AP Statistics Quiz: Introducing Statistics Are Variables Related

What this quiz covers

This quiz focuses on Introducing Statistics Are Variables Related, giving you a quick way to practice the rules, question types, and explanations that matter most for AP 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. the sample size of three rolls is too small to determine if a pattern is meaningful or just due to random chance. (correct answer)
2. the theoretical probability of rolling doubles needs to be calculated before any such claims can be made.
3. the dice may not have been rolled in a properly randomized manner to ensure independent outcomes.
4. the results from one player's turn cannot be generalized to the population of all players of the game.

Explanation: Apparent streaks or patterns often occur in small samples due to random variation. Concluding that the dice are lucky based on only three rolls is premature because this small sample does not provide enough evidence to rule out random chance as the explanation. This directly relates to the concept that apparent patterns may not be meaningful.

Question 2

1. How much corn does the farm typically yield in an average year, and what is the standard deviation?
2. What is the average amount of spring rainfall in the town over the last 20 years?
3. Is there a statistically significant relationship between the amount of spring rainfall and the corn crop yield? (correct answer)
4. Does higher spring rainfall directly cause an increase in the farm's corn crop yield?

Explanation: The farmer's observation is about a potential relationship between two quantitative variables: rainfall and yield. The most appropriate statistical question seeks to determine if this relationship is statistically significant, meaning it's unlikely to be due to random chance. The other options focus on single variables or jump to a conclusion about causation.

Question 3

1. The relationship is likely due to a lurking variable, such as higher temperatures in summer, which is associated with both variables. (correct answer)
2. The sample size of months is likely too small to draw any conclusions about a relationship between the two variables.
3. The data for ice cream sales are likely skewed to the right, which makes any analysis of a relationship with drownings invalid.
4. There is no statistical evidence of an association because drowning is a rare event, making its count data unreliable for analysis.

Explanation: The observed association between ice cream sales and drowning incidents is best explained by a lurking variable. Higher temperatures in the summer months lead to both increased ice cream consumption and more swimming activities, which in turn leads to more drowning incidents. This illustrates that association does not imply causation.

Question 4

1. What is the average blood pressure of pet owners in the study?
2. Can the observed difference in blood pressure be attributed to pet ownership, or are other lifestyle factors potentially involved? (correct answer)
3. Is the distribution of blood pressure for the group of non-pet owners in the study significantly skewed?
4. How can an experiment be designed to prove that owning a pet is the direct and sole cause of lower blood pressure?

Explanation: The finding of an association between pet ownership and lower blood pressure leads to the question of causation versus confounding. The most important follow-up question is whether other variables (lurking or confounding variables), such as exercise habits or overall health consciousness, could explain the difference. This question frames the next step in the investigation.

Question 5

1. The observed pattern is probably just a random occurrence and no real association exists between temperature and profit.
2. The profit data reported by the company may not have been adjusted for inflation, which could invalidate the association.
3. The association could be due to a lurking variable, such as the summer tourist season, which is related to both temperature and sales. (correct answer)
4. The true relationship between the average monthly temperature and the company's profit is likely nonlinear, not linear.

Explanation: A lurking variable is a common explanation for an observed association between two variables. In this case, the tourist season is a plausible lurking variable: it occurs during months with high temperatures, and it also independently drives up sales and profit. This provides a sound statistical alternative to the direct causal claim.

Question 6

1. Determining if the player changed their batting stance or training regimen during these seasons.
2. Calculating the precise slope of the least-squares regression line of batting average versus season.
3. Considering whether this amount of variation could be due to natural, random fluctuation in athletic performance. (correct answer)
4. Checking if the five-season period is a long enough duration to establish a definitive long-term career pattern.

Explanation: An apparent pattern in data, especially with a small number of data points, may simply be the result of random chance. A key statistical task is to determine whether the observed trend is statistically significant or if it could plausibly be explained by random variability. The other options relate to potential causes or calculations, but the core statistical question is about the nature of the variation.

Question 7

1. The law of large numbers, which states that with many more flips, the proportion of heads should get closer to 0.5.
2. The fact that for a fair coin, any specific sequence of 10 flips is just as likely to occur as any other specific sequence. (correct answer)
3. The necessity of using a control group, which would involve flipping a coin that is known to be perfectly fair.
4. The central limit theorem, which describes the shape of the sampling distribution for the proportion of heads in repeated samples.

