Introducing Statistics: Are Variables Related
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AP Statistics › Introducing Statistics: Are Variables Related
A statistician would caution against this conclusion primarily because...
the results from one player's turn cannot be generalized to the population of all players of the game.
the dice may not have been rolled in a properly randomized manner to ensure independent outcomes.
the theoretical probability of rolling doubles needs to be calculated before any such claims can be made.
the sample size of three rolls is too small to determine if a pattern is meaningful or just due to random chance.
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.
Which of the following is the most appropriate question to formalize this observation for a statistical study?
Is there a statistically significant relationship between the amount of spring rainfall and the corn crop yield?
Does higher spring rainfall directly cause an increase in the farm's corn crop yield?
How much corn does the farm typically yield in an average year, and what is the standard deviation?
What is the average amount of spring rainfall in the town over the last 20 years?
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.
The official suggests that high ice cream sales are causing more drownings. What is the most likely statistical issue with this conclusion?
The sample size of months is likely too small to draw any conclusions about a relationship between the two variables.
The data for ice cream sales are likely skewed to the right, which makes any analysis of a relationship with drownings invalid.
The relationship is likely due to a lurking variable, such as higher temperatures in summer, which is associated with both variables.
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.
Based on this finding, which statistical question is most appropriate for a follow-up investigation?
Is the distribution of blood pressure for the group of non-pet owners in the study significantly skewed?
What is the average blood pressure of pet owners in the study?
How can an experiment be designed to prove that owning a pet is the direct and sole cause of lower blood pressure?
Can the observed difference in blood pressure be attributed to pet ownership, or are other lifestyle factors potentially involved?
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.
An executive claims that warmer weather makes customers more willing to spend money. A statistician suggests other possibilities. Which is the most likely statistical alternative explanation?
The true relationship between the average monthly temperature and the company's profit is likely nonlinear, not linear.
The observed pattern is probably just a random occurrence and no real association exists between temperature and profit.
The profit data reported by the company may not have been adjusted for inflation, which could invalidate the association.
The association could be due to a lurking variable, such as the summer tourist season, which is related to both temperature and sales.
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.
What is a key statistical consideration when deciding if this upward trend is meaningful?
Calculating the precise slope of the least-squares regression line of batting average versus season.
Considering whether this amount of variation could be due to natural, random fluctuation in athletic performance.
Determining if the player changed their batting stance or training regimen during these seasons.
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.
What statistical principle is most important to consider before concluding the coin is biased?
The central limit theorem, which describes the shape of the sampling distribution for the proportion of heads in repeated samples.
The fact that for a fair coin, any specific sequence of 10 flips is just as likely to occur as any other specific sequence.
The law of large numbers, which states that with many more flips, the proportion of heads should get closer to 0.5.
The necessity of using a control group, which would involve flipping a coin that is known to be perfectly fair.
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.
A gym manager records, for 28 members, the number of minutes they exercise per workout (x) and their resting heart rate (y, beats per minute). The manager is interested in whether exercise time and resting heart rate appear to be related. Do the data suggest the variables are related?
Yes; the points show a generally negative association, so longer workouts tend to go with lower resting heart rates.
No; the points form a perfect horizontal line, so there is no relationship.
Yes; because longer workouts cause resting heart rate to be exactly the same for everyone.
No; since there are a few points that don’t follow the trend, there is no association.
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.
A teacher records, for 20 students, the number of absences during a semester (x) and the student’s final course percentage (y). The teacher is interested in whether absences and final grade appear to be related. Do the data suggest the variables are related?
Yes; the points show a generally negative association, so more absences tend to go with lower final grades.
No; the points show no association because grades vary at each absence count.
Yes; the points show a generally positive association, so more absences tend to go with higher grades.
Yes; because absences directly cause every student’s grade to drop by the same amount.
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.
A real estate agent samples 26 homes and records the home’s size (x, in square feet) and its sale price (y, in thousands of dollars). The agent wants to know whether size and price appear to be related. Do the data suggest the variables are related?
Yes; the points show a generally negative association, so larger homes tend to sell for less.
No; because there is at least one expensive small home, there is no relationship.
Yes; because increasing square footage causes the sale price to increase by a fixed amount.
Yes; the points show a generally positive association, so larger homes tend to sell for more.
No; the points show no clear pattern, so size and price are unrelated.
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.