# Grade 9 Statistics & Probability Worksheet: Study Design
Section A: Multiple Choice Questions
Q1: A researcher wants to study the effect of sleep on test scores in high school students. Which of the following is the explanatory variable? (a) The high school students (b) The amount of sleep students get (c) The test scores (d) The time of day the test is taken
Solution:
Ans: (b) Explanation: The explanatory variable (or independent variable) is the variable that we think explains or causes changes in another variable. In this study, the amount of sleep is hypothesized to affect test scores, making sleep the explanatory variable and test scores the response variable.
Q2: A study finds that students who carry lighters are more likely to develop lung cancer. The researchers conclude that carrying lighters causes lung cancer. What is the most likely issue with this conclusion? (a) The sample size was too small (b) There is a confounding variable (smoking) (c) The study was not conducted in a laboratory (d) The response variable was incorrectly identified
Solution:
Ans: (b) Explanation: A confounding variable is a variable that influences both the explanatory and response variables, making it appear that they are related when they may not be. In this case, smoking is a confounding variable because smokers are more likely to carry lighters AND more likely to develop lung cancer. The lighters themselves do not cause cancer.
Q3: Which of the following is an example of an observational study? (a) Randomly assigning students to study with music or in silence, then comparing test scores (b) Surveying students about their study habits and comparing their GPAs (c) Giving half the students a new teaching method and keeping the other half with the old method (d) Flipping a coin to decide which students receive tutoring
Solution:
Ans: (b) Explanation: In an observational study, researchers observe and collect data without manipulating or assigning treatments to subjects. Option (b) involves surveying existing behaviors without intervention. Options (a), (c), and (d) all involve random assignment of treatments, which are characteristics of experiments.
Q4: A researcher divides 100 students into groups based on grade level (freshmen, sophomores, juniors, seniors) and then randomly selects 10 students from each group. What sampling method is being used? (a) Simple random sampling (b) Stratified random sampling (c) Cluster sampling (d) Systematic sampling
Solution:
Ans: (b) Explanation:Stratified random sampling involves dividing the population into groups (called strata) based on a characteristic, then randomly sampling from each group. Here, grade level forms the strata. This differs from simple random sampling (where every individual has equal chance) and cluster sampling (where entire groups are randomly selected).
Q5: What is the main advantage of using a control group in an experiment? (a) It increases the sample size (b) It allows comparison to see the effect of the treatment (c) It eliminates all bias from the study (d) It makes the study easier to conduct
Solution:
Ans: (b) Explanation: A control group receives no treatment or a standard treatment, providing a baseline for comparison. This allows researchers to determine whether the treatment group shows different results due to the treatment itself. Without a control group, it would be impossible to know if observed changes were due to the treatment or other factors.
Q6: A pharmaceutical company tests a new drug by having participants take either the real drug or a sugar pill, but neither the participants nor the doctors know who receives which. This is an example of: (a) Single-blind experiment (b) Double-blind experiment (c) Observational study (d) Stratified sampling
Solution:
Ans: (b) Explanation: A double-blind experiment is one where neither the subjects nor the researchers know who is receiving the treatment versus the placebo. This prevents bias from affecting the results. In a single-blind experiment, only the subjects don't know which treatment they receive. The sugar pill mentioned is a placebo.
Q7: Which of the following would introduce bias into a survey about school lunch quality? (a) Randomly selecting 50 students from the entire school (b) Only surveying students who are eating in the cafeteria (c) Using stratified sampling by grade level (d) Ensuring all students have an equal chance of being selected
Solution:
Ans: (b) Explanation:Sampling bias occurs when the sample is not representative of the population. Surveying only students in the cafeteria creates bias because students who don't like the food are more likely to bring lunch from home and won't be included. This is called voluntary response bias or selection bias. Options (a), (c), and (d) all describe proper random sampling methods.
Q8: A researcher wants to determine if a new teaching method improves math scores. She randomly assigns 60 students to two groups: 30 use the new method and 30 use the traditional method. What is the purpose of random assignment? (a) To ensure the groups are the same size (b) To balance out other variables that might affect the results (c) To make the study easier to analyze (d) To guarantee that the new method will work
Solution:
Ans: (b) Explanation:Random assignment helps ensure that the treatment and control groups are similar in all respects except for the treatment itself. This balances out potential confounding variables (like prior math ability, study habits, etc.) across both groups, making it more likely that any differences in results are due to the treatment rather than pre-existing differences between groups.
Section B: Fill in the Blanks
Q9: In an experiment, the variable that is manipulated or controlled by the researcher is called the __________ variable.
Solution:
Ans: explanatory (or independent) Explanation: The explanatory variable (also called the independent variable) is the variable that researchers change or control to observe its effect on the response variable. It "explains" changes in the outcome.
Q10: A sampling method where every member of the population has an equal chance of being selected is called __________ sampling.
