is n 30? Is n 30? Is n 30? yes no yes no yes no Give up Give up
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 What is the 95th percentile of the t distribution when df = 10? The table tells us that P(t > 1.812) = 0.05, when df = 10. Thus, the 95th percentile is 1.812.
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. So, what is the 5th percentile of t when df = 10? What is the 90th 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. 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). 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.
Examples Enter tcdf(1.812, E99, 10). Enter tcdf(-E99, -1.812, 10). The result is 0.0500. Enter tcdf(-E99, -1.812, 10). Thus, P(t > 1.812) = 0.05 when there are 10 degrees of freedom (n = 11). Find P(t > 2.4) when df = 12.
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. Either use Z or “give up.”
Hypothesis Testing with t Continue with Example 10.1, p. 616. Step 3: Find the value of the test statistic. The test statistic is now Step 4: Compute t fromx, , s, and n.
Example Step 5: Find the p-value. We must look it up in the t table, or use tcdf on the TI-83. Step 6: Make the decision regarding H0 and H1. Step 7: State the conclusion about cigarettes and carbon monoxide.
Example H0: = 15 mg H1: < 15 mg = 0.05. p-value = P(t < -2.608) = tcdf(-E99, -2.608, 24) = 0.007711. Reject H0. The carbon-monoxide content of cigarettes today is less than it was in the past.
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. Enterx. 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, 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. P(T > 2.5), df = 14. P(-3 < T < 3), df = 7. P(T < 3.5), df = 5. Use the TI-83 to find the following t-distribution percentiles. 95th percentile, df = 25. 20th percentile, df = 21. Endpoints of the middle 80%, df = 12.