Explanation: The core of the student's concern is the seemingly non-random pattern. However, for a fair coin, every possible sequence of 10 flips has the same probability of occurring ($(0.5)^{10}$). The sequence HHHHH TTTTT is just as likely as the more 'random-looking' sequence HTTHT HHTHT. Recognizing this helps distinguish between a truly biased process and an outcome that seems unusual but is consistent with a random process.

Question 8

1. Yes; the points show a generally negative association, so longer workouts tend to go with lower resting heart rates. (correct answer)
2. No; the points form a perfect horizontal line, so there is no relationship.
3. Yes; because longer workouts cause resting heart rate to be exactly the same for everyone.
4. No; since there are a few points that don't follow the trend, there is no association.
5. Yes; the points show a generally positive association, so longer workouts tend to go with higher resting heart rates.

Explanation: This question examines the relationship between exercise duration and resting heart rate. The correct answer identifies a negative association - longer workouts tend to go with lower resting heart rates, which makes biological sense. Choice C incorrectly claims exercise causes everyone to have the same heart rate. Choice D wrongly concludes that a few exceptions eliminate any association. When analyzing scatterplots, look for the overall direction of the point cloud: negative associations show a downward trend from left to right. Individual variations don't negate the overall pattern.

Question 9

1. Yes; the points show a generally negative association, so more absences tend to go with lower final grades. (correct answer)
2. No; the points show no association because grades vary at each absence count.
3. Yes; because absences directly cause every student's grade to drop by the same amount.
4. Yes; the points show a generally positive association, so more absences tend to go with higher grades.
5. No; since one student with many absences still earned a high grade, there is no relationship.

Explanation: This question asks about the relationship between student absences and final grades. The correct answer identifies a negative association - more absences tend to go with lower grades. Choice C wrongly claims direct causation and identical effects for every student. Choice E incorrectly reasons that one exception (high grade despite many absences) eliminates any relationship. When analyzing educational data, negative associations are common between absences and performance. Focus on the overall downward trend rather than expecting every student to follow the pattern exactly.

Question 10

1. No; the points show no clear pattern, so size and price are unrelated.
2. Yes; because increasing square footage causes the sale price to increase by a fixed amount.
3. Yes; the points show a generally positive association, so larger homes tend to sell for more. (correct answer)
4. No; because there is at least one expensive small home, there is no relationship.
5. Yes; the points show a generally negative association, so larger homes tend to sell for less.

Explanation: This question tests recognition of positive association between home size and sale price. The correct answer identifies that larger homes tend to sell for more, showing a positive association. Choice B incorrectly states causation and claims a fixed increase amount. Choice D commits the error of thinking one exception (expensive small home) disproves the entire relationship. When examining real estate data, we expect positive associations between size and price, but remember that association allows for variability - it describes tendency, not absolute rules. Look for the overall upward trend in the data.

Question 11

1. Yes; the points show a generally negative association, so more fertilizer tends to go with shorter plants.
2. No; the points show no clear pattern, suggesting little to no association. (correct answer)
3. Yes; there is a generally positive association, so more fertilizer tends to go with taller plants.
4. Yes; because fertilizer causes height to increase for every plant by the same amount.
5. No; since some plants with low fertilizer are tall, there cannot be any relationship.

Explanation: This question asks about the relationship between fertilizer amount and plant height. The correct answer states there is no clear pattern, indicating little to no association between the variables. Choice D incorrectly claims causation and suggests every plant responds identically. Choice E makes the common error of thinking that any exception (tall plants with low fertilizer) disproves a relationship entirely. When data points are scattered without a clear upward or downward trend, we conclude the variables show little to no association. Remember that "no association" means no consistent pattern, not that every point must be identical.

Question 12

1. No; the points are widely scattered with no trend, so temperature and electricity use are unrelated.
2. Yes; the points show a generally positive association, so higher temperatures tend to go with greater electricity use. (correct answer)
3. Yes; because higher temperature causes electricity use to increase on every day with no exceptions.
4. Yes; the points prove a perfect linear relationship with no variability.
5. No; since a few cool days still had high usage, there can't be any association.

Explanation: This question examines temperature and electricity use in an office building. The correct answer recognizes a positive association - higher temperatures tend to go with greater electricity use, likely due to air conditioning. Choice C incorrectly states causation and claims no exceptions exist. Choice E wrongly concludes that a few cool days with high usage eliminate any association. When analyzing environmental data, look for overall trends while understanding that other factors (like special events) can create exceptions. Association describes the general pattern, not a rule without exceptions.

Question 13

1. Is there an association between a student's typical seating location and their test score? (correct answer)
2. Does sitting in the front of the classroom cause a student's test score to increase significantly?
3. What is the average test score for all students who choose to sit in the front of the classroom?
4. Are the test scores for the students in this particular class approximately normally distributed?

