Statistics Quiz: Fitting Linear Functions To Data

What this quiz covers

This quiz focuses on Fitting Linear Functions To Data, giving you a quick way to practice the rules, question types, and explanations that matter most for Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Because $r=0.95$, salary can be predicted exactly from years at the company for every employee.
2. Because $r=0.95$, each additional year at the company increases salary by $0.95$ dollars.
3. There is a very strong negative linear association between years at the company and salary, meaning employees who stay longer tend to earn less.
4. There is a very strong positive linear association between years at the company and salary, but this does not by itself prove that time at the company causes salary to increase. (correct answer)

Explanation: Very high r values near 1 indicate a strong positive linear association, suggesting close alignment in the data. For years at the company and salary, r=0.95 means longer tenure is strongly linked to higher pay. Yet, this doesn't prove causation; performance or promotions might drive both. Errors often involve assuming perfect prediction, as in choice C, or confusing r with slope, like in choice D. Even with r=0.95, some variation exists around the trend. r only captures linear relationships, missing potential plateaus. Interpreting r thoughtfully prevents overgeneralization in workplace data.

Question 2

1. There is a strong positive linear association between absences and GPA, meaning more absences tend to be associated with higher GPAs.
2. Because $r=-0.74$ is close to $-1$, the relationship must be perfectly linear and GPA can be predicted exactly from absences.
3. Because $r=-0.74$, increasing absences will always decrease GPA by 0.74 points.
4. There is a strong negative linear association between absences and GPA, meaning more absences tend to be associated with lower GPAs, but this does not prove absences cause GPA to change. (correct answer)

Explanation: The value of r indicates the direction and strength of linear association: negative values mean that as one variable increases, the other tends to decrease. With r=-0.74, there's a strong negative linear association between absences and GPA, so more absences are linked to lower GPAs. Importantly, this does not establish causation; absences might not directly cause lower GPAs, as lurking variables like motivation could influence both. Misinterpretations often include assuming r gives an exact rate of change, like in choice C, or thinking a value near -1 means perfect predictability, as in choice D. In reality, r=-0.74 allows for some prediction but with variability around the trend line. Remember, r assesses only linear relationships and doesn't account for outliers or non-linear effects. Interpreting r correctly helps avoid overstating its implications in educational data.

Question 3

1. There is a moderate positive linear association between calories and BMI, though this does not imply that higher calories cause higher BMI. (correct answer)
2. There is a weak negative linear association between calories and BMI, so higher calorie intake tends to be associated with lower BMI.
3. Because $r=0.41$, the relationship between calories and BMI must be very strong and nearly perfectly linear.
4. Because $r=0.41$, increasing calories will always increase BMI.

Explanation: The correlation r ranges from -1 to 1, with magnitudes around 0.4 typically indicating a moderate linear association. Here, r=0.41 shows a moderate positive link between calories consumed and BMI, meaning higher calorie intake tends to be associated with higher BMI. Crucially, this does not imply causation; diet quality or exercise could confound the relationship. Common mistakes include assuming causation or perfect linearity, as in choices C and D, which overstate r's meaning. Instead, r=0.41 suggests some predictive power but with considerable scatter in the data. r focuses solely on linear trends, potentially missing complex nutritional dynamics. Understanding these nuances prevents misapplying correlation in health contexts.

Question 4

1. There is a moderate negative linear association between practice problems and quiz time: students who complete more practice problems tend to have shorter quiz times, though this does not show that practice causes faster times. (correct answer)
2. Because $r=-0.52$, quiz time will always decrease when a student completes more practice problems.
3. There is a moderate positive linear association between practice problems and quiz time: students who complete more practice problems tend to take longer on the quiz.
4. Because $r=-0.52$, there is no relationship at all between practice problems and quiz time.

Explanation: A moderate negative r, like -0.52, means a fair inverse linear association without being overwhelmingly strong. Here, more practice problems tend to link with shorter quiz times, but not definitively. Correlation isn't causation; innate ability might affect both variables. Misinterpretations include assuming positive direction, as in choice B, or guaranteed effects, like in choice C. Low |r| doesn't mean no relationship, countering choice D, but indicates moderate predictability. r ignores non-linear patterns or outliers. Proper analysis avoids these pitfalls in educational research.

