Statistical Reports

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Answer Key & Explanations

Statistical Reports · Grade 11 · Worksheet 1

1. Ava is evaluating a health news article that claims a new dietary supplement improves memory. The article cites a study where 85% of participants showed improved test scores after taking the supplement for six months. However, Ava notices the study was conducted with only 12 participants, all of whom were employees of the company that produces the supplement. Additionally, the article does not mention a control group or any blinding procedure. Identify at least two major statistical flaws in this study and explain why the claim might be misleading. Answer: Small sample size (12 participants) and selection bias (all participants are employees of the supplement company) are major flaws; the lack of a control group and blinding also weaken the claim. Solution: Identify the sample size. The study used only 12 participants. A small sample size increases the risk that results are due to chance and not representative of the larger population.
   Full step-by-step solution

   Step 1: Identify the sample size. The study used only 12 participants. A small sample size increases the risk that results are due to chance and not representative of the larger population. This makes the claim less reliable. Step 2: Identify potential bias. All participants were employees of the company producing the supplement. This introduces selection bias because employees may feel pressure to report positive results, or they may already have a positive attitude toward the product. Their results are unlikely to represent the general public. Step 3: Note missing elements. The study lacks a control group, so there is no way to compare the supplement's effect against a placebo or no treatment. Without blinding, participants and researchers may unconsciously influence outcomes (placebo effect or observer bias). Step 4: Conclusion. The claim that 85% of participants improved is misleading due to small sample size, selection bias, lack of a control group, and no blinding. The study design is too flawed to support the article's claim. Answer: Small sample size (12 participants) and selection bias (all participants are employees of the supplement company) are major flaws; the lack of a control group and blinding also weaken the claim.

2. A study claims that 72% of students at a high school prefer online learning. The survey was conducted by asking 27 students in an advanced computer science class. Evaluate the validity of this claim. Answer: The claim is likely invalid due to selection bias, small sample size, and lack of randomization. Solution: Identify the sample size. The survey used 27 students, which is a small fraction of a typical high school population (often hundreds or thousands).
   Full step-by-step solution

   Step 1: Identify the sample size. The survey used 27 students, which is a small fraction of a typical high school population (often hundreds or thousands). A sample size of 27 may not be large enough to accurately represent the entire school. Step 2: Identify potential bias. The students were from an advanced computer science class. This group is likely more comfortable with technology and online learning than the average student. This creates selection bias, as the sample does not represent all students (e.g., those who struggle with technology or prefer in-person interaction). Step 3: Evaluate the claim. The claim states that 72% of all students prefer online learning, but the data comes from a biased, non-random sample. Therefore, the claim is not valid for the entire school population. A proper survey would require a random, representative sample of students from all grades and courses. Conclusion: The claim is invalid due to selection bias and insufficient sample size.

3. A study claims that 85% of teenagers prefer online learning. The survey was conducted by a website selling online courses, with a sample size of 50 teenagers who visited their site. Evaluate the validity of this claim. Answer: The claim is likely invalid due to selection bias, small sample size, and conflict of interest. Solution: Identify potential bias. The survey was conducted by a website selling online courses, which has a financial interest in promoting online learning. A sample size of 50 is relatively small for making a claim about millions of teenagers.
   Full step-by-step solution

   Step 1: Identify potential bias. The survey was conducted by a website selling online courses, which has a financial interest in promoting online learning. This creates a conflict of interest and potential bias in how the survey was designed or reported. Step 2: Examine the sample source. The sample consists of teenagers who visited the website. These teenagers already have an interest in online learning (they visited the site), so they are not representative of all teenagers. This is selection bias. Step 3: Evaluate sample size. A sample size of 50 is relatively small for making a claim about millions of teenagers. A larger, more diverse sample would be needed for a reliable conclusion. Step 4: Conclusion. Due to selection bias (sample from website visitors), conflict of interest (company selling online courses), and small sample size (50), the claim of 85% preference is not valid. The true percentage among all teenagers is likely much lower.

