1. Chapter 12: Comparing Independent Means
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2. In Chapter 12: 12.1 Paired and Independent Samples
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In Chapter 12:
12.1 Paired and Independent Samples
12.2 Exploratory and Descriptive Statistics
12.3 Inference About the Mean Difference
12.4 Equal Variance t Procedure (Optional)
12.5 Conditions for Inference
12.6 Sample Size and Power
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Basic Biostat

3. Types of Samples Single sample. One group; no concurrent control group
Paired samples. Two samples; data points uniquely matched
Two independent samples. Two samples, separate (unrelated) groups.
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4. What Type of Sample?
Measure vitamin content in loaves of bread and see if the average meets national standards
Compare vitamin content of loaves immediately after baking versus content in same loaves 3 days later
Compare vitamin content of bread immediately after baking versus loaves that have been on shelf for 3 days
Answers:
1 = single sample
2 = paired samples
3 = independent samples
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5. Experimental vs. Observational Groups
Independent samples can
Experimental –an intervention or treatment is assigned as part of the study protocol
Non-experimental (observational) – groups defined by a innate characteristics or self-selected exposure
“Two Groups” by Pieter Bruegel the Elder (c – 1569)
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6. Do means from these populations differ? If so, by how much?
Illustrative Data*
* Data set WCGS.sav (p. 49)
Type A personality men (n = 20) 233, 291, 312, 250, 246, 197, 268, 224, 239, 239, 254, 276, 234, 181, 248, 252, 202, 218, 212, 325
Type B personality men (n = 20) 344, 185, 263, 246, 224, 212, 188, 250, 148, 169, 226, 175, 242, 252, 153, 183, 137, 202, 194, 213
Do means from these populations differ? If so, by how much?
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7. Notation Statistics (sample) Group 1 n1 s1 Group 2 n2 s2
Parameters (population)
Group 1 N1 µ1 σ1
Group 2 N2 µ2 σ2
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8. Illustrative Data Cholesterol levels (mg / dL)
Group n
mean
std dev 1 20
245.05
36.64 2
210.30
48.34
Type A men in the sample have higher average cholesterol by 35 mg/dL
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9. Standard Error
To address this question, calculate the standard error of the mean difference:
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10. Degrees of Freedom Two ways to estimate degrees of freedom:
dfWelch [complex formula on p. 244 of text]
dfconserv. = the smaller of (n1 – 1) or (n2 – 1)
For the illustrative data:
dfWelch = 35.4 (via SPSS)
dfWelch = 35.4 (via SPSS)
dfconserv. = smaller of (n1–1) or (n2 – 1)
= 20 – 1 = 19
dfconserv. = smaller of (n1–1) or (n2 – 1)
= 20 – 1 = 19
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11. (point estimate) ± (t)(SE)
(1 – α)100% CI for µ1–µ2
Note:
(point estimate) ± (t)(SE)
margin of error
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12. Comparison of CI Formulas
Type of sample
Point estimate
df for t* SE
Single
n – 1
Paired
Independent
smaller of n1−1 or n2−1
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13. Example
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14. Interpretation
The CI interval aims for µ1 − µ2 with (1– α)100% confidence
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15. Hypothesis Test Test claim of “no difference in populations”
Note: widely different sample means can arise just by chance
Null hypothesis: H0: μ1 – μ2 = 0 (equivalently H0: μ1 = μ2)
Alternative hypothesis Ha: μ1 – μ2 ≠ 0 (two-sided) OR Ha: μ1 – μ2 > 0 (“right-sided”) OR Ha: μ1 – μ2 < 0 (“left-sided”)
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16. Test Statistic
dfWelch= 35.4 (via SPSS)
dfconserv. = 19
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17. P-value via Table C tstat = 2.56 with 19 df
One-tailed P between .01 and .005
Two-tailed P between .02 and .01 (i.e., less than .02)
.01 < P < .02 provides good evidence against H0  observed difference is statistically significant
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18. SPSS Response variable (chol) in one column
Explanatory variable (group) in a different column
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19. SPSS Output Equal Variance Not Assumed Preferred method (§12.3)
Equal variance t procedure (§12.4)
Equal Variance Not Assumed Preferred method (§12.3)
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20. Summary of independent t test
H0: μ1 –μ2 = 0
C. P-value from Table C or computer (Interpret in usual fashion)
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21. Hypothesis Test with the CI
H0: μ1 – μ2 = 0 can be tested at α-level of significance with the (1 – α)100% CI
Example: 95% CI for μ1 – μ2 = (6.4 to 63.1)  excludes μ1 – μ2 = 0
 Significant difference at α = .05
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22. Hypothesis Test with the CI
99% CI for μ1 – μ2 is (-2.2 to 71.7), which includes μ1 – μ2 = 0
 Not Significant at α = .01
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