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Student’s t Distribution

Student’s t Distribution. Lecture 45 Section 10.2 Tue, Apr 13, 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 .

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Student’s t Distribution

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  1. Student’s t Distribution Lecture 45 Section 10.2 Tue, Apr 13, 2004

  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. x is N(, /n). • The trouble is,x is not N(, s/n) unless the sample size is large enough, say n  30.

  4. What if  is Unknown? • Ifx is not N(, s/n), then (x – )/(s/n) is not N(0, 1). • If it is not N(0, 1) , then what is it? • It has a distribution called Student’s t distribution. t = (x – )/(s/n).

  5. 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!!!

  6. The t Distribution • The shape of the t distribution is very similar to the shape of the standard normal distribution. • 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.

  7. 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.

  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.

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

  10. Table IV – t Percentiles • Table IV gives certain percentiles of t for certain degrees of freedom. • 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.

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

  12. The t Distribution on the TI-83 • The TI-83 will find probabilities for the t distribution (but not percentiles). • Press DISTR. • Select tcdf and press ENTER. • tcdf( appears in the display. • Enter the lower endpoint. • Enter the upper endpoint.

  13. The t Distribution on the TI-83 • Enter the number of degrees of freedom. • Press ENTER. • The result is the probability.

  14. 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).

  15. Hypothesis Testing with t • We should use t 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.

  16. 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 t = (x – 0)/(s/n). • Step 4: Find the p-value. • We must look it up in the t table, or use tcdf on the TI-83.

  17. Example • Re-do Example 10.1 under the assumption that  is unknown.

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

  19. Hypothesis Testing on the TI-83 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.

  20. Hypothesis Testing on the TI-83 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.

  21. Hypothesis Testing on the TI-83 When  is Unknown • The sample standard deviation. • The sample size.

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

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

  24. Assignment • Page 585: Exercises 7, 8, 9*, 10*, 11*, 13*, 15*, 16*. • Page 606: Exercises 34*, 35*, 38*, 40, 41*. * Show of the steps of the hypothesis test.

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