Statistical Reports

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Answer Key & Explanations

Statistical Reports · Grade 11 · Worksheet 3

1. Liam is reading a news article that claims a new study shows 80% of teenagers prefer a particular brand of sneakers. The study surveyed 20 teenagers at a local skate park. Evaluate the validity of the claim made by the study based on the information provided. Answer: The claim is likely invalid due to significant sampling bias and a very small sample size. Solution: Identify the claim: 80% of all teenagers prefer a particular brand of sneakers. The sample size is 20 teenagers. Evaluate sample size.
   Full step-by-step solution

   Step 1: Identify the claim: 80% of all teenagers prefer a particular brand of sneakers. The sample size is 20 teenagers. Step 2: Evaluate sample size. A sample of 20 is extremely small for making a generalization about a population as large and diverse as all teenagers. This introduces high sampling variability. Step 3: Evaluate sampling bias. The survey was conducted at a local skate park. Teenagers at a skate park may have different preferences (e.g., more likely to prefer skateboarding brands or specific sneaker styles) than the general teenage population. This is a clear example of convenience sampling and selection bias. Step 4: Conclusion. Because the sample is both very small and highly biased, the claim that 80% of all teenagers prefer that brand is not valid. The study's results cannot be generalized to the broader teenage population.

2. Mere is reading a news report claiming that a new study shows 85% of high school students prefer online learning over traditional classrooms. The study surveyed 20 students from a single advanced mathematics class at a private school that has fully integrated online learning tools. Evaluate the validity of this claim, identifying potential sources of bias, issues with sample size, and whether the conclusion is justified. Answer: The claim is not valid due to severe selection bias (only one advanced math class at a private school with online tools), insufficient sample size (20 students), and lack of generalizability to all high school students. Solution: Identify the claim. The report claims 85% of high school students prefer online learning, based on a survey of 20 students from one advanced math class at a private school with integrated online tools.
   Full step-by-step solution

   Step 1: Identify the claim. The report claims 85% of high school students prefer online learning, based on a survey of 20 students from one advanced math class at a private school with integrated online tools. Step 2: Evaluate sample size. A sample of 20 students is very small for making a claim about millions of high school students. With such a small sample, the margin of error is large, and the results are not statistically reliable. Step 3: Identify bias. The sample suffers from severe selection bias: (a) Only one class at one school, not a random sample of schools. (b) The class is an advanced mathematics class, which may attract students who are more comfortable with technology. (c) The private school has fully integrated online learning tools, meaning these students have a positive experience with online learning that may not be typical. This creates a biased sample that does not represent all high school students. Step 4: Assess generalizability. The claim is about 'all high school students,' but the sample is from a very specific, non-representative group. The results cannot be generalized to students in public schools, different socioeconomic backgrounds, or those without access to online tools. Step 5: Conclusion. The claim is not valid. The small sample size and extreme selection bias mean the reported 85% likely reflects only the preferences of that particular group, not the broader population. A valid study would require a much larger, randomly selected sample from diverse schools and backgrounds.

3. A study claims that 61% of students at a high school prefer online learning. The study surveyed 16 students from an advanced mathematics class. Evaluate the validity of this claim. Answer: The claim is likely invalid due to a biased sample and small sample size. Solution: Identify the sample. The study surveyed 16 students from an advanced mathematics class. This is a convenience sample, not a random sample of all students.
   Full step-by-step solution

   Step 1: Identify the sample. The study surveyed 16 students from an advanced mathematics class. This is a convenience sample, not a random sample of all students. Step 2: Consider bias. Students in an advanced math class may have different preferences than the general student body (e.g., they might be more comfortable with technology or have different academic attitudes). This introduces selection bias. Step 3: Evaluate sample size. A sample of 16 students is very small for making a claim about an entire high school population. With such a small sample, the margin of error is large, and the results are not reliable. Step 4: Conclusion. The claim that 61% of all students prefer online learning is not valid because the sample is biased (only advanced math students) and too small (n=16) to generalize to the entire school. The study should use a larger, random sample from all grade levels and classes. The answer is: The claim is likely invalid due to a biased sample and small sample size.