Explanation: The teacher's observation suggests a potential relationship between two variables: seating location and test score. The most appropriate initial statistical question is to ask whether an association exists between these two variables. The other options either presume causation, focus on a single variable, or describe the distribution of one variable.

Question 14

1. The data likely contain influential outliers, such as countries with very high chocolate consumption, that are skewing the results.
2. The observed association might be coincidental or explained by other factors, such as a country's overall wealth and development. (correct answer)
3. The method for counting Nobel laureates may be inconsistent across different countries, leading to measurement errors.
4. The sample of countries may not be a simple random sample and therefore might not be representative of all countries in the world.

Explanation: The main issue with the conclusion is the classic statistics mantra: association does not imply causation. A lurking variable, such as national wealth, is a plausible explanation for the association. Wealthier countries may have both higher chocolate consumption and more resources for advanced education and research, leading to more Nobel laureates. The other options are valid statistical concerns but do not address the primary flaw in the causal conclusion.

Question 15

1. Concluding that increased advertising spending directly causes higher daily revenue for the company.
2. Formulating a hypothesis and selecting a statistical test to determine if there is a significant association between advertising spending and revenue. (correct answer)
3. Calculating the average daily revenue for the company over the entire period of the data collection.
4. Recommending that the marketing department increase its advertising spending to guarantee higher revenue.

Explanation: The observation of a pattern is the preliminary step. A formal statistical investigation begins by framing the question in terms of a testable hypothesis about the association between the two variables (advertising spending and revenue) and then choosing an appropriate method to test it. The other options jump to a conclusion, analyze a single variable, or make a business recommendation without sufficient evidence.

Question 16

1. The presence of public libraries in a city creates an environment that leads to an increase in its violent crime rate.
2. The analysis is flawed because crime rate is a continuous variable while the number of libraries is a discrete variable.
3. A lurking variable, such as the city's population size, is associated with both the number of libraries and the rate of violent crime. (correct answer)
4. The data were not collected from a simple random sample of cities, so no claim of association can be made from this study.

Explanation: This is a classic example of a lurking variable. Larger cities tend to have both more libraries and more crime than smaller cities. The city's population is the lurking variable that explains the observed association between libraries and crime. The association is not causal.

Question 17

1. The researcher's primary interest and the overall motivation level of the students in the study.
2. The total number of students included in the study and their average grade point average.
3. The number of hours worked per week by a student and that student's grade point average. (correct answer)
4. The type of high school the student attends and the category of part-time job the student holds.

Explanation: A study about a relationship between two variables requires identifying those two variables. In this case, the researcher is investigating how 'number of hours worked per week' and 'grade point average' are related for each student in the study.

Question 18

1. By how much does a positive blog review cause the company's stock price to increase on average?
2. Is there a predictable relationship between the publication of positive blog reviews and the stock's price movements? (correct answer)
3. What is the average price of the stock on days when there are no positive blog reviews published about the company?
4. Should investors adopt a strategy of buying the stock every time the blog publishes a positive review?

Explanation: This question appropriately frames the investigation around exploring a 'predictable relationship' or 'association.' It avoids making a causal claim. Option A assumes causation. Option C focuses on a single condition (no reviews) rather than a relationship. Option D is about an investment strategy, not the statistical relationship itself.

Question 19

1. What is the overall rate of pneumonia diagnosis for all patients at the clinic during the winter months?
2. What is the financial cost of the flu shot compared to the average cost of treating a case of pneumonia?
3. Are there other health-related behaviors or pre-existing conditions that differ between the group that gets a flu shot and the group that does not? (correct answer)
4. What are the specific medical criteria used by the clinic's doctors to diagnose a patient with pneumonia?

Explanation: This is an observational study, so it is crucial to consider potential confounding variables. People who choose to get a flu shot may also engage in other healthy behaviors (e.g., better diet, more exercise) or be in a different health status, and these factors could be the real reason for the lower rate of pneumonia. Investigating these other differences is key to understanding the observed association.

Question 20

1. The year the Olympic games were held and the winning time in the 100-meter dash. (correct answer)
2. The number of competitors in the 100-meter dash and the winning time for that race.
3. The host city of the Olympic games and the winning time in the 100-meter dash.
4. The world population in a given year and the winning time in the 100-meter dash that year.

Explanation: The passage describes a trend 'for several decades,' which implies that time is one of the variables. The other variable is the quantity that is changing over time, which is the 'winning time.' Therefore, the two variables being related are the year and the winning time.