Statistical Reports

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Answer Key & Explanations

Statistical Reports · Grade 11 · Worksheet 2

1. Matiu reads a news report claiming that a new after-school tutoring program improved student grades. The report states: 'In a survey of 8 students who participated in the program, 100% reported an improvement in their math grades. Therefore, the program is effective for all students.' Evaluate the validity of this claim, identifying at least two specific issues with the study's methodology or conclusions. Answer: The claim is invalid due to small sample size, sampling bias, lack of control group, and overgeneralization. Solution: Identify the sample size. The survey used only 8 students. A sample of 8 is very small and cannot represent the entire population of students.
   Full step-by-step solution

   Step 1: Identify the sample size. The survey used only 8 students. A sample of 8 is very small and cannot represent the entire population of students. Step 2: Recognize sampling bias. The students were participants in the tutoring program, so they are likely motivated or already doing well, creating a biased sample. Step 3: Note the lack of a control group. Without comparing to students who did not participate, we cannot attribute the improvement to the program. Step 4: Identify overgeneralization. The claim extends the result to 'all students,' which is not supported by data from only 8 students. Therefore, the claim is invalid due to small sample size, sampling bias, lack of control group, and overgeneralization.

2. Liam is designing a suspension bridge and needs to model the cable shape. The cable follows a parabolic curve described by the function f(x) = 2x² - 12x + 16, where x represents the horizontal distance from the left tower in meters. Liam needs to find the minimum height of the cable above the water level. At what horizontal distance does this minimum height occur? Answer: 3 Solution: We are given the function f(x) = 2x² - 12x + 16, which is a parabola opening upwards because the coefficient of x² is positive (2 > 0). The minimum height occurs at the vertex of the parabola.
   Full step-by-step solution

   We are given the function f(x) = 2x² - 12x + 16, which is a parabola opening upwards because the coefficient of x² is positive (2 > 0). The minimum height occurs at the vertex of the parabola. Step 1: Identify the vertex formula for a parabola in the form f(x) = ax² + bx + c. The x-coordinate of the vertex is given by: x = -b / (2a) Step 2: From f(x) = 2x² - 12x + 16, we have: a = 2 b = -12 c = 16 Step 3: Substitute a and b into the vertex formula: x = -(-12) / (2 * 2) x = 12 / 4 x = 3 Step 4: Interpretation The minimum height of the cable occurs at x = 3 meters from the left tower. Thus, the horizontal distance where the minimum height occurs is 3.

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 study claims that 88% of students at a high school prefer online learning. The survey was conducted by asking 15 students in a computer science class. Evaluate the validity of this claim, identifying any potential issues with sample size, bias, or generalizability. Answer: The claim is likely invalid due to small sample size (15), selection bias (computer science students may favor online learning), and lack of random sampling; the result cannot be generalized to the entire school. Solution: Identify the sample size. The survey used only 15 students. A sample of 15 is very small for a school population, leading to high sampling variability and low reliability.
   Full step-by-step solution

   Step 1: Identify the sample size. The survey used only 15 students. A sample of 15 is very small for a school population, leading to high sampling variability and low reliability. Step 2: Identify potential bias. The survey was conducted in a computer science class. Students in this class are likely more comfortable with technology and may have a stronger preference for online learning compared to the general student body. This is selection bias. Step 3: Consider generalizability. Because the sample is not random and is drawn from a specific subgroup, the results cannot be generalized to all students at the school. The claim that 88% of all students prefer online learning is not supported by this data. Step 4: Conclusion. The study's claim is invalid due to small sample size, selection bias, and lack of random sampling. The reported percentage likely overestimates the true preference for online learning among the entire student population.

5. A study claims that 88% 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 selection bias and small sample size. Solution: Identify the claim: 88% of all students at the high school prefer online learning. Examine the sample: Only 25 students were surveyed, all from the computer club.
   Full step-by-step solution

   Step 1: Identify the claim: 88% of all students at the high school prefer online learning. Step 2: Examine the sample: Only 25 students were surveyed, all from the computer club. Step 3: Identify bias: Computer club members are more likely to be comfortable with technology and online platforms, so they are not representative of the entire student body. This is selection bias. Step 4: Consider sample size: A sample of 25 out of possibly hundreds or thousands of students is very small, leading to high margin of error and low reliability. Step 5: Conclusion: The claim is invalid because the sample is biased (only computer club members) and too small to generalize to the whole school. The reported 88% likely overestimates the true preference for online learning among all students.

6. 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.

7. 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.