1. 16/23/2016Inference about µ1 Chapter 17 Inference about a Population Mean

2. 26/23/2016Inference about µ2 σ not known In practice, we do not usually know population standard deviation σ Therefore, we cannot calculate σ x-bar Instead, we calculate this standard error of the mean:

3. 36/23/2016Inference about µ3 t Procedures Because σ is now known, we do NOT use z statistics. Instead, we use this t statistic T procedures are based on Student’s t distribution

4. 46/23/2016Inference about µ4 Student’s t Distributions A “family” of distributions Each family member has different degrees of freedom (df) More area in their tails than Normal distributions (fatter tails) As df increases, s becomes a better estimate of σ and the t distributions becomes more Normal t with more than 30 df  very similar to z

5. 56/23/2016Inference about µ5 t Distributions

6. 66/23/2016 Inference about µ6 Table C “t Table” Table entries = t* critical values Rows = df; Columns = probability levels Familiarize yourself with the t table in the “Tables and Formulas for Moore” handoutTables and Formulas for Moore

7. 76/23/2016 Inference about µ 7 Using Table C Question: What t critical value should I use for 95% confidence when df = 7? Answer: t* = 2.365

8. 86/23/2016Inference about µ8 Confidence Interval for μ t* is the critical value with df = n−1 and C level of confidence Lookup in Table C

9. 9 Example Statement : What is the population mean µ birth weight of the SIDS population? Data: We take an SRS of n = 10 from the population of SIDS babies and retrieve their birth certificates. This was their birth weights (grams): 2998, 3740, 2031, 2804, 2454, 2780, 2203, 3803, 3948, 2144 Plan: We will calculate the sample mean and standard deviation. We will then calculate and interpret the 95% CI for µ.

10. 10 Example (Solution) We are 95% confident population mean µ is between 2375 and 3406 gms.

11. 116/23/2016Inference about µ11 One-Sample t Test (Hypotheses) Draw simple random sample of size n from a large population having unknown mean µ Test null hypothesis H 0 : μ = μ 0 where μ 0 ≡ stated value for the population mean –μ 0 changes from problem to problem –μ 0 is NOT based on the data –μ 0 IS based on the research question The alternative hypothesis is: –H a : μ > μ 0 (one-sided looking for a larger value) OR –H a : μ < μ 0 (one-sided looking for a smaller value) OR –H a : μ ≠ μ 0 (two-sided)

12. 126/23/2016Inference about µ12 One-Sample t Test One-sample t statistic: P-value = tail beyond t stat (use Table C)

13. 13Basics of Significance Testing13 P-value: Interpretation P-value (interpretation) Smaller-and-smaller P- values indicate stronger-and-stronger evidence against H 0 Conventions:.10 < P < 1.0  evidence against H 0 not significant.05 < P ≤.10  evidence against H 0 marginally signif..01 < P ≤.05  evidence against H 0 significant P ≤.01  evidence against H 0 highly significant

14. 146/23/2016Inference about µ14 Statement: We want to know whether there is good evidence for weight change in a particular population. We take an SRS on n = 10 from this population and find the following changes in weight (lbs). Example: “Weight Gain” 2.0, 0.4, 0.7, 2.0, −0.4, 2.2, −1.3, 1.2, 1.1, 2.3 Calculate: Do data provide significant evidence for a weight change?

15. 156/23/2016Inference about µ15 Example “Weight Gain” (Hypotheses) Under null hypothesis, no weight gain in population H 0 : μ = 0 Note: µ 0 = 0 in this particular example One-sided alternative, weight gain in population. H a : μ > 0 Two-sided alternative hypothesis, weight change: H a : μ ≠ 0

16. 166/23/2016Inference about µ16 Example (Test Statistic)

17. 176/23/2016 Inference about µ 17 Example (P-value) Table C, row for 9 df t statistic (2.70) is between t* = 2.398 (P = 0.02) and t* = 2.821 (P = 0.01) One-sided P-value is between.01 and.02:.01 < P <.02

18. 18 Two-tailed P-value For two-sided H a, P-value = 2 × one- sided P In our example, the one-tailed P-value was between.01 and.02 Thus, the two-tailed P value is between.02 and.04

19. 19 Interpretation Interpret P-value in context of claim made by H 0 In our example, H 0 : µ = 0 (no weight gain) Two-tailed P-value between.02 and.04 Conclude: significant evidence against H 0

20. 206/23/2016Inference about µ20 Paired Samples Responses in matched pairs Parameter μ now represents the population mean difference

21. 216/23/2016Inference about µ21 Example: Matched Pairs Pollution levels in two regions (A & B) on 8 successive days Do regions differ significantly? Subtract B from A = last column Analyze differences DayABA – B 12.921.841.08 21.880.950.93 35.354.261.09 43.813.180.63 54.693.441.25 64.863.691.17 75.814.950.86 85.554.471.08

22. 226/23/2016Inference about µ22 Hypotheses: H 0 : μ = 0 (note: µ 0 = 0, representing no mean difference) H a : μ > 0 (one-sided) H a : μ ≠ 0 (two-sided) Test Statistic: Example: Matched Pairs

23. 236/23/2016Inference about µ23 P-value: Table C  7 df row t statistic is greater than largest value in table: t* = 5.408 (upper p = 0.0005). Thus, one-tailed P < 0.0005 Two-tailed P = 2 × one-tailed P-value: P < 0.001 Conclude: highly significant evidence against H 0 Illustration (cont.)

24. 246/23/2016Inference about µ24 Air pollution data: n = 8, x-bar = 1.0113, s = 0.1960 df = 8  1 = 7 For 95% confidence, use t* = 2.365 (Table C) 95% Confidence Interval for µ 95% confidence population mean difference µ is between 0.847 and 1.175

25. 256/23/2016Inference about µ25  The confidence interval seeks population mean difference µ (IMPORTANT)  Recall the meaning of “confidence,” i.e., the ability of the interval to capture µ upon repetition  Recall from the prior chapter that the confidence interval can be used to address a null hypothesis Interpreting the Confidence Interval

26. 266/23/2016Inference about µ26 Normality Assumption t procedures require Normality, but they are robust when n is “large” Sample size less than 15: Use t procedures if data are symmetric, have a single peak with no outliers. If data are highly skewed, avoid t. Sample size at least 15: Use t procedures except in the presence of strong skewness. Large samples: Use t procedures even for skewed distributions when the sample is large (n ≥ ~40)

27. 276/23/2016Inference about µ27 Can we use a t procedure? Moderately sized dataset (n = 20) w/strong skew. t procedures cannot be trusted

28. 286/23/2016Inference about µ28 Word lengths in Shakespeare’s plays (n ≈ 1000) The data has a strong positive skew but since the sample is large, we can use t procedures.

29. 296/23/2016Inference about µ29 Can we use t? The distribution has no clear violations of Normality. Therefore, we trust the t procedure.