Chapter 11: Inference About a Mean

1. Chapter 11: Inference About a Mean

2. In Chapter 11: 11.1 Estimated Standard Error of the Mean 11.2 Student’s t Distribution 11.3 One-Sample t Test 11.4 Confidence Interval for μ 11.5 Paired Samples 11.6 Conditions for Inference 11.7 Sample Size and Power

3. §11.1 Estimated Standard Error of the Mean • We rarely know population standard deviation σ instead, we calculate sample standard deviations sand use this as an estimate of σ • We then use s to calculate this estimated standard error of the mean: • Using s instead of σ adds a source of uncertainty z procedures no longer apply  use t procedures instead

4. §11.2 Student’s t distributions • A family of distributions identified by “Student” (William Sealy Gosset) in 1908 • t family members are identified by their degrees of freedom, df. • t distributions are similar to z distributions but with broader tails • As df increases → t tails get skinnier → t become more like z

6. t table (Table C) Use Table C to look up t values and probabilities Table C: Entries  t values Rows  df Columns  probabilities

7. Left tail: Pr(T9 < -1.383) = 0.10 Right tail: Pr(T9 > 1.383) = 0.10 Understanding Table C Let tdf,p ≡ a t value with df degrees of freedom and cumulative probability p. For example, t9,.90 = 1.383

8. A. Hypotheses. H0: µ = µ0 vs. Ha: µ ≠ µ0 (two-sided) [Ha: µ < µ0 (left-sided) or Ha: µ > µ0 (right-sided)] B. Test statistic. C. P-value. Convert tstat to P-value [table C or software]. Small P strong evidence against H0 D. Significance level (optional). See Ch 9 for guidelines. §11.3 One-Sample t Test

9. One-Sample t Test: Example Statement of the problem: • Do SIDS babies have lower than average birth weights? • We know from prior research that the mean birth weight of the non-SIDs babies in this population is 3300 grams • We study n = 10 SIDS babies, determine their birth weights, and calculate x-bar = 2890.5 and s = 720. • Do these data provide significant evidence that SIDs babies have different birth weights than the rest of the population?

10. One-Sample t Test: Example • H0: µ = 3300 versus Ha: µ ≠ 3300 (two-sided) B. Test statistic C. P = 0.1054 [next slide]Weak evidence against H0 D. (optional) Data are not significant at α = .10

11. |tstat| = 1.80 Converting the tstat to a P-value tstat P-value via Table C. Wedge |tstat| between critical value landmarks on Table C. One-tailed 0.05 < P < 0.10 and two-tailed 0.10 < P < 0.20. tstat P-value via software. Use a software utility to determine that a t of −1.80 with 9 df has two-tails of 0.1054.

12. Two-sided P-value associated with a t statistic of -1.80 and 9 df

13. §11.4 Confidence Interval for µ • Typical point “estimate ± margin of error” formula • tn-1,1-α/2 is from t table (see bottom row for conf. level) • Similar to z procedure except uses s instead of σ • Similar to z procedure except uses tinstead of z • Alternative formula:

14. Confidence Interval: Example 1 Let us calculate a 95% confidence interval for μ for the birth weight of SIDS babies.

15. Confidence Interval: Example 2 Data are “% of ideal body weight” in 18 diabetics: {107, 119, 99, 114, 120, 104, 88, 114, 124, 116, 101, 121, 152, 100, 125, 114, 95, 117}. Based on these data we calculate a 95% CI for μ.

