Statistical Process
A researcher creates a bar graph to compare the proportion of high school freshmen, sophomores, juniors, and seniors who participate in extracurricular activities. Which step of the statistical investigative process does this represent?
(A) Formulate questions | (B) Collect data | (C) Analyze data | (D) Interpret results
(C) Analyze data
A restaurant manager prints a survey link on every receipt and offers customers a chance to enter a raffle if they go online and complete it. What type of sampling method is being used?
(A) Simple random sample | (B) Convenience sample | (C) Voluntary response sample | (D) Stratified sample
(C) Voluntary response sample
In an experiment testing a new hair growth cream, half of the subjects receive the new cream and the other half receive a cream with no active ingredients that looks and feels identical. What is the main purpose of giving the second group the inactive cream?
(A) To ensure random assignment | (B) To eliminate the need for replication | (C) To serve as a control group for comparison | (D) To guarantee double-blinding
(C) To serve as a control group for comparison
If a researcher wants to establish a cause-and-effect relationship between two variables, which of the following MUST be part of the study design?
(A) Random selection of subjects from the population | (B) Random assignment of subjects to treatment groups | (C) A sample size of at least 1,000 participants | (D) A voluntary response survey
(B) Random assignment of subjects to treatment groups
In a simulation, a $p$-value represents the probability of getting a sample result as extreme as, or more extreme than, the observed result purely by chance, assuming the null model is true. True or False?
True
Identify the correct sequence of the four steps in the statistical investigative process:
Analyze Data
Formulate Questions
Interpret Results
Collect Data
Sequence is 2 --> 4 --> 1 --> 3 (Formulate Questions --> Collect Data --> Analyze Data --> Interpret Results)
A police officer in uniform walks around a high school library and asks individual students, "Have you ever driven over the speed limit?" What primary source of bias is present in this scenario?
(A) Nonresponse bias | (B) Response bias | (C) Voluntary response bias | (D) Undercoverage
(B) Response bias
A study evaluates whether listening to classical music while studying improves vocabulary retention. 60 student volunteers are randomly split into two groups: Group A listens to classical music while studying a word list for 30 minutes; Group B studies the same word list in complete silence for 30 minutes. All students take the same vocabulary quiz immediately afterward in the same quiet room.
Identify the four principles of experimental design present in this experiment:
Comparison: ____________________
Randomization: ____________________
Control: ____________________
Replication: ____________________
Comparison: Classical music group vs. Quiet group.
Randomization: Random assignment of the 60 volunteers to the two groups.
Control: Same study time (30 min), same word list, same testing room and test.
Replication: 30 students per treatment group.
A study followed 2,000 adults for ten years and recorded their daily coffee consumption and heart health. The researchers concluded that drinking 2+ cups of coffee daily decreases the risk of heart disease by 15%.
(a) Is the headline "Coffee Decreases Risk of Heart Disease" justified? Explain.
(b) Identify one potential confounding variable and explain how it connects to both coffee drinking and heart health.
a) No. This was an observational study (no treatments were randomly assigned), so cause-and-effect cannot be established.
(b) Exercise/Fitness level: People who exercise more regularly might also drink more coffee/water and have better heart health.
A factory claims that only 5% of its batteries are defective. A quality auditor tests a random sample of 80 batteries and finds that 8 of them (10%) are defective. The auditor runs 100 trials of a simulation assuming the true defective rate is 5%. Out of 100 simulated trials, 6 trials produced a sample result of 10% or higher.
(a) What is the estimated p-value from this simulation?
(b) Is there convincing evidence at the 5% significance level (alpha = 0.05) that the factory's defective rate is higher than 5%? Explain.
(a) p-value = 6 / 100 = 0.06 (6%).
(b) No. Since 0.06 > 0.05, the result is not statistically significant. A 10% defective rate could reasonably happen by chance (6% of the time) even if the true rate is 5%.
A school district administrator wants to know if high school seniors who take an online personal finance course have higher financial literacy scores than those who take a traditional classroom course.
(a) Formulate a clear, researchable investigative question for this scenario.
(b) Explain how the administrator could design a study that allows them to generalize the findings to all seniors in the district AND establish cause-and-effect. (If this is not practically possible, explain why).
(a) "Do high school seniors in our district who are randomly assigned to an online personal finance course score significantly higher on a financial literacy test than those assigned to a traditional classroom course?"
(b) To generalize to all seniors, you must select a random sample of district seniors. To show cause-and-effect, you must randomly assign them to online vs. classroom courses. Note: In practice, forcing students into specific course formats can be logistically or ethically difficult.
