- 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?
Answer: The association could be due to a lurking variable, such as the summer tourist season, which is related to both temperature and sales.
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.
- Before concluding that the flu shot helps protect against pneumonia, which question is most important to investigate from a statistical standpoint?
Answer: 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?
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.
- 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?
Answer: Yes; the points show a generally negative association, so more absences tend to go with lower final grades.
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.
- Which of the following is the most appropriate statistical question to investigate based on this observation?
Answer: Is there an association between a student's typical seating location and their test score?
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.
- This initial observation would most likely lead to a statistical investigation focused on which of the following questions?
Answer: Is there a measurable association between tree height and trunk diameter for the trees in this forest?
The ecologist observed a potential relationship between two quantitative variables: height and trunk diameter. The logical next step is to frame a statistical question that investigates this relationship, which involves asking if there is a measurable association between them.
- Which of the following best describes the two variables of interest for this study?
Answer: The number of hours worked per week by a student and that student's grade point average.
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.
- A statistician would caution against this conclusion primarily because...
Answer: the sample size of three rolls is too small to determine if a pattern is meaningful or just due to random chance.
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.
- The official suggests that high ice cream sales are causing more drownings. What is the most likely statistical issue with this conclusion?
Answer: The relationship is likely due to a lurking variable, such as higher temperatures in summer, which is associated with both variables.
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?
Answer: Can the observed difference in blood pressure be attributed to pet ownership, or are other lifestyle factors potentially involved?
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.
- 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?
Answer: Yes; the points show a generally negative association, so longer workouts tend to go with lower resting heart rates.
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.
- An observation of this trend is the first step in investigating a relationship between which two variables?
Answer: The year the Olympic games were held and the winning time in the 100-meter dash.
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.
- Which of the following is the most appropriate question to formalize this observation for a statistical study?
Answer: Is there a statistically significant relationship between the amount of spring rainfall and the corn crop yield?
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.
- Why would it be statistically premature to conclude that the new pizza recipe's popularity is the reason for the increased sales?
Answer: The one-day increase in sales could be due to random day-to-day variation in the number of students buying lunch.
A conclusion based on a single day's data is weak because it doesn't account for natural, random fluctuation. The number of students buying lunch varies daily for many reasons. To attribute the increase to the new pizza, one would need to show a consistent pattern over time that is unlikely to be due to this random chance.
- 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?
Answer: Yes; the points show a generally positive association, so larger homes tend to sell for more.
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.
- What is the primary statistical reason to be cautious about concluding that eating more chocolate helps one win a Nobel prize?
Answer: The observed association might be coincidental or explained by other factors, such as a country's overall wealth and development.
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.
- Which of the following represents a valid statistical question about whether the observed pattern in the sample is meaningful?
Answer: Is the association between commute time and salary in this sample simply a result of random sampling variability?
When an association is observed in a sample, it is crucial to ask whether this pattern reflects a real association in the population or if it could have appeared in the sample just by chance (i.e., due to sampling variability). This question directly addresses the core issue of whether the pattern is 'random or not.'
- What is a key statistical consideration when deciding if this upward trend is meaningful?
Answer: Considering whether this amount of variation could be due to natural, random fluctuation in athletic performance.
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?
Answer: 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 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. - An environmental scientist records, for 22 days, the day's high temperature (x, in °F) and the amount of electricity used in a small office building (y, in kWh). The scientist wants to know whether temperature and electricity use appear to be related. Do the data suggest the variables are related?
Answer: Yes; the points show a generally positive association, so higher temperatures tend to go with greater electricity use.
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.
- A botanist measures, for 24 plants of the same species, the amount of fertilizer applied each week (x, in grams) and the plant's height after 6 weeks (y, in centimeters). The botanist wants to know whether fertilizer amount and height appear to be related. Do the data suggest the variables are related?
Answer: No; the points show no clear pattern, suggesting little to no association.
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.
