- 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?
Answer: Whether a change in exercise hours causes a change in resting heart rate.
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 scatterplot shows a moderate, positive, non-linear association between two quantitative variables, X and Y. Which of the following statements is most consistent with this description?
Answer: The points are somewhat scattered around a gentle curve that goes up from left to right.
'Moderate' strength implies the points are somewhat, but not perfectly, close to the pattern. 'Positive' direction means the pattern goes up from left to right. 'Non-linear' form means the pattern is a curve. This statement combines all three characteristics correctly.
- A scatterplot of property values versus square footage for homes in a large city shows two distinct groups of points. One group consists of smaller, less expensive homes and is clustered in the lower-left area of the plot. The other group consists of larger, more expensive homes clustered in the upper-right area. This type of unusual feature is known as:
Answer: Clustering, which indicates that there are distinct subgroups within the data.
The presence of distinct groups or clouds of points in a scatterplot is referred to as clustering. It often suggests that the data comes from different subgroups that may have different characteristics.
- A scatterplot of the number of years of education versus annual income for a random sample of adults shows a positive association. What does the phrase "positive association" imply in this context?
Answer: Adults with higher levels of education tend to have higher annual incomes, on average.
In statistics, an association describes a general trend or tendency in the data, not a deterministic rule that applies to every single individual. A positive association means that, on average, higher values of one variable are paired with higher values of the other.
- 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?
Answer: Non-linear, because the points clearly follow a curved pattern.
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 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:
Answer: A moderate, positive, linear association.
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.
- A teacher creates a scatterplot of the number of absences for students in a class versus their final exam scores. The plot shows a general downward trend, but the points are widely scattered. Which of the following provides the best description of the association?
Answer: A weak, negative, linear association.
A 'downward trend' indicates a negative direction. 'Widely scattered' points indicate a weak strength. Assuming the general trend is straight, the form is linear. Therefore, the best description is a weak, negative, linear association.
- What is the primary purpose of constructing a scatterplot in a statistical analysis involving two variables?
Answer: To visually examine the relationship or association between two quantitative variables recorded for the same individuals.
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.
- The athletic department of a university collects data on the heights (in inches) and weights (in pounds) of all its varsity athletes. This dataset is best described as:
Answer: A bivariate quantitative dataset, as two numerical measurements are recorded for each athlete.
The data consists of two variables (height and weight) for each individual. Since both height and weight are numerical measurements, this is a bivariate quantitative dataset.
- 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?
Answer: The explanatory variable is the number of hours studied, and the response variable is the grade point average.
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 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?
Answer: Linear, indicating the points follow a straight-line pattern.
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?
Answer: Strong, because the points closely follow a clear linear pattern.
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.
- 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?
Answer: A negative association, because an increase in one variable is related to a decrease in the other.
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 coffee shop owner wants to see if the daily high temperature affects the sales of hot coffee. Over a period of several weeks, she records the daily high temperature and the number of hot coffees sold. Which of the following is the most appropriate setup for a scatterplot?
Answer: Explanatory variable: daily high temperature; Response variable: number of hot coffees sold.
The owner is investigating how temperature might affect or predict sales. Therefore, daily high temperature is the explanatory variable, and the number of hot coffees sold is the response variable.
- An automotive engineer examined a scatterplot relating the weight of a car (in pounds) to its fuel efficiency (in miles per gallon). The plot showed a fairly straight pattern sloping downwards, with most points reasonably close to the pattern. One car, a very heavy vehicle, had much lower fuel efficiency than the pattern suggested. Which is the most complete description?
Answer: The association is moderate to strong, negative, and roughly linear, with one potential outlier.
A complete description of a scatterplot should address direction, form, strength, and any unusual features. The description 'moderate to strong, negative, and roughly linear, with one potential outlier' covers all four of these essential components.
- A city's environmental protection agency wants to determine if there is a relationship between the amount of rainfall (in inches) and the level of a certain pollutant (in parts per million) in the local river. They plan to use rainfall to help predict pollutant levels. Which variable should be plotted on the horizontal axis of the scatterplot?
Answer: The amount of rainfall, because it is the explanatory variable being used to make predictions.
By statistical convention, the explanatory (or predictor) variable is plotted on the horizontal (x) axis, and the response variable is plotted on the vertical (y) axis. Here, rainfall is used to explain or predict pollutant levels.
- A student creates a scatterplot of their classmates' heights versus their birth months (coded 1 for January, 2 for February, etc.). The points on the scatterplot show no discernible pattern and appear as a random cloud. What conclusion can be drawn about the association?
Answer: There is no apparent association between height and birth month.
A scatterplot where the points form a random cloud with no discernible pattern (no upward or downward trend, no clear curve) indicates that there is no apparent association between the two 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?
Answer: A bivariate quantitative dataset.
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.
- When describing a scatterplot of two quantitative variables, what is meant by an "unusual feature" such as an outlier?
Answer: A point that deviates from the overall pattern of the relationship shown in the other points.
In the context of a bivariate relationship, an outlier is a data point that does not fit the general pattern established by the majority of the data points. It stands apart from the overall trend.