Question 5

1. Because $r=-0.91$, the resale price of a car can be predicted exactly from its age.
2. There is a strong positive linear association between car age and resale price: older cars tend to sell for more.
3. Because $r=-0.91$, each additional year of age lowers the resale price by $0.91$ dollars.
4. There is a strong negative linear association between car age and resale price: older cars tend to have lower resale prices, though this does not establish a cause-and-effect relationship. (correct answer)

Explanation: r values near -1 signify a strong negative linear association, where increases in one variable correspond to decreases in the other. For car age and resale price, r=-0.91 indicates that older cars tend to have much lower prices, reflecting a strong downward trend. However, this association doesn't prove causation; factors like mileage or condition might also affect price. A frequent misinterpretation is equating r with the exact slope, as in choice C, which wrongly suggests a $0.91 decrease per year. Another error is assuming perfect prediction from a high |r|, like in choice D, but even strong correlations leave room for variation. r only captures linear patterns, so non-linear depreciation curves might not be fully represented. Proper interpretation emphasizes the strength and direction without implying cause or exactness.

Question 6

1. Because $r=0.67$, each additional ad increases load time by exactly 0.67 seconds.
2. There is a moderate to strong negative linear association between ads shown and load time, meaning pages with more ads tend to load faster.
3. Because $r=0.67$, load time can be predicted perfectly from the number of ads.
4. There is a moderate to strong positive linear association between ads shown and load time, meaning pages with more ads tend to have longer load times, but this does not prove the ads cause the longer load times. (correct answer)

Explanation: r values around 0.7 suggest a moderate to strong positive linear association, where both variables tend to increase together. With r=0.67 for ads and load time, more ads are associated with longer load times. This doesn't establish causation; page complexity might contribute to both. Common errors include misinterpreting the direction, as in choice B, or assuming r gives exact changes, like in choice C. Predictions from r=0.67 will have some error, countering choice D's perfect prediction claim. r assesses only linearity, potentially overlooking other influences. Emphasizing these points aids in understanding web performance data.

Question 7

1. Because $r=0.82$, studying more hours causes students’ scores to increase.
2. There is a strong negative linear association between hours studied and test score, meaning students who study more tend to score lower.
3. There is a strong positive linear association between hours studied and test score, though this does not by itself show that studying causes higher scores. (correct answer)
4. Because $r=0.82$ is not 1, hours studied cannot be used to predict test scores at all.

Explanation: The correlation coefficient r quantifies the strength and direction of the linear association between two variables, ranging from -1 to 1, where values near 1 indicate a strong positive relationship. In this scenario, r=0.82 suggests a strong positive linear association, meaning students who study more hours tend to have higher test scores. However, a key point is that correlation does not imply causation; the association does not prove that studying causes better scores, as other factors like prior knowledge could be at play. Common misinterpretations include assuming causation, as in choice C, or dismissing any predictive value because r is not exactly 1, as in choice D. Instead, r=0.82 indicates that hours studied can help predict test scores reasonably well, but not perfectly. It's also important to remember that r only measures linear relationships and may miss non-linear patterns. Overall, interpreting r requires considering both its magnitude and sign while avoiding overstatements about cause and effect.

Question 8

1. There is a weak (or no) linear association between study time and quiz score. (correct answer)
2. Because $r=0.12$, quiz score can be predicted exactly from study time.
3. Studying longer causes quiz scores to increase.
4. There is a strong positive linear association between study time and quiz score.

Explanation: The correlation coefficient r indicates the degree of linear association between variables, where values near 0 suggest little to no linear relationship, regardless of other patterns. For r = 0.12, there is a weak or no linear association between study time and quiz scores, so study time doesn't linearly predict scores well. A key misinterpretation is equating correlation with causation; even if positive, it wouldn't mean studying causes better scores without further evidence. Another common error is believing any r allows exact predictions, but low r means high variability. This low r highlights that linear models may not capture the relationship effectively.

Question 9

1. Because $r=-0.64$, resting heart rate can be predicted exactly from minutes of exercise.
2. More minutes of exercise per week cause resting heart rate to decrease.
3. There is a moderate negative linear association between minutes of exercise and resting heart rate. (correct answer)
4. There is a moderate positive linear association between minutes of exercise and resting heart rate.

Explanation: The correlation coefficient r quantifies the strength and direction of the linear relationship between two variables, with negative values indicating that as one variable increases, the other tends to decrease. Here, r = -0.64 suggests a moderate negative linear association between minutes of exercise and resting heart rate, implying that more exercise is associated with lower heart rates. It's important not to confuse this with causation; correlation does not mean exercise causes the heart rate change, as lurking variables might exist. A frequent misinterpretation is assuming r enables exact predictions, but |r| = 0.64 means there's still considerable scatter around the linear trend. Understanding r helps in recognizing patterns without overinterpreting the data as deterministic.

Question 10

1. Building more parks causes average home prices to increase.
2. There is a strong negative linear association between number of parks and average home price.
3. There is a strong positive linear association between number of parks and average home price. (correct answer)
4. Because $r=0.91$, average home price is determined exactly by the number of parks.

