Chapter 7: Inference for Distributions

Section 7.2: Comparing Two Means

Two-sample problems

Two-sample problems:

• compare the responses to two treatments or to compare the characteristics of two populations
• Have a separate sample from each treatment or each population

Comparing two population means

Assumptions for comparing two means:

• We have two SRSs from two distinct populations. The samples are independent and NOT matched.
• Both populations are normally distributed. Use stemplots or histograms to check.
• μ and σ are unknown.

Let x₁ be the variable we measure from the first population and x₂ be the variable we measure from the first population:

Population	Variable	Mean	s.d.
1	x1	μ1	σ1
2	x2	μ2	σ2

We use sample means and s.d. to estimate the parameters:

Population	Sample Size	Sample Mean	Sample s.d.
1	n1	1	s1
2	n2	2	s2

Two–sample t procedures

Suppose we want to infer something about the difference in population means (μ₁ – μ₂) from the difference in sample means (₁ – ₂).

The sampling distribution of ₁ – ₂ has:

• mean of μ₁ – μ₂
• s.d. approximated by

The two–sample t–statistic is: .

If the two population distributions are both normal, then the two–sample t–statistic is normally distributed [i.e., N(0, 1)].

Unfortunately, the two-sample t–statistic does not have a t distribution. Even so, the two–sample t–statistic is used with the critical values from the t distribution. The book describes two options to do this (see pp. 394–395), but we will rely on the TI83 to calculate the p–value.

First, a level C CI for μ is x-bar ± t^*×SE. The TI83 can calculate the two–sample t CI!

t–test for a population mean:

The null hypothes1s can be stated in two ways:

1. H₀: μ₁ = μ₂
2. H₀: μ₁ – μ₂ = 0

1. State the hypotheses:
   • one-tailed tests:
     1. 
     2. 
   • two-tailed test:
     
2. Calculate the P–value
3. Make your conclusion

Recall: H₀ and H_a always refer to the population and NOT to a particular outcome. It is often easier (and more appropriate) to state H₀ and H_a before looking at the data.

Robustness again

A statistical inference procedure is called robust if the probability calculations do not change very much when the assumptions of the procedure are violated.

• two–sample t procedures are are more robust than one–sample t procedures when the populations are not symmetric
• two–sample t procedures are quite robust when the sample sizes are equal and when the populations have the same shape
• when the populations have different shapes, larger sample sizes are needed
• n₁ + n₂ < 15: use t procedures if the data are close to normal
• 15 < n₁ + n₂ < 40: t procedures can be used except in the presence of outliers or strong skewness
• n₁ + n₂ > 40: The t procedures can be used even for clearly skewed distribution