4. log₃(27) + cos(π/3) = ? Answer: 3.5 Solution: Evaluate log₃(27). We need to find the exponent such that 3^x = 27. Since 3^3 = 27, log₃(27) = 3.
   Full step-by-step solution

   Step 1: Evaluate log₃(27). We need to find the exponent such that 3^x = 27. Since 3^3 = 27, log₃(27) = 3. Step 2: Evaluate cos(π/3). π/3 radians is 60 degrees. The cosine of 60 degrees is 1/2. Step 3: Add the results: 3 + 1/2 = 3.5 The answer is 3.5.

5. log₂(16) + cos(π) = ? Answer: 3 Solution: Evaluate log₂(16). This asks: 2 to what power equals 16? Since 2^4 = 16, log₂(16) = 4.
   Full step-by-step solution

   Step 1: Evaluate log₂(16). This asks: 2 to what power equals 16? Since 2^4 = 16, log₂(16) = 4. Step 2: Evaluate cos(π). On the unit circle, at angle π radians (180°), the x-coordinate is -1. So cos(π) = -1. Step 3: Add the results: 4 + (-1) = 3. The answer is 3.

6. log₂(32) + 2sin(π/6) = ? Answer: 6 Solution: Evaluate log₂(32). Since 2^5 = 32, log₂(32) = 5. Evaluate 2sin(π/6).
   Full step-by-step solution

   Step 1: Evaluate log₂(32). Since 2^5 = 32, log₂(32) = 5. Step 2: Evaluate 2sin(π/6). Since sin(π/6) = 1/2, 2 * (1/2) = 1. Step 3: Add the results: 5 + 1 = 6. The answer is 6.

7. A study claims that 67% of students prefer online learning. The study surveyed 25 students from a single advanced mathematics class. Evaluate the validity of this claim. Answer: The claim is likely invalid due to small sample size and selection bias. Solution: Identify the sample size: 25 students. This is a very small sample for making a general claim about all students. Identify the sampling method: The survey was conducted in a single advanced mathematics class.
   Full step-by-step solution

   Step 1: Identify the sample size: 25 students. This is a very small sample for making a general claim about all students. Step 2: Identify the sampling method: The survey was conducted in a single advanced mathematics class. This introduces selection bias because students in an advanced math class may have different preferences than the general student population (e.g., they may be more comfortable with technology). Step 3: Evaluate the claim: The claim that 67% of all students prefer online learning is not supported by this data. The sample is too small (n=25) to generalize to a large population, and the sample is not representative due to the specific class chosen. Therefore, the claim is likely invalid. The answer is: The claim is likely invalid due to small sample size and selection bias.

8. A study claims that 73% of students at a high school prefer online learning. The study surveyed 25 students from an advanced mathematics class. Evaluate the validity of this claim. Answer: The claim is likely invalid due to biased sampling and small sample size. Solution: Identify the sample size. The study surveyed only 25 students, which is a very small sample for a high school with potentially hundreds or thousands of students.
   Full step-by-step solution

   Step 1: Identify the sample size. The study surveyed only 25 students, which is a very small sample for a high school with potentially hundreds or thousands of students. A small sample size increases the margin of error and reduces reliability. Step 2: Identify potential bias. The sample was taken from an advanced mathematics class. Students in an advanced math class may have different preferences and characteristics compared to the general student population (e.g., they may be more academically inclined, have different study habits, or different access to technology). This introduces selection bias. Step 3: Evaluate the claim. The claim that 73% of all students prefer online learning is based on a non-representative sample. The results cannot be generalized to the entire school. The claim is invalid because the sample is biased and too small to support such a broad conclusion. The answer is: The claim is likely invalid due to biased sampling and small sample size.