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Relative Frequencies

Grade 11 · Statistics · Worksheet 2

  1. Noah is analyzing a survey of 250 residents in a coastal town to study support for a new conservation policy. The survey found that 85 residents strongly support the policy. What is the relative frequency of residents who strongly support the policy, and what does this relative frequency represent in the context of the survey? Answer: ______________
  2. Aroha is analyzing survey data from 175 students at her school to understand preferences for after-school activities. She finds that 63 students prefer sports over arts and music. Based on this relative frequency, if the entire school has 875 students, how many students would be expected to prefer sports? Answer: ______________
  3. In a survey of 240 Grade 11 students, Noah found that 156 students prefer studying in the evening. What is the relative frequency of students who prefer evening study, and what does this value mean in the context of the survey? Answer: ______________
  4. In a survey of 120 Grade 11 students, Hana found that 48 students preferred studying mathematics in the morning. What is the relative frequency of students who prefer morning mathematics study, and what does this value represent in this context? Answer: ______________
  5. Isabella is analyzing the results of a survey at her school about participation in extracurricular activities. She collected data from 320 students and found that 84 of them participate in at least two extracurricular activities. What is the relative frequency of students who participate in at least two extracurricular activities? Interpret this relative frequency in the context of the school's student body of 1,200 students. Answer: ______________
  6. Emma is analyzing the results of a survey on social media usage among high school students. She finds that the relative frequency of students who spend more than 3 hours per day on social media is 0.35. If the school has a total of 750 students, how many students would she expect to spend more than 3 hours per day on social media? Answer: ______________
  7. A right circular cone has a height of 12 cm and a base radius of 5 cm. A plane cuts the cone parallel to its base, creating a smaller cone at the top with height 4 cm. What is the volume of the frustum (the remaining portion) between the two parallel planes? Answer: ______________
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Answer Key & Explanations

Relative Frequencies · Grade 11 · Worksheet 2

  1. Noah is analyzing a survey of 250 residents in a coastal town to study support for a new conservation policy. The survey found that 85 residents strongly support the policy. What is the relative frequency of residents who strongly support the policy, and what does this relative frequency represent in the context of the survey? Answer: 0.34 Solution: Identify the number of favorable outcomes: 85 residents strongly support the policy. Identify the total number of observations: 250 residents surveyed. Calculate relative frequency: 85 / 250.
    Full step-by-step solution

    Step 1: Identify the number of favorable outcomes: 85 residents strongly support the policy. Step 2: Identify the total number of observations: 250 residents surveyed. Step 3: Calculate relative frequency: 85 / 250. Step 4: Simplify the fraction: 85 / 250 = 17 / 50. Step 5: Convert to decimal: 17 / 50 = 0.34. Step 6: Interpretation: The relative frequency of 0.34 means that, based on the survey data, approximately 34% of the residents in the town strongly support the new conservation policy. This also represents the probability that a randomly selected resident from the survey strongly supports the policy. The answer is 0.34.

  2. Aroha is analyzing survey data from 175 students at her school to understand preferences for after-school activities. She finds that 63 students prefer sports over arts and music. Based on this relative frequency, if the entire school has 875 students, how many students would be expected to prefer sports? Answer: 315 Solution: Calculate the relative frequency from the sample. Relative frequency = Number of students who prefer sports / Total sample size Relative frequency = 63 / 175 Simplify the fraction.
    Full step-by-step solution

    Step 1: Calculate the relative frequency from the sample. Relative frequency = Number of students who prefer sports / Total sample size Relative frequency = 63 / 175 Step 2: Simplify the fraction. 63/175 = 9/25 (since both 63 and 175 are divisible by 7) Step 3: Apply this relative frequency to the larger population. Expected number = Relative frequency x Total school population Expected number = (9/25) x 875 Step 4: Perform the calculation. (9/25) x 875 = 9 x (875/25) = 9 x 35 = 315 The expected number of students who prefer sports is 315.

  3. In a survey of 240 Grade 11 students, Noah found that 156 students prefer studying in the evening. What is the relative frequency of students who prefer evening study, and what does this value mean in the context of the survey? Answer: 0.65 Solution: Identify the number of students who prefer evening study: 156. Identify the total number of students surveyed: 240. Calculate the relative frequency: 156 ÷ 240.
    Full step-by-step solution

    Step 1: Identify the number of students who prefer evening study: 156. Step 2: Identify the total number of students surveyed: 240. Step 3: Calculate the relative frequency: 156 ÷ 240. Step 4: Perform the division: 156 ÷ 240 = 0.65. Step 5: Interpret the meaning: The relative frequency of 0.65 means that 65% of the surveyed Grade 11 students prefer studying in the evening. This indicates that a majority of the students in this sample have a preference for evening study, or equivalently, for every 100 students surveyed, about 65 prefer studying in the evening.

