Statistical Reports

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Statistical Reports · Grade 11 · Worksheet 3

1. A news report claims that 90% of high school students in a city get at least 8 hours of sleep per night. The claim is based on a survey of 20 students who were approached outside a local gym at 6:00 AM. Evaluate the validity of this claim, considering potential sources of bias and the adequacy of the sample size. Answer: The claim is likely invalid due to severe sampling bias (surveying at a gym at 6 AM selects unusually active early risers) and an inadequate sample size (n=20 is too small to represent an entire city's student population). Solution: Identify the sample: 20 students at a gym at 6:00 AM. Identify potential bias: The sample is severely biased because it only includes students who are at a gym very early in the morning.
   Full step-by-step solution

   Step 1: Identify the sample: 20 students at a gym at 6:00 AM. Step 2: Identify potential bias: The sample is severely biased because it only includes students who are at a gym very early in the morning. These students are likely more health-conscious and may have different sleep habits than the average student. Students who sleep less or have different schedules would not be present. Step 3: Evaluate sample size: A sample size of 20 is extremely small for a claim about an entire city's student population. For a reliable estimate, hundreds or thousands of randomly selected students would be needed. Step 4: Conclusion: The claim is invalid due to both sampling bias (non-representative group) and inadequate sample size. The report should not be trusted.

2. A study claims that 82% of students at a high school prefer online learning. The study surveyed 50 students who were all members of the school's computer club. Evaluate the validity of this claim. Answer: The claim is likely invalid due to sampling bias; the sample is not representative of the entire student population. Solution: Identify the sample size: 50 students. This is a moderate sample size, but not necessarily large enough to generalize to the entire school population.
   Full step-by-step solution

   Step 1: Identify the sample size: 50 students. This is a moderate sample size, but not necessarily large enough to generalize to the entire school population. Step 2: Identify the sampling method: The survey only included members of the computer club. This introduces selection bias because computer club members may have a stronger preference for online learning than the average student. Step 3: Evaluate representativeness: The sample is not random and does not represent all students (e.g., athletes, artists, students not in clubs). Therefore, the claim that 82% of all students prefer online learning is not supported by the data. Step 4: Conclusion: The claim is invalid due to sampling bias. A valid conclusion would require a random sample of all students at the school. The answer is: The claim is likely invalid due to sampling bias; the sample is not representative of the entire student population.

3. Sophia is evaluating a news report that claims 'A new study of 12 students found that 100% of participants improved their test scores after using a new study app. Therefore, the app is guaranteed to improve test scores for all students.' Identify at least two major statistical flaws in this claim and explain why the conclusion is not valid. Answer: Sample size is too small (12 students) and the sample is not representative of all students; also, the claim of 'guaranteed' improvement ignores variability and potential bias. Solution: Identify the sample size. The study used only 12 students. A sample of 12 is very small and may not represent the diversity of all students (different ages, backgrounds, learning styles, etc.).
   Full step-by-step solution

   Step 1: Identify the sample size. The study used only 12 students. A sample of 12 is very small and may not represent the diversity of all students (different ages, backgrounds, learning styles, etc.). Step 2: Consider sampling bias. The report does not say how the 12 students were chosen. If they were volunteers or from a specific class, they may not be representative. Step 3: Examine the claim of 100% improvement. With only 12 students, even one or two students not improving would change the percentage significantly. Small samples are more sensitive to chance. Step 4: The word 'guaranteed' is inappropriate. Statistics cannot guarantee outcomes for all individuals; they can only suggest trends or probabilities. Step 5: Conclusion: The study has serious flaws including insufficient sample size, potential sampling bias, and overgeneralization. The claim is not valid. Final answer: Two major flaws are the very small sample size (12 students) and the lack of representative sampling, plus the overconfident language of 'guaranteed' improvement.

4. A survey claims that 85% of students at a high school prefer online learning. The survey was conducted by asking 50 students leaving the school's computer lab. Evaluate the validity of this claim. Answer: The claim is likely invalid due to selection bias; the sample is not representative of the entire student population. Solution: Identify the claim: 85% of students prefer online learning. Examine the sampling method: The survey was conducted on 50 students leaving the computer lab. This is a convenience sample, not random.
   Full step-by-step solution

   Step 1: Identify the claim: 85% of students prefer online learning. Step 2: Examine the sampling method: The survey was conducted on 50 students leaving the computer lab. This is a convenience sample, not random. Step 3: Identify bias: Students leaving the computer lab are more likely to be comfortable with technology and may prefer online learning. This introduces selection bias—the sample overrepresents students who already use computers. Step 4: Evaluate sample size: 50 students out of a typical high school population (e.g., 1000) is only 5%. While not necessarily too small, the bias is the main issue. Step 5: Conclusion: The claim is invalid because the sample is not representative. The results cannot be generalized to all students. A random sample across different locations and times would be needed. The answer is: The claim is likely invalid due to selection bias; the sample is not representative of the entire student population.

5. Mere is reviewing a news report that claims a new after-school tutoring program improved student math scores. The report states: 'In a study of 35 students who participated in the program, 100% of students showed improvement in their math grades. Therefore, the program is guaranteed to improve math scores for all students.' Mere needs to evaluate the validity of this claim. Identify at least two statistical issues with the report's claim and explain why the conclusion is not justified. Answer: The report has multiple issues: (1) sample size of 35 is too small to generalize to all students; (2) no control group to compare against; (3) selection bias (students who chose tutoring may already be motivated); (4) no mention of how improvement was measured or if it was statistically significant; (5) correlation does not imply causation—other factors (e.g., extra study time, better teachers) could explain improvement. Solution: Identify the sample size. The study used 35 students. A sample of 35 is very small to represent all students, leading to high variability and low generalizability.
   Full step-by-step solution

   Step 1: Identify the sample size. The study used 35 students. A sample of 35 is very small to represent all students, leading to high variability and low generalizability. Step 2: Check for a control group. The report mentions no comparison group of students who did not receive tutoring. Without a control group, we cannot know if the improvement was due to the program or other factors (e.g., natural learning over time, better teaching, student motivation). Step 3: Look for bias. Students who volunteer for a tutoring program may be more motivated or have supportive parents, so they are not representative of all students. This is selection bias. Step 4: Examine the measurement. The report says 'showed improvement' but does not define how improvement was measured (e.g., grade change, test scores, teacher assessment). Without a clear metric, the claim is vague. Step 5: Consider causation vs. correlation. Even if all 35 students improved, that does not prove the program caused the improvement. Other variables could be responsible. Conclusion: The claim is not justified due to small sample size, lack of control group, selection bias, vague measurement, and confusion of correlation with causation. Therefore, the report's conclusion is invalid.

6. A regular hexagon is inscribed in a circle with radius 8 cm. The hexagon is divided into 6 congruent equilateral triangles by drawing lines from the center to each vertex. What is the exact area of one of these equilateral triangles? Answer: 16√3 Solution: In a regular hexagon inscribed in a circle, each side equals the radius. So side length = 8 cm. Each triangle formed by connecting the center to adjacent vertices is equilateral with side length 8 cm.
   Full step-by-step solution

   Step 1: In a regular hexagon inscribed in a circle, each side equals the radius. So side length = 8 cm. Step 2: Each triangle formed by connecting the center to adjacent vertices is equilateral with side length 8 cm. Step 3: The area of an equilateral triangle with side length s is (s²√3)/4. Step 4: Substitute s = 8: Area = (8²√3)/4 = (64√3)/4 = 16√3. The exact area of one equilateral triangle is 16√3 cm².