Comparing Means in Statistical Analysis: Independent and Paired Samples - Prof. Michele Gu, Study notes of Statistics

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Stat 538 Biostatistical Methods I for Public Health and Medical Sciences

Lecture 9

Michele Guindani

University of New Mexico Albuquerque 2008-

Two-Sample Inferences for Means

• In many cases, we may be interested in comparing observations coming from two separate samples (e.g., male/female; two different ethnic groups; people from two different States...)

o How do we compare the two samples?

Paired Versus Independent Samples.

We consider two populations, say populations 1 and 2, and we would like to compare their (unknown) population means, μ 1 and μ 2. Of course, inferences on the unknown population means are based on samples from each population.

• Independent samples, where the sample taken from population 1 are not related to the observations selected from population 2.
• Paired or dependent samples, where experimental units are paired based on factors related or unrelated to the variable measured.

Two Independent Samples: CI and Test with equal

variances

• Assumption The populations have normal frequency curves, with equal population s.d., i.e. σ 1 = σ 2. o Let (n 1 , Y 1 , s 1 ) and (n 2 , Y 2 , s 2 ) be the sample sizes, means and standard deviations from the two samples. o To build CIs and do hipothesis testing we need to determine tcrit and ts.

Two Independent Samples: CI and Test with equal

variances

• Assumption The populations have normal frequency curves, with equal population s.d., i.e. σ 1 = σ 2. o Let (n 1 , Y 1 , s 1 ) and (n 2 , Y 2 , s 2 ) be the sample sizes, means and standard deviations from the two samples. o To build CIs and do hipothesis testing we need to determine tcrit and ts. o The critical value tcrit for CI and tests is obtained in usual way from a t-table with df = n 1 + n 2 − 2.

Pooled variance estimator

• Here the pooled variance estimator,

s^2 pooled =

(n 1 − 1)s^21 + (n 2 − 1)s^22 n 1 + n 2 − 2

is our best estimate of the common population variance.

• The pooled estimator of variance is a weighted average of the two sample variances, with more weight given to the larger sample. If n 1 = n 2 then s^2 pooled is the average of s^21 and s^22.

Two Independent Samples: CI and Test with unequal

variances

o In case the populations are still normal, but they presumably have different standard deviations, we use the Satterthwaite’s Method.