- The scatterplot above shows the relationship between two variables. A line of best fit has been drawn. Based on the plot, what is the best prediction of y when x=7, and how reliable is that prediction?
Answer: y≈16; reliable prediction within the data range.
Using the line of best fit (slope ≈ 2, y-intercept ≈ 2), at x = 7 the predicted y is approximately 2(7) + 2 = 16. Since x = 7 falls within the observed data range and the points cluster tightly around the line, this prediction is reliable. Choice B incorrectly suggests x = 7 is outside the data range. Choice C applies an invalid reasoning. Choice D ignores the clear linear pattern shown. - A researcher is analyzing a set of data points representing a student's test scores over several weeks. The ordered pairs are (Week 1, 75), (Week 2, 78), (Week 3, 82), (Week 4, 85), and (Week 5, 89). Which statement best characterizes the relationship between the week number and the test score?
Answer: An approximately positive linear relationship, with scores generally increasing over time.
The scores are consistently increasing. From week to week, the increases are +3, +4, +3, and +4. While not perfectly constant, the rate of change is very similar from week to week, suggesting a strong or approximately positive linear relationship. Real-world data often exhibits such minor variations.
- If a line of best fit for a scatterplot has the equation y=−3x+10, but most of the data points are very far from this line, how would the linear relationship be described?
Answer: A weak negative linear relationship.
The equation y=−3x+10 has a negative slope (-3), so the relationship is negative. However, the fact that 'most of the data points are very far from this line' indicates that the linear model is not a good fit for the data. This describes a weak relationship. A strong or perfect relationship would have points very close to the line. - A data set contains the points (10, 50), (20, 45), (30, 40), and (40, 60). Which statement best describes the relationship among these points?
Answer: The points show a negative linear trend that is disrupted by an outlier.
The first three points (10, 50), (20, 45), and (30, 40) follow a perfect negative linear pattern: as x increases by 10, y decreases by 5. The fourth point (40, 60) breaks this pattern significantly. It deviates from the expected value of (40, 35) and is therefore an outlier that disrupts the linear trend.
- Which of the following scenarios is LEAST likely to be modeled by a linear relationship?
Answer: The population of a bacterial colony that doubles in size every hour.
A population that doubles every hour is exhibiting exponential growth, not linear growth. In a linear relationship, the amount of increase would be constant for each time interval, whereas here the amount of increase grows larger each hour. The other scenarios describe situations with a constant rate of change, which are linear.
- A dataset relates two variables, x and y. Which of the following observations provides the strongest evidence for a linear relationship between x and y?
Answer: For any two points in the dataset, the rate of change between y and x is approximately constant.
A linear relationship is defined by a constant rate of change (slope). If the change in y divided by the change in x is approximately the same for all pairs of points, the relationship is linear. Option A describes any increasing function, not just linear ones. Option B describes a proportional relationship (y=kx), which is a specific type of linear relationship, but C is more general and fundamental. Option D does not define a functional relationship. - A researcher collects data on the age of a car in years (x) and its resale value (y). The data points are (1, $20000), (3, $16000), (5, $12000), (7, $8000). What is the rate of change per year that defines this linear relationship?
Answer: The value decreases by $2,000 for every year the car ages.
To find the rate of change (slope), we can use any two points. Using (1, 20000) and (3, 16000), the change in y is $16000 - $20000 = -$4000. The change in x is 3 - 1 = 2 years. The rate of change is -$4000 / 2 years = -$2000 per year.
- A real estate agent notices that for a particular neighborhood, the selling price of a house tends to increase as its square footage increases. If the data points representing this relationship on a scatterplot are widely spread out around a general upward trend, which term best describes the relationship?
Answer: A weak positive linear relationship.
The trend is upward ('price...tends to increase as...square footage increases'), so the relationship is positive. Because the points are 'widely spread out,' the relationship is weak. A strong relationship would have points clustered tightly around a line.
- Using the scatterplot shown above, which of the following is the best estimate of the y-intercept of the line of best fit?
Answer: 3
The line of best fit passes through approximately (2, 5) and (10, 13). Its slope is (13−5)/(10−2)=1, and extending the line back to x=0 gives y≈3. A student who reads the leftmost data point's y-value as the intercept (the point (2, 5) lies above the intercept) might mistakenly choose a different value. C (9) could come from averaging all the y-values. D (12) could come from confusing a point on the line with the intercept. A (−3) reverses the sign. - A teacher plots the number of hours a student reports studying (x-variable) against their final exam score (y-variable). The data points generally show that students who study more get higher scores. However, one point, (2, 98), is far from the general trend of the other data. Most students who studied for 2 hours scored around 70. How should this data point be interpreted?
Answer: As an outlier that weakens an otherwise positive linear relationship.
