Math 37 - Lecture 23

Math 37 - Lecture 23

Two Sample Comparisons, cont.

Last time: Procedures to compare two population proportions.

Works the same way for quantitative data in two independent groups.

Example To find out if increasing the amount of calcium in our diet reduces blood pressure, researchers designed a randomized comparative experiment. Twenty-one healthy men were randomly divided: 10 received a calcium supplement and 11 received a placebo. Response is decrease in systolic pressure after 12 weeks.

Group 1(calcium): 7, -4, 18, 17, -3, -5, 1, 10, 11, -2

Group 2(placebo:) -1, 12, -1, -3, 3, -5, 5, 2, -11, -1, -3

1. Numerical Summary

Sample	Sample size	Sample mean	Sample Standard Dev.
1	n₁ = 10	₁=	s₁=
2	n₂ = 11	₂=	s₂=

2. Graphical Summary

- Back to back stemplots (small samples) (p. 11)

- Side by side boxplots (larger samples) (p. 49)

3. Inference

Is this outcome good evidence that calcium decreases blood pressure in the entire population of healthy men more than a placebo does?

Parameter:

Statistic:

Null Hypothesis:

Alternative Hypothesis:

Assume Null hypothesis is true, sampling distribution of the statistic:

Shape:

(If both populations are normal)

Center:

Variability:

Test Statistic:

Degrees of freedom

Approximation: min(n₁-1, n₂-1) (conservative)

or Exact value: Minitab

Confidence interval for m₁-m₂:

Technical assumptions:

Independent SRSs
Normal Populations

If populations are similar, reasonable even for n₁=n₂=5

If populations have different shapes, need larger samples.

In n₁+n₂<15, use only if data close to Normal
If n₁+n₂>15, use unless strong skewness or outliers
If n₁+n₂>40 can be safely used, even if strongly skewed

Example - Who was the better homerun hitter in terms of distance of the homerun, McGwire or Sosa?

Parameter:

Null hypothesis:

Alternative hypothesis:

Numerical Summary:

	n	(ft)	s (ft)
McGwire	66	407.48	38.14
Sosa	70	422.67	47.45

Is this difference in average distances statistically significant?

Are the technical assumptions satisfied?

Example - A control group of 53 community members (who had not reported UFO experiences) had a mean IQ of 100.6 with a standard deviation of 12.3. The sample mean IQ of the 25 people in the student who claimed to have had an intense experience with a UFO was 101.6; the standard deviation of these IQ's was 8.9.

- Test whether the mean IQ's of community members and UFO sighters differ significantly.

- Also find a confidence interval for the difference in population means and explain what the interval reveals.