Question

Classify each as independent or dependent samples. a. Heights of identical twins b. Test scores of the same students in English and psychology c. The effectiveness of two different brands of aspirin on two different groups of people d. Effects of a drug on reaction time of two different groups of people, measured by a before-and-after test e. The effectiveness of two different diets on two different groups of individuals

Short answer

a. Dependent, b. Dependent, c. Independent, d. Dependent, e. Independent

Step-by-step solution

Understand Independent vs. Dependent Samples

Independent samples occur when the selection of elements in one sample does not affect the selection of elements in another sample. Dependent samples (also called paired or matched samples) involve related elements where a natural pairing exists. For example, measurements before and after treatment on the same individuals create dependent samples.

Classify Each Scenario (Part a)

For part a, 'Heights of identical twins', since we are considering twins, each pair of heights is naturally matched. Therefore, this involves dependent samples as pairs are related to each other.

Classify Each Scenario (Part b)

Part b involves 'Test scores of the same students in English and psychology'. Here, the test scores are from the same students in both subjects, so each student's scores in both English and psychology are related. Thus, these are dependent samples.

Classify Each Scenario (Part c)

In part c, 'the effectiveness of two different brands of aspirin on two different groups of people', the groups receiving each brand are distinct and separate, with no direct relationship between the two groups. Thus, these are independent samples.

Classify Each Scenario (Part d)

Part d considers 'effects of a drug on reaction time of two different groups of people, measured by a before-and-after test'. Even though they are two different groups, a before-and-after measure typically creates dependent samples, as each person's reaction times are related before and after the drug.

Classify Each Scenario (Part e)

Finally, in part e, the 'effectiveness of two different diets on two different groups of individuals' involves separate groups each following a different diet with no direct relation between individual measurements across groups, hence these are independent samples.

Key concepts

Independent Samples

Independent samples are groups where the individuals in one sample have no connection or influence on the selection of individuals in another sample. This is an important concept in statistics because it allows for comparisons without worrying about interdependencies. When the samples are independent, each sample is treated as a standalone with no expectation of correlation.

In practice, independent samples are useful when comparing results from different groups. For example:

• The effectiveness of two different brands of aspirin on separate groups, as mentioned in the exercise. Each group is unique with no shared participants.
• Comparing the heights of children from two different schools.

These situations rely on the principle that the observations or elements in one group should not influence those in the other.

Dependent Samples

Dependent samples, also known as related or paired samples, involve a natural association between the two sets of data. This relationship means that each data pair originates from the same source or is naturally matched in some way. This connection is key in many statistical analyses where relationships between variables are examined.

This scenario is common when evaluating data where measuring outcomes on the same subjects or related groups is necessary. Examples include:

• Test scores of students taken before and after an educational intervention.
• Heights of identical twins, as in the exercise.

The dependency allows researchers to assess changes over time or differences in naturally paired subjects, offering valuable insights into related critical factors.

Statistical Classification

Statistical classification involves categorizing data based on distinct criteria to enable analysis. In the context of sample selection, it refers to determining whether samples are independent or dependent, as well as classifying other traits that might affect data relationships.

The process of classification aids in choosing the right statistical tests and models. Some common classifications include:

• Identifying independent versus dependent samples to select appropriate test models.
• Classifying data based on categorical variables like gender or age groups when analyzing survey results.

Proper classification ensures that the analyses yield meaningful and accurate interpretations. Understanding the nature of the sample relationships forms a cornerstone of effective statistical analysis.

Paired Samples

The concept of paired samples hinges on samples where each data point in one set has a corresponding match in the other set. Paired samples are often utilized in before-and-after studies or in settings where comparisons are made using the same subjects under different conditions.

Paired samples are particularly valuable when examining outcomes such as:

• Assessing improvement in test scores for the same students before and after a tutoring program.
• Studying physiological changes in patients pre- and post-medication.

The pairing of samples aids in reducing variability, allowing a more precise assessment of the factors at play. By controlling for individual differences, paired sample analyses can deliver robust findings about the effect of interventions or changes.