Chapter 11 Inference for Distributions

1. Chapter 11Inference for Distributions AP Statistics 11.1 – Inference for the Mean of a Population

2. Really just σ is unknown • Often the case in practice • When σ is known • When σ is unknown • Whoa, way different!

3. Standard Error Becomes This is just Standard error of the Sampling mean

4. t-distributions • Cousins of the z-distribution (Normal) • Conditions for inference about a mean • Random? – to generalize about the population • Normal? – Verify if the sampling distribution about the mean is approximately normal. • N>=10n? - Independent? • t(k) distribution where k = n – 1 degrees of freedom • S has n-1 degrees of freedom

5. t(k) distributions • Similar to Normal curve; symmetric, single peaked, bell shaped • Spread of t-dist. is greater than z-dist. • As degrees of freedom increase, the t(k) density curve approaches the normal curve more closely. • (s estimates more accurately as n increases) • t* uses upper tail probabilities (look at table) • Y1=normalpdf(x) • Y2=tpdf(x,df)

6. Using the t* table • What critical value t* would you use for a t distribution with 18 degrees of freedom having probability 0.9 to the left of t*?

7. Using the t* table • What t* value would you use to construct a 95% confidence interval with mean and an SRS of n = 12?

8. Using the t* table • What t* value would you use to construct a 80% confidence interval with mean and an SRS of n = 56?

9. t-CI’s & t-tests • 1-sample t-interval VS. 1-sample t-test

12. Normal Probability Plot

14. T(9) distribution

15. Data Software Packages

16. Matched Pairs t-procedures • Comparative Studies are more convincing than single-sample investigations • To compare the responses of the two treatments in a matched pairs design, apply the 1-sample t-procedures to the Observed DIFFERENCES!

19. Statistical Software Packages (con’t)

23. Robustness of t-procedures • A CI or Significance Test is called robust if the confidence level or P-value does not change very much when assumptions of the procedure or violated. • Outliers? – Like and s, the t-procedures are strongly influenced by outliers.

24. Quite Robust when No Outliers Sample size increases  CLT  more robust!

25. Using the t-procedure • SRS is more important than normal (except in the case of small samples) • n < 15, use t-procedures if the data are close to normal • n ≥ 15, use t-procedures except in presence of outliers or strong skewness • Large samples (roughly n ≥ 40), t-procedures can be used even for clearly skewed distributions • p. 636-637 - histograms

26. The power of the t-test • Power measures ability to detect deviations from the null hypothesis Ho • Higher power of a test is important! • Usually assume a fixed level of significance, α= 0.05

28. Here we go again. . . Power! • Director hopes that n=20 teachers will detect an average improvement of 2 pts in the mean listening score. Is this realistic? • Hypotheses? • Test against the alternative =2 when n=20. • Impt: Must have a rough guess of the size of to compute power! = 3 (from past samples)