Chapter 12: Comparing Independent Means

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# Chapter 12: Comparing Independent Means - PowerPoint PPT Presentation

Chapter 12: Comparing Independent Means. 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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### 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

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.
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

1 = single sample

2 = paired samples

3 = independent samples

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. 1525 – 1569)

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?

Illustrative DataCholesterol levels (mg / dL)

Type A men in the sample have higher average cholesterol by 35 mg/dL

Standard Error

To address this question, calculate the standard error of the mean difference:

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

(1 – α)100% CI for µ1–µ2

Note:

(point estimate) ± (t)(SE)

margin of error

Interpretation

The CI interval aims for µ1− µ2 with (1– α)100% confidence

HypothesisTest
• 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”) ORHa: μ1 – μ2 < 0 (“left-sided”)
Test Statistic

dfWelch= 35.4 (via SPSS)

dfconserv. = 19

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
SPSS

Response variable (chol) in one column

Explanatory variable (group) in a different column

Summary of independent t test
• H0: μ1 –μ2 = 0

C. P-value from Table C or computer(Interpret in usual fashion)

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

Hypothesis Test with the CI
• H0: μ1 – μ2 = 0
• 99% CI for μ1 – μ2 is (-2.2 to 71.7), which includes μ1 – μ2 = 0

 Not Significant at α = .01