Wiki Page, Math 13 - Perez Sandbox : Module 25 - Wrap Up "Inference for a Difference between Population Means"

Wiki Page
2026-01-17T21:15:44-08:00

Let's Summarize

Hypothesis tests and confidence intervals for two means can answer research questions about two populations or two treatments that involve quantitative data.

In "Inference for a Difference between Population Means," we focused on studies that produced two independent samples. We gather one set of measurements from each of the two samples. This is in contrast to a matched pairs design where we gather two sets of measurements from one sample or there is a natural way to pair individual data points in one sample with the individual data points in the other sample.

The hypotheses for two population means do not involve a variable value; rather, we test a statement about the relationship between the population means, e.g. is one mean greater than the other?

The null hypothesis, , is a statement of “no effect” or “no difference.”

: , which is the same as :

The alternative hypothesis, , takes one of the following three forms.

: , which is the same as :

: , which is the same as :

: , which is the same as :

As usual, how we collect the data determines whether we can use it in the inference procedure. We have our usual two requirements for data collection.

Samples must be random in order to remove or minimize bias.

Sample must be representative of the population in question.

We use the two-sample hypothesis test and confidence interval when the following conditions are met:

The two random samples are independent.

The variable is normally distributed in both populations. If this is not known, samples of more than 30 will have a difference in sample means that can be modeled adequately by the T-distribution.

As we discussed in "Hypothesis Test for a Population Mean," T-procedures are robust even when the variable is not normally distributed in the population. Therefore, if checking normality in the populations is impossible, then we look at the distribution in the samples. If a histogram or dotplot or boxplot of the data in each sample does not show extreme skew or outliers, we take it as a sign that the variable is not heavily skewed in the populations, and we use the inference procedure.

We use StatCrunch to conduct the hypothesis test or find the confidence interval for the difference in two population means.

For those who like formulas, we shared the following formulas but we did not practice using them. We will use StatCrunch for these tedious and complex calculations. That said, the formulas themselves are not surprising and follow the pattern we have already noted previously.

Formulas:

The confidence interval for is sample statistic ± margin of error. The sample statistic is the difference in sample means. The margin of error is based on a number of standard errors.

Hypothesis test for : is sample error divided by standard error.