- Check the assumptions for a two-sample [latex]t[/latex] confidence interval for population mean.
- Calculate and explain a confidence interval for the difference between two population means.
Assumptions and Conditions for Confidence Interval for Difference in Population Means
When you are using a confidence interval to estimate a population parameter, there are a few assumptions and conditions that you should review before proceeding. When you are interested in estimating a difference in population means using data from independent samples, you will use a two-sample [latex]t[/latex] confidence interval.
conditions for confidence interval for difference in population means
The conditions that you need to check for the two-sample [latex]t[/latex] confidence interval are:
- The samples are independent.
- Each sample is a random sample from the corresponding population of interest, or it is reasonable to regard the sample as if it were a random sample. It is reasonable to regard the sample as a random sample if it was selected in a way that should result in the sample being representative of the population. If the data are from an experiment, you just need to check that there was random assignment to experimental groups—this substitutes for the random sample condition and also results in independent samples.
- For each population, the distribution of the variable that was measured is approximately normal, or the sample size for the sample from that population is large. Usually, a sample of size [latex]30[/latex] or more is considered “large.” If a sample size is less than [latex]30[/latex], you should look at a plot of the data from that sample (a dotplot, a boxplot, or, if the sample size isn’t really small, a histogram) to make sure that the distribution looks approximately symmetric and that there are no outliers.
Notice the last two conditions are the same as those for the one-sample [latex]t[/latex] confidence interval. You just have to remember to check them for each of the two samples and to make sure that you have independent samples.
A simplified estimate of the degrees of freedom for an independent two sample [latex]t[/latex]-interval is [latex]df=n_1+n_2-2[/latex]. The statistical tool uses a more complex process to determine the degrees of freedom and [latex]t[/latex]-critical value. To view the [latex]t[/latex]-critical value in the statistical tool, you can toggle the “Show t-score for Margin of Error” option on in the left-hand menu. This will reveal both the degrees of freedom and the [latex]t[/latex]-critical used in calculating the margin of error.