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Student’s t Distribution

Student’s t Distribution. Lecture 36 Section 10.2 Fri, Nov 11, 2005. 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 .

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Student’s t Distribution

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  1. Student’s t Distribution Lecture 36 Section 10.2 Fri, Nov 11, 2005

  2. 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 .

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

  4. What if  is Unknown? • If it is notN(0, 1) , then what is it?

  5. Student’s t Distribution • It has a distribution called Student’s t distribution. • The t distribution was discovered by W. S. Gosset in 1908. • See http://mathworld.wolfram.com/Studentst-Distribution.html

  6. 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 .

  7. 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.

  8. 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).

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

  10. Is  known? yes no Is the population normal? TBA yes no Is n 30? yes no Give up Decision Tree

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

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

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

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

  15. 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 Decision Tree

  16. 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 Decision Tree

  17. 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 Decision Tree

  18. 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 Decision Tree

  19. 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 Decision Tree

  20. 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.

  21. 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?

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

  23. 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.

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

  25. Example • Enter tcdf(1.812, E99, 10). • The result is 0.0500. • Enter tcdf(-E99, -1.812, 10). • The result is 0.0500. • Thus, P(t > 1.812) = 0.05 when there are 10 degrees of freedom (n = 11).

  26. 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.”

  27. 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.

  28. Example • Re-do Example 10.1, p. 616, (by hand) under the assumption that  is unknown.

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

  30. 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.

  31. 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.

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

  33. Let’s Do It! • Let’s Do It! 10.3, p. 630 – Study Time. • Let’s Do It! 10.4, p. 631 – pH Levels.

  34. Homework • 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.

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