Solution:
Ans: simple random Explanation:Simple random sampling ensures that each individual in the population has an equal probability of selection, which helps create a representative sample and reduces sampling bias.
Q11: When researchers observe and measure variables without imposing treatments, they are conducting a(n) __________ study.
Solution:
Ans: observational Explanation: An observational study involves collecting data without manipulating variables or assigning treatments. Researchers simply observe what naturally occurs, which means they cannot establish cause-and-effect relationships as strongly as in experiments.
Q12: A harmless, inactive substance given to the control group to mimic the treatment is called a __________.
Solution:
Ans: placebo Explanation: A placebo is used in experiments to control for the placebo effect, where participants may show improvement simply because they believe they are receiving treatment. This allows researchers to isolate the actual effect of the treatment.
Q13: A variable that influences both the explanatory and response variables, potentially creating a false association between them, is called a __________ variable.
Solution:
Ans: confounding (or lurking) Explanation: A confounding variable (also called a lurking variable) can make it appear that two variables are related when they are not, or hide a true relationship. Identifying and controlling for confounding variables is crucial in study design.
Q14: The entire group of individuals about which we want information is called the __________.
Solution:
Ans: population Explanation: The population is the complete set of individuals or objects we are interested in studying. Because it is often impractical to study an entire population, researchers select a sample, which is a subset of the population.
Section C: Word Problems
Q15: A school principal wants to survey students about extending the school day. She decides to survey every 10th student who enters the building in the morning until she has 50 responses. Identify the sampling method used and explain one potential problem with this approach.
Solution:
Ans: Final Answer: The sampling method is systematic sampling. One potential problem is that students who arrive late to school (possibly those with transportation issues, part-time jobs, or less engagement) will be underrepresented or excluded entirely, creating sampling bias. This means the sample may not accurately represent the opinions of all students.
Q16: A researcher wants to test whether listening to classical music while studying improves test performance. She recruits 80 volunteers and randomly assigns 40 to study with classical music and 40 to study in silence. Both groups study the same material for the same amount of time, then take identical tests. Identify the explanatory variable, response variable, and explain why this is an experiment rather than an observational study.
Solution:
Ans: Final Answer: The explanatory variable is whether students listen to classical music while studying (music or silence). The response variable is the test performance (test scores). This is an experiment rather than an observational study because the researcher actively imposed a treatment by randomly assigning students to either listen to music or study in silence, rather than simply observing their natural study habits.
Q17: A medical researcher wants to study the effectiveness of a new medication for reducing blood pressure. She has 120 participants with high blood pressure. Describe how she should design a double-blind, randomized controlled experiment, including the use of random assignment, a control group, and a placebo.
Solution:
Ans: Final Answer: The researcher should randomly assign the 120 participants into two groups of 60 each. One group (the treatment group) receives the new medication, while the other group (the control group) receives a placebo (a pill that looks identical but contains no active medication). To make it double-blind, neither the participants nor the researchers measuring blood pressure should know who receives the real medication versus the placebo until after the study concludes. This prevents bias in both reporting symptoms and measuring outcomes.
Q18: A company wants to survey its 5,000 employees about job satisfaction. The company has four departments: Sales (1,500 employees), Production (2,000 employees), Administration (1,000 employees), and Research (500 employees). They want to use stratified random sampling to select 200 employees. How many employees should be randomly selected from each department? Show your calculations.
The company should randomly select 60 from Sales, 80 from Production, 40 from Administration, and 20 from Research for a total of 200 employees.
Q19: A news website posts an online poll asking "Should the voting age be lowered to 16?" and receives 10,000 responses, with 85% saying "yes." The website reports that most people support lowering the voting age. Explain why this conclusion may not be valid, identifying the type of bias present.
Solution:
Ans: Final Answer: This conclusion is not valid because the poll suffers from voluntary response bias (also called self-selection bias). Only people with strong opinions, particularly those who favor lowering the voting age (likely younger people who use the website), are motivated to respond to the online poll. This sample is not representative of the general population. Additionally, there may be undercoverage bias because older adults who may oppose the measure might be less likely to visit the website or participate in online polls.
Q20: A researcher studying the relationship between exercise and heart health observes 500 adults over 5 years. She finds that people who exercise regularly have fewer heart problems than those who don't exercise. She concludes that exercise prevents heart disease. Explain why she cannot make this causal conclusion based on this observational study, and suggest what type of study would be needed to establish causation.
Solution:
Ans: Final Answer: The researcher cannot make a causal conclusion because this is an observational study, not an experiment. There may be confounding variables that influence both exercise habits and heart health. For example, people who exercise regularly might also have healthier diets, lower stress levels, better access to healthcare, or genetic factors that reduce heart disease risk. We cannot determine which factor(s) actually cause the better health outcomes. To establish causation, a randomized controlled experiment would be needed, where participants are randomly assigned to exercise or not exercise (though this would be ethically problematic and practically difficult for a long-term study).
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