Preparing for the next class

In the next in-class activity, you will need to be able to calculate the mean of a  difference and identify the differences between independent and dependent samples.

Dependent Samples vs. Independent Samples

Previously, you learned how to create confidence intervals and conduct hypothesis tests  with a single variable. You also learned how to compare means or proportions from two  samples. Some statistical studies use samples from more than one population. In order  to compare the difference between two populations, it is important to identify if the  samples are dependent (paired) or independent. Dependent and independent sample hypothesis tests are used to answer questions about the difference between two population means.

For dependent (paired) samples, the same variable is recorded for each sample, and  there is a logical way to pair the observations from one sample with the observations in  the other sample. In contrast, when samples are independently selected, the same  variable is measured for both samples, but there is no logical way to pair an observation  from one sample with a particular observation from the other sample.

For an example of paired samples, consider an investigation on the effectiveness of  hypnosis in reducing pain. The variable could be the pain level of a patient, and it could  be measured “before” hypnosis and then again “after” hypnosis for the same patient.  This would result in two samples, one “before” pain measurement and one “after” pain  measurement, and there would be a logical pairing of the “before” measurement with  the “after” measurement for the same person. This form of pairing, often referred to as  “pre/post,” is not the only situation where paired samples can be used. Other cases  involve using “natural pairs,” such as twins, siblings, or couples. In either case, it is not  reasonable that the measurement from one sample is not related to the measurement in  the second sample.

Questions 1–7: Use the previous information to determine if the following situations  would result in dependent or independent samples.

Comparing Means from Two Dependent (Paired) Samples

We will use the individual differences, [latex]d[/latex], between each pair as our sample. A dependent or paired t-test compares the mean of the differences, [latex]\mu_{d}[/latex], to a  hypothesized value, which is often 0. Thus, a dependent t-test is the same as a one sample t-test performed on the difference variable, [latex]d[/latex].

When thinking about the difference variable, we need to use a different calculation for the standard deviation of the estimate. The standard deviation of the difference in the  sample means, [latex]\bar{x}_{1}-\bar{x}_{2}[/latex] is NOT the same as the standard deviation of the difference variable, denoted using [latex]s_{d}[/latex].

Since a dependent t-test is the same as a one-sample t-test on the mean of the difference variable, the assumptions for a paired t-test are the same as those discussed  in In-Class Activity 13.B for a single sample hypothesis test for means.

Conditions for a One-Sample t-Test

1. The sample is a random sample from the population of interest or it is  reasonable to regard the sample as random. It is reasonable to regard the  sample as a random sample if it was selected in a way that should result in a  sample that is representative of the population.
2. 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 30 or more is considered to be  “large.” If a sample size is less than 30, 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.

In summary, where [latex]k[/latex] is the value of the null hypothesis, we have:

The notations for the summary statistics used to compare paired populations/samples  are shown in the following table. We will use[latex]d[/latex] to represent the difference variable.