Chapter 11 Inference About a Mean 5192021 1
Chapter 11: Inference About a Mean 5/19/2021 1
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 5/19/2021 2
σ not known • Prior chapter: σ was known before collecting data z procedures used to help infer µ • When σ NOT known, calculate sample standard deviations s and use it to calculate this standard error: 5/19/2021 3
Additional Uncertainty • Using s instead of σ adds uncertainty to inferences can NOT use z procedures • Instead, rely on Student’s t procedures 5/19/2021 The Normal distribution doesn’t fit well William Sealy Gosset (1876– 1937) 4
Student’s t distributions • Family of probability distributions • Family members identified by degrees of freedom (df) • Similar to “Z”, but with broader tails • As df increases → tails get skinnier → t become like z 5/19/2021 A t distribution with infinite degrees of freedom is a Standard Normal Z distribution 5
Table C (t table) Rows df Columns probabilities Entries t values Notation: tcum_prob, df = t value Example: t. 975, 9 = 2. 262 5/19/2021 6
One-Sample t Test Objective: test a claim about population mean µ Conditions : • Simple Random Sample • Normal population or “large sample” 5/19/2021 7
Hypothesis Statements • Null hypothesis H 0: µ = µ 0 where µ 0 represents the pop. mean expected by the null hypothesis • Alternative hypotheses Ha: µ < µ 0 (one-sided, left) Ha: µ > µ 0 (one-sided, right) Ha: µ ≠ µ 0 (two-sided) 5/19/2021 8
Example • Do SIDS babies have lower average birth weights than a general population mean µ of 3300 gms? • H 0: µ = 3300 • Ha: µ < 3300 (onesided) or Ha: µ ≠ 3300 (twosided) 5/19/2021 9
One-Sample t Test Statistic where This t statistic has n – 1 degrees of freedom 5/19/2021 10
2998 3740 2031 2804 2454 2780 2203 3803 3948 2144 5/19/2021 Example (Data) SRS n = 10 birth weights (grams) of SIDS cases 11
Example Testing H 0: µ = 3300 5/19/2021 12
P-value via Table C • Bracket |tstat| between t critical values • For |tstat| = 1. 80 with 9 df Table C. Upper-tail P 0. 25 df = 9 0. 703 |tstat| = 1. 80 0. 20 0. 15 0. 10 0. 05 0. 025 0. 883 1. 100 1. 383 1. 833 2. 262 Thus One-tailed: 0. 05 < P < 0. 10 Two-tailed: 0. 10 < P < 0. 20 5/19/2021 13
For a more precise P-value use a computer utility Here’s output from the free utility Sta. Table Graphically: 5/19/2021 14
Interpretation • Testing H 0: µ = 3300 gms • Two-tailed P >. 10 • Conclude: weak evidence against H 0 • The sample mean (2890. 5) is NOT significantly different from 3300 5/19/2021 15
(1− α)100% CI for µ where 5/19/2021 16
Same Data Interpretation: Population mean µ is between 2375 and 3406 grams with 95% confidence 5/19/2021 17
§ 11. 5 Paired Samples • Two samples • Each data point in one sample uniquely matched to a data point in the other sample • Examples of paired samples – “Pre-test/post-test” – Cross-over trials – Pair-matching 5/19/2021 18
Example • • Does oat bran reduce LDL cholesterol? Start half of subjects on CORNFLK diet Start other half on OATBRAN Two weeks LDL cholesterol Washout period Cross-over to other diet Two weeks LDL cholesterol 5/19/2021 19
Oat bran data LDL cholesterol mmol 5/19/2021 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 20
Within-pair difference “DELTA” • Let DELTA = CORNFLK - OATBRAN • First three observations in OATBRAN data: ID CORNFLK OATBRAN -------1 4. 61 3. 84 2 6. 42 5. 57 3 5. 40 5. 85 DELTA ----0. 77 0. 85 -0. 45 etc. All procedures are now directed toward difference variable DELTA 5/19/2021 21
Exploratory and descriptive stats DELTA: 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 f|4 |-0*|2 |+0*|01 |+0 t|33 |+0 f| |+0 s|6677 |+0. |88 × 1 LDL (mmol) 5/19/2021 subscript d denotes “difference” 22
95% CI for µd 95% confident population mean difference µd is between 0. 105 and 0. 656 mmol/L 5/19/2021 23
Hypothesis Test • Claim: oat bran diet is associated with a decline (one-sided) or change (two-sided) in LDL cholesterol. • Test H 0: µd = µ 0 where µ 0 = 0 Ha: µd > µ 0 (one-sided) Ha: µ ≠ µ 0 (two-sided) 5/19/2021 24
Paired t statistic 5/19/2021 25
P-value via Table C |tstat| = 3. 043 Table C. Upper-tail P df = 11 . 01. 005. 0025 2. 718 3. 106 3. 497 Thus One-tailed: . 005 < P <. 01 Two-tailed: . 01 < P <. 02 5/19/2021 26
P-value via Computer 5/19/2021 27
SPSS Output: “Oat Bran” 5/19/2021 28
Interpretation • Testing H 0: µ = 0 • Two-tailed P = 0. 011 • Good reason to doubt H 0 • (Optional) The difference is “significant” at α =. 05 but not at α =. 01 5/19/2021 My P value is smaller than yours! 29
The Normality Condition • t Procedures require Normal population or large samples • How do we assess this condition? • Guidelines. Use t procedures when: – Population Normal – population symmetrical and n ≥ 10 – population skewed and n ≥ ~45 (depends on severity of skew) 5/19/2021 30
Can a t procedures be used? Skewed small sample avoid t procedures 5/19/2021 31
Can a t procedures be used? Mild skew in moderate sample t OK 5/19/2021 32
Can a t procedures be used? Skewed moderate sample avoid t 5/19/2021 33
Sample Size and Power Methods: (1) n required to achieve m when estimating µ (2) n required to test H 0 with 1−β power (3) Power of a given test of H 0 5/19/2021 34
Power • • • α ≡ alpha (two-sided) Δ ≡ “difference worth detecting” = µa – µ 0 n ≡ sample size σ ≡ standard deviation Φ(z) ≡ cumulative probability of Standard Normal z score 5/19/2021 35
Power: SIDS Example • Let α =. 05 and z 1 -. 05/2 = 1. 96 • Test: H 0: μ = 3300 vs. Ha: μ = 3000. Thus: Δ ≡ µ 1 – µ 0 = 3300 – 3000 = 300 • n = 10 and σ ≡ 720 (see prior SIDS example) Use Table B to look up cum prob Φ(-0. 64) =. 2611 5/19/2021 36
Power: Illustrative Example 5/19/2021 37
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