  4. In a survey of 120 Grade 11 students, Hana found that 48 students preferred studying mathematics in the morning. What is the relative frequency of students who prefer morning mathematics study, and what does this value represent in this context? Answer: 0.4 Solution: Identify the number of students who prefer morning mathematics study: 48 students Identify the total number of students surveyed: 120 students Calculate the relative frequency: 48 ÷ 120 = 0.4 Interpret the result: A relative frequency of 0.4 means that 40% of the surveyed Grade 11 students…
    Full step-by-step solution

    Step 1: Identify the number of students who prefer morning mathematics study: 48 students Step 2: Identify the total number of students surveyed: 120 students Step 3: Calculate the relative frequency: 48 ÷ 120 = 0.4 Step 4: Interpret the result: A relative frequency of 0.4 means that 40% of the surveyed Grade 11 students prefer studying mathematics in the morning.

  5. Isabella is analyzing the results of a survey at her school about participation in extracurricular activities. She collected data from 320 students and found that 84 of them participate in at least two extracurricular activities. What is the relative frequency of students who participate in at least two extracurricular activities? Interpret this relative frequency in the context of the school's student body of 1,200 students. Answer: 0.2625 Solution: Identify the number of students with the characteristic: 84 students participate in at least two extracurricular activities. Identify the total sample size: 320 students. Calculate the relative frequency: 84 / 320.
    Full step-by-step solution

    Step 1: Identify the number of students with the characteristic: 84 students participate in at least two extracurricular activities. Step 2: Identify the total sample size: 320 students. Step 3: Calculate the relative frequency: 84 / 320. Step 4: Simplify the fraction: divide numerator and denominator by 4 to get 21/80. Step 5: Convert to decimal: 21/80 = 0.2625. Step 6: Interpret: The relative frequency 0.2625 means that about 26.25% of the students in the sample participate in at least two extracurricular activities. For the entire school of 1,200 students, we would expect approximately 0.2625 * 1200 = 315 students to participate in at least two extracurricular activities, assuming the sample is representative. The answer is 0.2625.

  6. Emma is analyzing the results of a survey on social media usage among high school students. She finds that the relative frequency of students who spend more than 3 hours per day on social media is 0.35. If the school has a total of 750 students, how many students would she expect to spend more than 3 hours per day on social media? Answer: 262.5 Solution: Identify the relative frequency of students who spend more than 3 hours per day on social media: 0.35. Identify the total number of students at the school: 750.
    Full step-by-step solution

    Step 1: Identify the relative frequency of students who spend more than 3 hours per day on social media: 0.35. Step 2: Identify the total number of students at the school: 750. Step 3: Multiply the total number of students by the relative frequency to find the expected count: 750 * 0.35. Step 4: Perform the multiplication: 750 * 0.35 = 262.5. Step 5: The expected number of students who spend more than 3 hours per day on social media is 262.5. Since we are dealing with an expected value from a relative frequency, it is acceptable to leave the answer as a decimal. The answer is 262.5.

  7. A right circular cone has a height of 12 cm and a base radius of 5 cm. A plane cuts the cone parallel to its base, creating a smaller cone at the top with height 4 cm. What is the volume of the frustum (the remaining portion) between the two parallel planes? Answer: 728π/3 Solution: Calculate the volume of the original cone. Volume of original cone = (1/3)πr²h = (1/3)π(5)²(12) = (1/3)π(25)(12) = 100π cm³ Find the radius of the smaller cone using similar triangles.
    Full step-by-step solution

    Step 1: Calculate the volume of the original cone. Volume of original cone = (1/3)πr²h = (1/3)π(5)²(12) = (1/3)π(25)(12) = 100π cm³ Step 2: Find the radius of the smaller cone using similar triangles. The height of the smaller cone is 4 cm, and the height of the original cone is 12 cm, so the scale factor is 4/12 = 1/3. The radius of the smaller cone = (1/3) × 5 = 5/3 cm. Step 3: Calculate the volume of the smaller cone. Volume of smaller cone = (1/3)πr²h = (1/3)π(5/3)²(4) = (1/3)π(25/9)(4) = (100π)/27 cm³ Step 4: Subtract to find the volume of the frustum. Volume of frustum = Volume of original cone - Volume of smaller cone = 100π - (100π)/27 = (2700π)/27 - (100π)/27 = (2600π)/27 = 728π/3 cm³ (after simplifying) The answer is 728π/3.