Statistical Reports

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Answer Key & Explanations

Statistical Reports · Grade 11 · Worksheet 2

1. log₂(8) + sin(π/6) = ? Answer: 3.5 Solution: Evaluate log₂(8). Since 2 raised to the power of 3 equals 8, log₂(8) = 3. Evaluate sin(π/6).
   Full step-by-step solution

   Step 1: Evaluate log₂(8). Since 2 raised to the power of 3 equals 8, log₂(8) = 3. Step 2: Evaluate sin(π/6). The sine of π/6 radians (which is 30 degrees) is 1/2. Step 3: Add the results: 3 + 1/2 = 3.5. The answer is 3.5.

2. Mere is evaluating a report from a local fitness company that claims their program helps people lose weight. The company surveyed 12 participants who completed their 8-week program. All 12 participants reported losing weight, with an average loss of 6 kg. The survey was conducted by emailing a link to a satisfaction questionnaire to all 120 people who had ever purchased the program, but only 12 responded. Evaluate the validity of the company's claim that the program is effective for weight loss, identifying any potential biases or issues with the data. Answer: The claim is not valid due to volunteer response bias, a very small sample size (12 out of 120), and lack of a control group. Solution: Identify the sample size. The company surveyed 120 people, but only 12 responded. This is a response rate of only 10%, which is very low.
   Full step-by-step solution

   Step 1: Identify the sample size. The company surveyed 120 people, but only 12 responded. This is a response rate of only 10%, which is very low. Step 2: Recognize volunteer response bias. People who had a positive experience (lost weight) are much more likely to respond to a satisfaction survey than those who had negative or neutral results. The 12 respondents are not representative of all 120 customers. Step 3: Consider the lack of a control group. Even if all 12 truly lost weight, there is no comparison group to show that the program caused the weight loss. Other factors (diet, exercise outside the program, natural variation) could be responsible. Step 4: Evaluate the small sample size. With only 12 participants, the results are not statistically reliable for making broad claims about the program's effectiveness. Step 5: Conclusion. The claim is not valid because the data suffers from severe volunteer response bias, a tiny sample size relative to the population, and no control group to establish causation. The report should not be accepted as evidence that the program works.

3. log₂(64) + cos(π/3) = ? Answer: 6.5 Solution: Evaluate log₂(64). Since 2^6 = 64, log₂(64) = 6. Evaluate cos(π/3).
   Full step-by-step solution

   Step 1: Evaluate log₂(64). Since 2^6 = 64, log₂(64) = 6. Step 2: Evaluate cos(π/3). π/3 radians is 60 degrees, and cos(60°) = 1/2 = 0.5. Step 3: Add the results: 6 + 0.5 = 6.5. The answer is 6.5.

4. A biologist is modeling the population growth of a colony of bacteria using the function P(t) = 500e^(0.03t), where t is time in hours. After how many hours will the population reach 1500 bacteria? Round your answer to the nearest tenth of an hour. Answer: 36.6 Solution: P(t) = 500 * e^(0.03t) We want to find t when P(t) = 1500. Set up the equation. 1500 = 500 * e^(0.03t) Divide both sides by 500 to isolate the exponential term.
   Full step-by-step solution

   We are given the population growth model: P(t) = 500 * e^(0.03t) We want to find t when P(t) = 1500. Step 1: Set up the equation. 1500 = 500 * e^(0.03t) Step 2: Divide both sides by 500 to isolate the exponential term. 1500 / 500 = e^(0.03t) 3 = e^(0.03t) Step 3: Take the natural logarithm of both sides to solve for the exponent. ln(3) = ln(e^(0.03t)) ln(3) = 0.03t * ln(e) Since ln(e) = 1, we have: ln(3) = 0.03t Step 4: Solve for t. t = ln(3) / 0.03 Step 5: Calculate numerical values. ln(3) ≈ 1.0986122887 t ≈ 1.0986122887 / 0.03 t ≈ 36.6204096233 Step 6: Round to the nearest tenth. t ≈ 36.6 hours Final answer: 36.6

5. 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.

6. Emma is analyzing the motion of a pendulum in her physics lab. The angle θ (in radians) of the pendulum from vertical as a function of time t (in seconds) is modeled by θ(t) = 0.4 cos(3t). She needs to determine the first positive time when the pendulum reaches its maximum angular displacement from vertical. What time should Emma calculate? Answer: π/6 Solution: For trigonometric functions modeling periodic motion, maximum displacement occurs when the trigonometric function reaches its extreme value. For cosine functions, this happens at specific points in the function's period.
   Full step-by-step solution

   For trigonometric functions modeling periodic motion, maximum displacement occurs when the trigonometric function reaches its extreme value. For cosine functions, this happens at specific points in the function's period. The general form of cosine functions and their properties can help determine when maximum values occur.

7. 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