4. Liam is analyzing the growth of a bacterial culture in his biology lab. The culture starts with 500 bacteria and doubles every 3 hours. He needs to determine the time it will take for the culture to reach 8,000 bacteria. Write your answer as an exact expression involving logarithms. Answer: 3 * (ln(16) / ln(2)) hours Solution: The bacteria start at 500 and double every 3 hours.
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Identify the exponential growth model** The bacteria start at 500 and double every 3 hours. So, the number of bacteria after \( t \) hours is: \[ N(t) = 500 \cdot 2^{t/3} \] --- **Step 2: Set up the equation for the target population** We want \( N(t) = 8000 \): \[ 500 \cdot 2^{t/3} = 8000 \] --- **Step 3: Divide both sides by 500** \[ 2^{t/3} = \frac{8000}{500} \] \[ 2^{t/3} = 16 \] --- **Step 4: Apply logarithms** Take natural logs of both sides: \[ \ln\left(2^{t/3}\right) = \ln(16) \] Using the power rule \( \ln(a^b) = b \ln(a) \): \[ \frac{t}{3} \cdot \ln(2) = \ln(16) \] --- **Step 5: Solve for \( t \)** \[ \frac{t}{3} = \frac{\ln(16)}{\ln(2)} \] \[ t = 3 \cdot \frac{\ln(16)}{\ln(2)} \] --- **Step 6: Final answer** The exact expression is: \[ t = 3 \cdot \frac{\ln(16)}{\ln(2)} \text{ hours} \] This matches the required form.

5. Emma is analyzing the motion of a pendulum in her physics lab. The angle θ (in radians) of the pendulum from its resting position at time t seconds is modeled by the function θ(t) = 0.4 cos(3t). She needs to determine the first positive time when the pendulum reaches its maximum angular displacement from the resting position. Answer: π/3 Solution: For a cosine function of the form A cos(Bt), the maximum displacement occurs when the cosine equals 1. This happens when the argument of the cosine function is an integer multiple of 2π.
   Full step-by-step solution

   For a cosine function of the form A cos(Bt), the maximum displacement occurs when the cosine equals 1. This happens when the argument of the cosine function is an integer multiple of 2π. The first positive time occurs when the argument equals 0, but we need to consider when the pendulum first reaches maximum displacement after starting its motion.

6. A study claims that 85% of students at a high school prefer online learning. The study surveyed 20 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 claim: 85% of all high school students prefer online learning.
   Full step-by-step solution

   Step 1: Identify the claim: 85% of all high school students prefer online learning. Step 2: Examine the sample: Only 20 students were surveyed, which is a very small sample size for a large population (e.g., hundreds or thousands of students). Step 3: Identify potential bias: The sample consists entirely of computer club members. These students likely have a stronger interest in technology and may be more comfortable with online learning than the average student. This introduces selection bias. Step 4: Evaluate validity: Because the sample is small and biased, the results cannot be generalized to the entire student body. The claim is not supported by the evidence. The answer is: The claim is likely invalid due to sampling bias; the sample is not representative of the entire student population.

7. A study claims that 73% of teenagers prefer online learning over traditional classrooms. The study surveyed 40 students from a single advanced mathematics class at a private high school. Evaluate the validity of this claim. Answer: The claim is likely invalid due to significant sampling bias and insufficient sample size. Solution: Identify the claim. The claim is that 73% of all teenagers prefer online learning. Step 2: Examine the sample.
   Full step-by-step solution

   Step 1: Identify the claim. The claim is that 73% of all teenagers prefer online learning. Step 2: Examine the sample. The sample consists of 40 students from one advanced math class at a private high school. This is a convenience sample, not a random sample of all teenagers. Step 3: Identify bias. Students in an advanced math class at a private school are not representative of all teenagers. They likely have different socioeconomic backgrounds, academic interests, and access to technology. This introduces selection bias. Step 4: Evaluate sample size. A sample size of 40 is small for making a claim about millions of teenagers. The margin of error would be large, and the results may not be reliable. Step 5: Conclusion. The claim is invalid because the sample is biased (only one class at one private school) and too small to generalize to the entire population of teenagers. The study does not provide a valid statistical basis for the claim.