16. §11.5 Paired Samples • Paired samples: Each point in one sample is matched to a unique point in the other sample • Pairs be achieved via sequential samples within individuals (e.g., pre-test/post-test), cross-over trials, and match procedures • Also called “matched-pairs” and “dependent samples”

17. Example: Paired Samples • A study addresses whether oat bran reduce LDL cholesterol with a cross-over design. • Subjects “cross-over” from a cornflake diet to an oat bran diet. • Half subjects start on CORNFLK, half on OATBRAN • Two weeks on diet 1 • Measures LDL cholesterol • Washout period • Switch diet • Two weeks on diet 2 • Measures LDL cholesterol

18. Example, Data Subject CORNFLK OATBRAN ---- ------- ------- 1 4.61 3.84 2 6.42 5.57 3 5.40 5.85 4 4.54 4.80 5 3.98 3.68 6 3.82 2.96 7 5.01 4.41 8 4.34 3.72 9 3.80 3.49 10 4.56 3.84 11 5.35 5.26 12 3.89 3.73

19. Calculate Difference Variable “DELTA” • Step 1 is to create difference variable “DELTA” • Let DELTA = CORNFLK - OATBRAN • Order of subtraction does not materially effect results (but but does change sign of differences) • Here are the first three observations: ID CORNFLK OATBRAN DELTA ---- ------- ------- ----- 1 4.61 3.84 0.77 2 6.42 5.57 0.85 3 5.40 5.85 -0.45 ↓ ↓↓↓ Positive values represent lower LDL on oatbran

20. Explore DELTAValues Here are all the twelve paired differences (DELTAs): 0.77, 0.85, −0.45, −0.26, 0.30, 0.86, 0.60, 0.62, 0.31, 0.72, 0.09, 0.16 Stemplot |-0|42|+0|0133|+0|667788×1 EDA shows a slight negative skew, a median of about 0.45, with results varying from −0.4 to 0.8.

21. Descriptive stats for DELTA Data (DELTAs): 0.77, 0.85, −0.45, −0.26, 0.30, 0.86, 0.60, 0.62, 0.31, 0.72, 0.09, 0.16 The subscript d will be used to denote statistics for difference variable DELTA

22. 95% Confidence Interval for µd A t procedure directed toward the DELTA variable calculates the confidence interval for the mean difference. “Oat bran” data:

23. Paired t Test • Similar to one-sample t test • μ0 is usually set to 0, representing “no mean difference”, i.e., H0: μ = 0 • Test statistic:

24. Paired t Test: Example“Oat bran” data A. Hypotheses. H0: µd = 0 vs. Ha: µd 0 B. Test statistic. C. P-value.P = 0.011 (via computer). The evidence against H0 is statistically significant. D. Significance level (optional). The evidence against H0 is significant at α = .05 but is not significant at α = .01

25. SPSS Output: Oat Bran data

26. §11.6 Conditions for Inference t procedures require these conditions: • SRS (individual observations or DELTAs) • Valid information (no information bias) • Normal population or large sample (central limit theorem)

27. The Normality Condition The Normality condition applies to the sampling distribution of the mean, not the population. Therefore, it is OK to use t procedures when: • The population is Normal • Population is not Normal but is symmetrical and n is at least 5 to 10 • The population is skewed and the n is at least 30 to 100 (depending on the extent of the skew)

28. Can a t procedures be used? • Dataset A is skewed and small: avoid t procedures • Dataset B has a mild skew and is moderate in size: use t procedures • Data set C is highly skewed and is small: avoid t procedure

29. §11.7 Sample Size and Power • Questions: • How big a sample is needed to limit the margin of error to m? • How big a sample is needed to test H0 with 1−βpower at significance level α? • What is the power of a test given certain conditions? • In this presentation, we cover only the last question

30. Power where: • α≡ (two-sided) alpha level of the test • Δ≡ “the mean difference worth detecting” (i.e., the mean under the alternative hypothesis minus the mean under the null hypothesis) • n ≡ sample size • σ≡ standard deviation in the population • Φ(z) ≡ the cumulative probability of z on a Standard Normal distribution [Table B] with .

31. Power: Illustrative Example SIDS birth weight example. Consider the SIDS illustration in which n = 10 and σ is assumed to be 720 gms. Let α = 0.05 (two-sided). What is the power of a test under these conditions to detect a mean difference of 300 gms?