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 measure in the second sample.
Questions 1-7: Use the previous information to determine if the following situations would result in dependent or independent samples.
1) A company that creates fishing accessories is researching two of their most popular fishing rods. The company collects a random sample of the number of sales for each fishing rod from 100 of their stores.
2) The N.C. Zoo is researching whether their animals are more active in the morning or in the evening. An employee at the zoo visits each habitat in the zoo and collects information for the study. The employee counts how many of each species is visible in the morning and then visits a second time to count how many of each species is visible during the evening.
3) A company that creates blood pressure medicine is researching the effectiveness of their blood pressure medicine. The company conducts a study in which volunteers are randomly assigned to two groups. One group will be given the new medication and the other group will continue to take their current blood pressure medicine.
4) The same company that creates blood pressure medicine is still researching the effectiveness of their blood pressure medicine. The company conducts a second study in which volunteers are all given the new medication. The blood pressure of each patient is measured before the study begins. The patients are all given the new medication for 6 weeks. The blood pressure of each patient is measured after the 6 week period.
5) A psychologist wants to know if children’s’ levels of anxiety are different if their parents are divorced. The psychologist decides to study 100 children from divorced parents and 100 children from non-divorced parents.
6) The quality control manager at a manufacturing plant is investigating the production rate of two machines that are built with the same materials and same design but manufactured at two different plants.
7) A statistics teacher wants to know if a curriculum is effective. The teacher conducts a pre-test, implements the curriculum and then conducts a post-test on the same group of students. The scores on the pre-test and post-test are used to compare the difference in understanding of statistics before and after students have completed the curriculum.
8) Suppose you want to study the effectiveness of a diet. Suppose that 8 people were randomly selected to participate in your study. The weight (lbs.) of each of the 8 participants is recorded before and after the diet in the following table. You know from past studies that body weight is approximately normally distributed.
Patient Before After
1 150 146
2 160 159
3 200 200
4 178 174
5 190 189
6 167 160
7 151 148
8 210 198
Mean (
Part A: What is the average weight before and after the diet? Fill in your answers in the table.
Part B: On average, how many pounds did the participants lose? In other words, what is the estimated difference between the mean weight after and before the diet?
Part C: Are the two samples independent or dependent?
9) Consider the previous example using this new table:
Patient Before After Difference
1 150 146 146-150 4
2 160 159 159-160 1
3 200 200
4 178 174
5 190 189
6 167 160
7 151 148
8 210 198 1
Mean (
Part A: How much weight did each individual lose? Complete the table by finding the difference in each participant’s weight (afterbefore).
Part B: Consider ONLY the difference variable. What is the average weight loss for the 8 participants?
Part C: How does your answer to Question 9, Part B compare to Question 8, Part B?
