Representing Relationships Between Two Quantitative Variables
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AP Statistics › Representing Relationships Between Two Quantitative Variables
A scatterplot relates the score on a pre-test to the score on a post-test for students in a statistics course. The points are scattered, but generally, students who scored higher on the pre-test also scored higher on the post-test. The relationship, however, is not very tight. The best description would be:
A strong, negative, linear association.
A weak, positive, non-linear association.
No association between pre-test and post-test scores.
A moderate, positive, linear association.
Explanation
The general trend that higher scores on one test are associated with higher scores on the other indicates a positive direction. The fact that the relationship is 'not very tight' suggests a moderate strength. Unless a curve is evident, a linear form is assumed.
What is the primary purpose of constructing a scatterplot in a statistical analysis involving two variables?
To visually examine the relationship or association between two quantitative variables recorded for the same individuals.
To display the distribution of a single quantitative variable, focusing on its shape, center, and spread.
To compare the distributions of a categorical variable across several different populations or groups.
To determine the precise probability of a specific outcome in a chance-based experiment.
Explanation
A scatterplot is a graphical tool specifically designed to show the relationship between two quantitative variables. Each point on the plot represents one individual's values for both variables.
A study is conducted to explore the relationship between two variables. For each of 100 participants, their daily screen time (in hours) and their score on a happiness scale (from 1 to 10) are recorded. How is this data best classified?
An experimental dataset designed to show a causal relationship between the variables.
A bivariate dataset with one quantitative variable and one categorical variable.
A bivariate quantitative dataset.
A set of two univariate datasets, one for screen time and one for happiness score.
Explanation
The data consists of two variables measured for each participant. Screen time (in hours) is quantitative. A numerical scale like the happiness score is also treated as quantitative. Therefore, the data are bivariate and quantitative.
A scientist plots the concentration of a chemical reactant over time during a reaction. The points on the scatterplot clearly follow a curve that decreases rapidly at first and then flattens out as time progresses. Which of the following is the best description of the form of this association?
Linear, because the relationship can be approximated with a straight line.
Weak, because the rate of change is not constant throughout the reaction.
Positive, because both time and concentration are positive measurements.
Non-linear, because the points clearly follow a curved pattern.
Explanation
The form of an association is its general shape. Since the points are described as following a curve rather than a straight line, the form is non-linear.
A scatterplot is created to show the relationship between the number of hours spent exercising per week and the resting heart rate for a group of adults. Which of the following can NOT be determined from viewing the scatterplot alone?
Whether there is a generally positive or generally negative trend in the data.
Whether a change in exercise hours causes a change in resting heart rate.
Whether the association between the two variables appears to be linear or curved.
Whether there are any individuals who deviate significantly from the overall pattern.
Explanation
A scatterplot can show an association or correlation between two variables, but it cannot establish causation. Proving causation requires a well-designed, controlled experiment, not just observational data shown in a scatterplot.
A meteorology student recorded data for 20 days in one city. For each day, the student recorded the high temperature (in °F) and the amount of electricity used that day (in kWh) for air conditioning in the student’s home. Each point in the scatterplot represents one day. Which statement best describes the relationship between temperature and electricity use?
There is a strong negative linear association: hotter days tend to use less electricity for air conditioning.
There is a strong association but it is clearly non-linear and U-shaped.
There is a strong positive linear association: hotter days tend to use more electricity for air conditioning.
Higher temperature causes higher electricity use for air conditioning on every day.
There is no association because electricity use varies across all temperatures.
Explanation
This question tests understanding of the relationship between temperature and electricity use for air conditioning. The correct answer is B because as temperature increases (moving right), electricity use increases (moving up), showing a positive association. The points cluster tightly around an upward-sloping pattern, indicating a strong linear relationship. This makes practical sense - hotter days require more air conditioning. Choice A incorrectly reverses the relationship, claiming hotter days use less electricity. When interpreting real-world scatterplots, consider whether the relationship makes logical sense based on your understanding of the variables.
A researcher wants to investigate if the number of hours a student studies per week can be used to predict their grade point average (GPA). In this study, what are the explanatory and response variables?
The explanatory variable is the grade point average, and the response variable is the number of hours studied.
The explanatory variable is the student, and the response variable is the school they attend.
Both the number of hours studied and the grade point average are response variables in this investigation.
The explanatory variable is the number of hours studied, and the response variable is the grade point average.
Explanation
The explanatory variable is the one used to explain or predict changes in the response variable. Here, study hours are used to predict GPA, making study hours the explanatory variable and GPA the response variable.
A scatterplot of data for the price of a used car versus its age in years shows that as the age of the car increases, its price tends to decrease. How would the direction of this association be described?
A negative association, because an increase in one variable is related to a decrease in the other.
A curvilinear association, because the price depreciation is not constant over time.
No association, because the relationship is expected and does not show a statistical pattern.
A positive association, because both price and age are positive values.
Explanation
A negative association occurs when higher values of one variable tend to be paired with lower values of another variable. As the age (one variable) increases, the price (the other variable) tends to decrease.
A biologist creates a scatterplot of the height versus the weight of a sample of adult elephants. The points on the plot appear to follow a pattern that is reasonably straight. How would the form of this relationship be described?
Linear, indicating the points follow a straight-line pattern.
Non-linear, indicating the points follow a curved pattern.
Positive, indicating that taller elephants tend to be heavier.
Strong, indicating the points are tightly clustered together.
Explanation
The 'form' of a relationship in a scatterplot refers to its general shape. If the points follow a pattern that is roughly a straight line, the form is described as linear.
An economist plots the gross domestic product (GDP) of a country against its adult literacy rate. The points on the scatterplot are very tightly clustered around a straight line with a positive slope. How would the strength of this association be described?
Weak, because economic data often contains significant variability.
Negative, because higher GDP can lead to a decrease in other societal factors.
Moderate, because a perfect relationship between these variables is not expected.
Strong, because the points closely follow a clear linear pattern.
Explanation
The 'strength' of an association refers to how closely the points in a scatterplot follow a particular form. When points are very tightly clustered around a line, the association is described as strong.