Explanation: The correlation coefficient r ranges from -1 to 1, with values above 0.8 often indicating a strong positive linear association. r = 0.91 suggests a strong positive linear association between number of parks and home prices, implying more parks link to higher prices. Misinterpreting this as causation is frequent; parks don't necessarily cause price increases, as affluent areas might afford more parks. Another error is thinking high r means exact determination, but variability remains. This high r underscores potential desirability factors in neighborhoods.

Question 11

1. There is a strong positive linear association between temperature and hot chocolate sales.
2. Because $r=-0.89$, temperature determines hot chocolate sales exactly every day.
3. There is a strong negative linear association between temperature and hot chocolate sales. (correct answer)
4. Hot chocolate sales cause the outside temperature to decrease.

Explanation: r, the correlation coefficient, assesses how closely two variables follow a linear pattern, with its sign showing the direction: positive for increasing together, negative for one increasing as the other decreases. With r = -0.89, there's a strong negative linear association between temperature and hot chocolate sales, meaning sales rise as temperature falls. However, this does not imply causation; higher sales don't cause temperature drops, and vice versa. People often misinterpret high |r| as meaning one variable exactly determines the other, but even r = -0.89 allows for some deviation from the line. The downward trend in the scatter plot aligns with this strong negative linear relationship.

Question 12

1. There is little to no linear association between distance from the lake and soil moisture. (correct answer)
2. Soil moisture causes locations to be farther from the lake.
3. There is a strong negative linear association between distance from the lake and soil moisture.
4. Because $r=-0.08$, soil moisture does not change at all as distance from the lake changes.

Explanation: The correlation coefficient r specifically evaluates linear associations, so even strong nonlinear patterns can yield low r values. Here, r = -0.08 indicates little to no linear association between distance from the lake and soil moisture, despite the curved pattern described. A misinterpretation is thinking low r means no relationship at all; there could be a nonlinear one, like the decreasing moisture curve. Correlation doesn't imply causation, so higher moisture doesn't cause proximity to the lake. Additionally, r near 0 doesn't mean no change; it just means no linear trend. This example shows r's limitation to linear relationships.

Question 13

1. There is a moderate negative linear association between tempo and number of plays.
2. There is a moderate positive linear association between tempo and number of plays. (correct answer)
3. Increasing a song’s tempo causes it to be played more often.
4. Because $r=0.43$, the number of plays can be predicted exactly from tempo.

Explanation: The correlation coefficient r indicates linear association strength, with |r| around 0.4 often moderate and positive sign meaning variables increase together. r = 0.43 shows a moderate positive linear association between song tempo and plays, with faster tempos slightly linked to more plays. A common misinterpretation is causation; higher tempo doesn't cause more plays, as popularity factors vary. Assuming r enables exact predictions is incorrect, especially with moderate values. The slight upward trend supports this moderate interpretation of r.

Question 14

1. There is a strong positive linear association: sessions with more time spent reading tend to have more pages read, but $r$ does not describe the exact rate of pages per minute. (correct answer)
2. There is a weak positive linear association: sessions with more time spent reading only slightly tend to have more pages read.
3. Because $r=0.95$, reading longer causes a person to read exactly 0.95 more pages each minute.
4. Because $r=0.95$, knowing the time spent reading allows you to predict the number of pages read with no error for every session.

Explanation: The correlation coefficient r = 0.95 indicates a very strong positive linear association between time spent reading and pages read. This means that reading sessions with more time tend to have proportionally more pages read, following a highly consistent linear pattern. The value 0.95 is very close to 1, indicating an extremely strong relationship. However, r does not tell us the actual rate of pages per minute - that would come from the slope of the regression line. Additionally, even with r = 0.95, there is still some variation in the data; we cannot predict pages read with perfect accuracy. The correlation describes the strength and direction of the linear relationship, not the specific rate of change.

Question 15

1. There is a strong negative linear association: students with more activities tend to spend much less time on homework.
2. There is a weak negative linear association: students with more activities tend to spend slightly fewer hours on homework, but the relationship is not strong. (correct answer)
3. Because $r=-0.28$, each additional activity reduces homework time by 0.28 hours per week.
4. Because $r=-0.28$, having more activities causes students to do less homework.

Explanation: The correlation coefficient r = -0.28 indicates a weak negative linear association between number of activities and homework hours. This means that students with more activities tend to spend slightly fewer hours on homework, but the relationship is not strong or consistent. Values of r between -0.3 and 0.3 are generally considered weak associations. The value -0.28 does not mean that each activity reduces homework by 0.28 hours - that would be the slope of a regression line. Additionally, correlation does not imply causation; we cannot conclude that having more activities causes less homework time. The weak correlation suggests many other factors influence homework time beyond just the number of activities.