The point (2, 98) deviates significantly from the general positive trend. Such a point is called an outlier. Outliers tend to weaken the strength of a correlation and can pull the line of best fit away from the other data points.
- The relationship between two variables is described as follows: 'As the x-variable increases by 1, the y-variable is multiplied by 3.' Which statement best classifies this relationship?
Answer: This is a positive non-linear relationship.
In a linear relationship, a constant amount is added to y for each unit increase in x. In this case, y is multiplied by a constant factor, which is the definition of an exponential relationship. For example, if we start at (1, 2), the next points would be (2, 6), (3, 18), etc. The amount of increase is not constant (+4, then +12), so it is non-linear.
- The scatterplot above was created from data collected over a range of x-values. Based on the plot, which of the following statements is best supported?
Answer: A linear model fits the data well only for smaller values of x, but not for larger values.
For smaller x-values (roughly x < 6), the points fall tightly along a straight line. For larger x-values, the points curve away from the line, showing that the relationship ceases to be linear beyond that point. This 'piecewise' pattern is a classic feature the test assesses. A ignores the curving at the right end; C reverses the pattern; D ignores the clear linear pattern at the left. - If a scatterplot shows data points that form a distinct, horizontal line, what does this imply about the relationship between the x-variable and the y-variable?
Answer: There is no linear relationship; the value of y does not change with x.
A horizontal line has a slope of zero. This means that as the x-variable changes, the y-variable remains constant. Therefore, there is no linear association or correlation between the two variables.
- If two variables have a strong positive linear correlation, which of the following statements must be true?
Answer: When one variable has a high value, the other variable is also likely to have a high value.
A strong positive linear correlation indicates that high values of one variable are associated with high values of the other. It does not prove causation (A is wrong). The relationship does not have to be perfect or have a slope of 1 (C and D are wrong), it just needs to show a strong upward trend.
- A scientist observes that for every 10 meters a diver descends into the ocean, the ambient pressure increases by approximately 1 atmosphere. Which of the following best describes the relationship between a diver's depth and the ambient pressure?
Answer: A strong positive linear relationship, where pressure increases with depth.
The pressure increases by a constant amount (1 atmosphere) for each constant interval of depth increase (10 meters). This constant rate of change is the definition of a linear relationship. Since pressure increases as depth increases, the relationship is positive and strong.
- A car depreciates in value over time. A study of a particular car model shows its value (y) after a certain number of years (x). The data collected are: (1, $25,000), (2, $22,500), (3, $20,000), (4, $17,500). However, the researcher made a typo and recorded the fifth data point as (5, $16,000) instead of the actual value of $15,000. How does this single incorrect point affect the description of the linear relationship?
Answer: It makes the negative linear relationship appear weaker than it actually is.
The actual data shows a perfect linear relationship where the value decreases by exactly $2,500 each year. The incorrect point (5, $16,000) deviates from this pattern (a decrease of only $1,500). This point is an outlier that does not fit the line, thus making the overall linear relationship appear less consistent and therefore weaker.
- Consider the following set of ordered pairs: (5, 20), (10, 15), (15, 10), (20, 5). Which statement accurately describes the linear relationship demonstrated by this data?
Answer: For every 5-unit increase in the x-variable, the y-variable decreases by 5.
Let's check the rate of change. From (5, 20) to (10, 15), x increases by 5 and y decreases by 5. From (10, 15) to (15, 10), x increases by 5 and y decreases by 5. This constant rate of change holds for the entire dataset, defining a perfect negative linear relationship.
- A scatterplot shows data points for two variables, X and Y. If all the data points lie on a vertical line, what can be concluded about the linear relationship between X and Y?
Answer: There is no defined linear relationship because the slope is undefined.
A vertical line means that the x-value is constant while the y-value changes. The slope of a vertical line (change in y / change in x) is undefined because the change in x is zero. In statistics, correlation is not defined for this case, and it is not considered a linear functional relationship.
- A dataset contains five points. Four of the points lie perfectly on the line y=2x+3. One point, an outlier, is located at (5, 5). How does this outlier affect the perceived relationship?
Answer: It weakens the otherwise perfect positive linear correlation.
The point (5, 5) does not fit the pattern. According to the line y=2x+3, when x=5, y should be 2(5)+3=13. The point (5, 5) is far from this expected value. An outlier that does not fit the pattern of the other data points will always weaken the correlation, making the relationship appear less linear than it is. - A data set shows a strong negative linear correlation between the number of winter coats sold and the average daily temperature. Which of the following is the most likely value for the correlation coefficient, r?
Answer: r=−0.88
A 'strong' correlation means the value of r is close to -1 or +1. A 'negative' correlation means the value of r must be negative. Of the choices, r=−0.88 is the only one that represents a strong negative linear relationship. r=0.91 is strong but positive. The other two values are very close to 0 and represent weak relationships.
