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It is important to determine if generalizations are valid. We can consider the following two things: Was the sample random? Did all members of the population have an equal chance of being selected? Is the sample size large enough based on the population size? (3 points) Valid Invalid The principal at a middle school wants to know the average number of clubs students participate in. She uses a random number generator from 0-500 representing all students. She chooses 100 numbers at random and surveys these students. She concludes that students participate on average in 2 clubs each. Jorge wants to know how many students in his school are going to the basketball game. Jorge asks 5 of his friends, and all of them say they are going. He concludes that all students will attend the basketball game. Nick wants to determine the most popular sport in his school of 500 students. He puts the names of all the students in the box, draws 100 of them, and surveys them all. He determines that soccer is the most popular sport in the school.

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Let’s evaluate the validity of the generalizations made by each individual based on the criteria provided.

  1. Principal's Survey

    • Was the sample random? Yes, she used a random number generator to select students.
    • Did all members of the population have an equal chance of being selected? Yes, since all students were represented by numbers from 0-500.
    • Is the sample size large enough based on the population size? A sample of 100 is a sufficient proportion of a population of 500.
    • Conclusion: Valid.

  2. Jorge's Survey

    • Was the sample random? No, he only asked his 5 friends, which is biased.
    • Did all members of the population have an equal chance of being selected? No, only his friends were asked.
    • Is the sample size large enough based on the population size? No, a sample of 5 out of 500 is too small and not representative.
    • Conclusion: Invalid.

  3. Nick's Survey

    • Was the sample random? Yes, he drew names from a box which included all students.
    • Did all members of the population have an equal chance of being selected? Yes, each student had an equal chance of being drawn.
    • Is the sample size large enough based on the population size? Yes, 100 is a good sample size from a population of 500.
    • Conclusion: Valid.

Summary:

  • Principal's Survey: Valid
  • Jorge's Survey: Invalid
  • Nick's Survey: Valid