Question 16

1. There is a moderate negative linear association: users with higher step counts tend to have lower resting heart rates, but this does not imply steps cause the lower heart rate. (correct answer)
2. There is a moderate positive linear association: users with higher step counts tend to have higher resting heart rates.
3. Because $r=-0.63$, resting heart rate decreases by 0.63 beats per minute for each additional step taken.
4. Because $r=-0.63$, step count predicts resting heart rate exactly for every user.

Explanation: The correlation coefficient r = -0.63 indicates a moderate negative linear association between daily step count and resting heart rate. This means that users with higher step counts tend to have lower resting heart rates, following a moderately consistent linear pattern. The magnitude 0.63 represents a moderate strength - stronger than weak but not as strong as values above 0.7 or below -0.7. However, correlation does not establish causation; while these variables are associated, r alone doesn't prove that taking more steps causes a lower heart rate. The value -0.63 is not a rate of change (beats per step), and it doesn't allow for perfect prediction of heart rate from step count.

Question 17

1. There is a strong negative linear association: commuters who live farther away tend to pay much less each month.
2. There is a strong positive linear association: commuters who live farther away tend to pay much more each month.
3. There is little to no linear association between distance from downtown and monthly transit cost in this sample. (correct answer)
4. Because $r=-0.05$, monthly transit cost decreases by $0.05 for each additional mile from downtown.

Explanation: The correlation coefficient r = -0.05 indicates essentially no linear association between distance from downtown and monthly transit cost. Values of r this close to 0 suggest that knowing how far someone lives from downtown provides virtually no information about their monthly transit costs. The relationship, if any exists, does not follow a linear pattern. This might seem counterintuitive, but could occur if transit pricing is based on zones rather than distance, or if other factors dominate cost determination. The value -0.05 is not a rate of change in dollars per mile. With r this close to zero, these variables are essentially unrelated in a linear sense.

Question 18

1. There is a strong positive linear association: as temperature increases, hot coffee sales tend to increase.
2. There is a strong negative linear association: as temperature increases, hot coffee sales tend to decrease, but this does not show temperature causes the change in sales. (correct answer)
3. Because $r=-0.74$, hot coffee sales decrease by 0.74 cups for every 1°F increase in temperature.
4. Because $r=-0.74$, higher temperatures will always lead to fewer hot coffees sold on every single day.

Explanation: The correlation coefficient r = -0.74 indicates a strong negative linear association between temperature and hot coffee sales. This means that as temperature increases, hot coffee sales tend to decrease, following a fairly consistent linear pattern. The negative sign tells us the direction of the relationship (inverse), while the magnitude 0.74 indicates the strength is fairly strong. However, correlation does not establish causation - while these variables are associated, r alone doesn't prove temperature causes the change in sales. The value -0.74 is not a rate of change (cups per degree), and it doesn't guarantee that every single warmer day will have fewer sales.

Question 19

1. There is a strong positive linear association: warmer days tend to be associated with more iced coffees sold. (correct answer)
2. There is a strong negative linear association: warmer days tend to be associated with fewer iced coffees sold.
3. Because $r=0.91$, each 1°F increase will increase iced coffee sales by exactly 0.91 coffees.
4. Warmer temperatures cause people to buy iced coffee, as shown by $r=0.91$.

Explanation: r, the correlation coefficient, assesses how closely two variables follow a linear pattern, with positive values indicating that as one increases, so does the other, and strength gauged by proximity to ±1. With r = 0.91, there is a strong positive linear association, suggesting warmer temperatures are associated with higher iced coffee sales. The value |r| = 0.91 denotes strong correlation, as it approaches 1. However, people often misinterpret r as proof of causation, like assuming temperature causes sales increases. Another common mistake is confusing r with the exact rate of change, such as expecting 0.91 more coffees per degree. r does not measure causation or predict individual outcomes precisely; it summarizes overall linear trends. The best interpretation highlights the association and avoids causal claims.

Question 20

1. There is a weak-to-moderate positive linear association: students who study more hours tend to have higher test scores. (correct answer)
2. There is a weak-to-moderate negative linear association: students who study more hours tend to have lower test scores.
3. Studying more hours causes a higher test score because the correlation is positive.
4. Because $r=0.47$, the score will always increase by exactly 0.47 points for each additional hour studied.

Explanation: The correlation coefficient r indicates the strength and direction of a linear relationship, where values between 0.3 and 0.5 often suggest weak-to-moderate positive association if positive. For r = 0.47, this means a weak-to-moderate positive linear link, with more study hours tending to accompany higher test scores. The moderate label fits as |r| is not strong (above 0.7) but not negligible. Misinterpretations frequently include inferring causation from correlation, such as claiming studying causes better scores. Others wrongly see r as a precise multiplier, like a 0.47-point increase per hour. Importantly, r only captures linear patterns and does not guarantee outcomes for individuals. Accurate explanations stress association without implying cause or exact effects.