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Chapter 17

Chapter 17. Inference about a Population Mean. 1. 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:. 2. t Procedures.

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Chapter 17

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  1. Chapter 17 Inference about a Population Mean Inference about µ 1

  2. σ not known In practice, we do not usually know population standard deviation σ Therefore, we cannot calculate σx-bar Instead, we calculate this standard error of the mean: Inference about µ 2

  3. 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 Inference about µ 3

  4. 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 Inference about µ 4

  5. t Distributions Inference about µ 5

  6. 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” handout Inference about µ 6

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

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

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

  11. One-Sample t Test (Hypotheses) • Draw simple random sample of size n from a large population having unknown mean µ • Test null hypothesis H0: μ = μ0where μ0≡ stated value for the population mean • μ0 changes from problem to problem • μ0 is NOT based on the data • μ0 IS based on the research question • The alternative hypothesis is: • Ha: μ > μ0 (one-sided looking for a larger value) OR • Ha: μ < μ0 (one-sided looking for a smaller value) OR • Ha: μ ≠ μ0 (two-sided) Inference about µ 11

  12. One-Sample t Test One-sample t statistic: P-value = tail beyond tstat(use Table C) 10/26/2014 Inference about µ 12

  13. P-value: Interpretation P-value (interpretation) Smaller-and-smaller P-values indicate stronger-and-stronger evidence against H0 Conventions: .10 < P < 1.0  evidence against H0 not significant .05 < P ≤ .10  evidence against H0 marginally signif. .01 < P ≤ .05  evidence against H0 significant P ≤ .01  evidence against H0 highly significant Basics of Significance Testing 13

  14. Example: “Weight Gain” 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). 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? Inference about µ 14

  15. Example “Weight Gain” (Hypotheses) • Under null hypothesis, no weight gain in populationH0: μ = 0 Note: µ0 = 0 in this particular example • One-sided alternative, weight gain in population. Ha: μ > 0 • Two-sided alternative hypothesis, weight change:Ha: μ ≠ 0 10/26/2014 Inference about µ 15

  16. Example (Test Statistic) Inference about µ 16

  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 10/26/2014 17 Inference about µ

  18. Two-tailed P-value • For two-sided Ha, 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. Interpretation • Interpret P-value in context of claim made by H0 • In our example, H0: µ = 0 (no weight gain) • Two-tailed P-value between .02 and .04 • Conclude: significant evidence against H0

  20. Paired Samples Responses in matched pairs Parameter μnow represents the population mean difference Inference about µ 20

  21. 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 Inference about µ 21

  22. Example: Matched Pairs Hypotheses: H0: μ = 0 (note: µ0 = 0, representing no mean difference) Ha: μ> 0 (one-sided) Ha: μ ≠ 0 (two-sided) Test Statistic: Inference about µ 22

  23. Illustration (cont.) 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 H0 Inference about µ 23

  24. 95% Confidence Interval for µ 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 population mean difference µ is between 0.847 and 1.175 Inference about µ 24

  25. Interpreting the Confidence Interval • 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 Inference about µ 25

  26. 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) Inference about µ 26

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

  28. 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. Inference about µ 28

  29. Can we use t? The distribution has no clear violations of Normality. Therefore, we trust the t procedure. Inference about µ 29

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