To evaluate customer satisfaction, a grocery store manager divides the day into four time blocks (Morning, Afternoon, Evening, Night). She then randomly selects 25 customers exiting the store during each time block to survey.
(a) What sampling technique did the manager use?
(b) Why is this method preferred over taking a simple random sample of 100 customers from the entire day?
(a) Stratified Random Sampling.
(b) It ensures representation from all times of day (morning, afternoon, evening, night), reducing variability compared to a pure SRS where one time period might be overrepresented by chance.
A city mayor wants to estimate the proportion of residents who support building a new bike lane downtown. Her team considers two plans:
Plan A: Stand outside a downtown coffee shop on Wednesday morning and survey 200 people.
Plan B: Obtain a master list of all registered households in the city, randomly select 200 households, and send a city employee to interview one adult from each selected household in person.
Compare both plans: Which plan avoids undercoverage bias, and which plan is most susceptible to convenience/location bias? Explain how Plan A's bias would likely skew the estimated support level compared to the true population proportion.
Plan B avoids undercoverage by using a complete master list of households.
Plan A uses convenience sampling (outside a downtown coffee shop).
Coffee shop visitors downtown are likely higher-income or urban commuters who favor downtown infrastructure (bike lanes) more than the general city population, leading Plan A to overestimate support.
An online university advertises: "Graduates of our online degree program earn an average of $15,000 more per year than high school graduates who do not attend college."
(a) Is this an observational study or an experiment?
(b) Explain how self-selection acts as a major confounding factor in evaluating whether the online degree caused the increase in earnings.
(a) Observational study.
(b) Students who choose to enroll in and complete college often possess higher baseline motivation, ambition, or socioeconomic advantages prior to entering college. These underlying traits (confounding variables) contribute to higher earnings, not just the degree itself.
A basketball player claims to be an 80% free-throw shooter. During a game, she shoots 20 free throws and only makes 11 (55%). Fans suspect her true free-throw percentage is lower than 80%.
A simulation of 200 trials (each simulating 20 shots with an 80% success rate) was conducted. The results are summarized below:
Simulated Shots Made (out of 20):
less or equal to 11 shots made: 7 trials
12 shots made: 11 trials
13 shots made: 24 trials
14+ shots made: 158 trials
(a) Calculate the simulated p-value for observing 11 or fewer made shots out of 20.
(b) Based on your p-value, write a complete conclusion in context regarding the player's claim.
(a) Simulated p-value = 7 / 200 = 0.035 or 3.5%.
(b) Because the p-value (3.5%) is less than 5%, there is convincing evidence that the player's true free-throw shooting percentage is less than 80%. Making 11 or fewer shots out of 20 is very unlikely to happen purely by chance for an 80% shooter.
A voter registry office mails a survey regarding local town budget cuts to a random sample of 1,000 registered voters. Only 120 voters complete and mail back the survey.
(a) Identify the specific type of bias present.
(b) Explain how this bias could lead to misleading conclusions about the town population's opinion.
(a) Nonresponse bias.
(b) The 880 people who did not respond may have very different opinions on budget cuts (e.g., indifferent or busy) than the 120 who took time to respond (who likely feel strongly for or against).
A company wants to test a new fertilizer designed to increase tomato yield (measured in total weight of tomatoes harvested per plant). They have 40 tomato plants available, but 20 are planted in sandy soil and 20 are planted in clay soil.
(a) Explain why a randomized block design blocking on soil type is better than a simple randomized experiment for this study.
(b) Outline how you would implement this randomized block design.
a) Soil type (sandy vs. clay) is a known variable that affects plant growth. Blocking isolates the variability caused by soil type so the true effect of the fertilizer can be isolated.
(b) Within the 20 plants in sandy soil, randomly assign 10 to receive fertilizer and 10 to receive control. Separately, within the 20 plants in clay soil, randomly assign 10 to receive fertilizer and 10 to receive control. Compare yields within each block.
Read the following news summary:
"Researchers recruited 50 volunteers suffering from chronic migraines from a local clinic. Participants were randomly assigned to either receive a new daily herbal supplement or a daily placebo pill. After 6 weeks, participants receiving the herbal supplement reported 60% fewer migraines than the placebo group. The researchers concluded: 'This herbal supplement cures migraines for all adults worldwide.'"
Identify two major flaws in the researchers' scope of inference regarding:
Generalizability (Population)
Causal Language / Conclusions
Generalizability Flaw: The sample consisted of 50 local clinic volunteers, not a random sample of all adults worldwide. Results cannot be generalized globally.
Causal Flaw: While the experiment shows the supplement reduced migraines compared to placebo, calling it a "cure for all adults" is an extreme overstatement not supported by a single 6-week trial.