Statistical Reports

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Answer Key & Explanations

Statistical Reports · Grade 11 · Worksheet 1

1. A right triangle is inscribed in a unit circle such that its hypotenuse is the circle's diameter. If one acute angle measures 30°, and the triangle's vertices are at coordinates (0,0), (1,0), and (x,y) on the circle, what is the exact area of the triangle? Answer: √3/4 Solution: When a right triangle is inscribed in a circle, Thale's theorem states that the hypotenuse is the diameter. For a unit circle, the diameter is 2.
   Full step-by-step solution

   When a right triangle is inscribed in a circle, Thale's theorem states that the hypotenuse is the diameter. For a unit circle, the diameter is 2. The sides opposite the acute angles can be found using sine and cosine functions. The area of any right triangle is half the product of its two shorter sides.

2. log₂(16) + 3cos(0) = ? Answer: 7 Solution: Evaluate log₂(16). Since 2^4 = 16, log₂(16) = 4. Evaluate 3cos(0).
   Full step-by-step solution

   Step 1: Evaluate log₂(16). Since 2^4 = 16, log₂(16) = 4. Step 2: Evaluate 3cos(0). cos(0) = 1, so 3 × 1 = 3. Step 3: Add the results: 4 + 3 = 7. The answer is 7.

3. Noah is evaluating a news report claiming that a new study shows 90% of teenagers prefer Brand X sneakers. The study surveyed 20 teenagers outside a Brand X store. Identify two significant issues with this study's methodology that could invalidate the claim, and explain how each issue affects the reliability of the conclusion. Answer: The two issues are: (1) Sampling bias: surveying only outside a Brand X store selects a pre-existing population of Brand X customers or enthusiasts, not a representative sample of all teenagers. (2) Small sample size: 20 teenagers is too small to generalize to the entire teenage population, leading to high variability and low statistical power. Both issues mean the claim is not valid for all teenagers. Solution: Identify the first issue - Sampling bias. The survey was conducted outside a Brand X store. A sample of only 20 teenagers is extremely small for making claims about a population of millions.
   Full step-by-step solution

   Step 1: Identify the first issue - Sampling bias. The survey was conducted outside a Brand X store. People who visit a Brand X store are more likely to already prefer Brand X or be interested in it. This means the sample is not a random or representative sample of all teenagers; it is biased toward Brand X supporters. This overestimates the true proportion of teenagers who prefer Brand X. Step 2: Identify the second issue - Small sample size. A sample of only 20 teenagers is extremely small for making claims about a population of millions. With such a small sample, the margin of error is very large, and the results are highly susceptible to random chance. For example, if even two or three respondents had different preferences, the percentage would change dramatically (e.g., from 90% to 85% or 95%). This makes the claim unreliable. Step 3: Conclusion. The combination of sampling bias and small sample size means the study's claim that 90% of teenagers prefer Brand X is not valid. The methodology fails to provide credible evidence for the conclusion.

4. log₂(8) + sin(π/2) = ? Answer: 4 Solution: Evaluate log₂(8) We ask: "2 raised to what power equals 8?" Since 2³ = 8, log₂(8) = 3. Evaluate sin(π/2) π/2 radians is 90 degrees. sin(90°) = 1.
   Full step-by-step solution

   Let's solve step by step. Step 1: Evaluate log₂(8) We ask: "2 raised to what power equals 8?" Since 2³ = 8, log₂(8) = 3. Step 2: Evaluate sin(π/2) π/2 radians is 90 degrees. sin(90°) = 1. So sin(π/2) = 1. Step 3: Add the results log₂(8) + sin(π/2) = 3 + 1 = 4. Final answer: 4

5. log₂(32) + sin(π/2) = ? Answer: 6 Solution: Evaluate log₂(32) We need to find the exponent such that 2 raised to that power equals 32. 32 = 2 × 2 × 2 × 2 × 2 = 2⁵ So log₂(32) = 5. Evaluate sin(π/2) π/2 radians is 90 degrees.
   Full step-by-step solution

   Let's solve step by step. Step 1: Evaluate log₂(32) We need to find the exponent such that 2 raised to that power equals 32. 32 = 2 × 2 × 2 × 2 × 2 = 2⁵ So log₂(32) = 5. Step 2: Evaluate sin(π/2) π/2 radians is 90 degrees. sin(90°) = 1. Step 3: Add the two results log₂(32) + sin(π/2) = 5 + 1 = 6. Final answer: 6

6. A study claims that 67% of students at a high school prefer online learning. The study surveyed 25 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: 25 students from the computer club. Step 2: Recognize bias: Computer club members are more likely to prefer online learning than the general student body, introducing selection bias.
   Full step-by-step solution

   Step 1: Identify the sample: 25 students from the computer club. Step 2: Recognize bias: Computer club members are more likely to prefer online learning than the general student body, introducing selection bias. Step 3: Evaluate sample size: 25 is small relative to a typical high school population (e.g., 1000+ students), making the result less reliable. Step 4: Conclusion: The claim is invalid because the sample is not random and does not represent all students. The answer is: The claim is likely invalid due to sampling bias; the sample is not representative of the entire student population.

7. log₃(81) + 2sin(π/6) = ? Answer: 5 Solution: Evaluate log₃(81). Since 3^4 = 81, log₃(81) = 4. Evaluate sin(π/6).
   Full step-by-step solution

   Step 1: Evaluate log₃(81). Since 3^4 = 81, log₃(81) = 4. Step 2: Evaluate sin(π/6). π/6 radians equals 30°, and sin(30°) = 1/2. Step 3: Multiply: 2 × sin(π/6) = 2 × 1/2 = 1. Step 4: Add the results: 4 + 1 = 5. The answer is 5.

8. Hana reads a news report claiming that '90% of dentists recommend Brand X toothpaste.' The report states that the survey was conducted by calling 15 dentists at a dental conference, and 14 of them recommended Brand X. Evaluate the validity of this claim, identifying any potential sources of bias or issues with the sample. Answer: The claim is likely invalid due to small sample size, selection bias (dentists at a conference may not represent all dentists), and potential sponsorship bias. Solution: Identify the claim: '90% of dentists recommend Brand X toothpaste.' The evidence is that 14 out of 15 dentists at a dental conference recommended it. Check sample size: 15 is a very small sample.
   Full step-by-step solution

   Step 1: Identify the claim: '90% of dentists recommend Brand X toothpaste.' The evidence is that 14 out of 15 dentists at a dental conference recommended it. Step 2: Check sample size: 15 is a very small sample. With such a small sample, the margin of error is large, and the result is not reliable for making a general claim about all dentists. Step 3: Check for selection bias: The sample was taken at a dental conference. Dentists who attend conferences may not represent all dentists (e.g., they might be more likely to be influenced by new products or sponsors). Also, the survey might have been conducted by the company that makes Brand X, leading to sponsorship bias. Step 4: Consider voluntary response bias: Only those who chose to respond were counted. The report does not say how many dentists were approached or if any refused. Step 5: Conclusion: The claim is not valid because of small sample size, selection bias, and potential sponsorship bias. The reported 90% is not representative of all dentists. Final answer: The claim is likely invalid due to small sample size, selection bias (dentists at a conference may not represent all dentists), and potential sponsorship bias.