Question
it is important to try to determine if generalizations are valid. We can consider the following two things:
what’s a 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?
Choose whether these are valid or invalid
The principal at a middle school wants to know the average number of club students participate in. She uses a random number generator from 0 to 500 representing all students. She chooses 100 numbers at random and surveys the students. She concludes the students participate on average in two clubs each
George wants to know how many students in his school are going to the basketball game. George asked five of his friends, and all of them say they are going. He concludes that all students will attend the basketball game.
Wants to determine the most popular sport in his school of 500 students. He put 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.
what’s a 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?
Choose whether these are valid or invalid
The principal at a middle school wants to know the average number of club students participate in. She uses a random number generator from 0 to 500 representing all students. She chooses 100 numbers at random and surveys the students. She concludes the students participate on average in two clubs each
George wants to know how many students in his school are going to the basketball game. George asked five of his friends, and all of them say they are going. He concludes that all students will attend the basketball game.
Wants to determine the most popular sport in his school of 500 students. He put 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.
Answers
GPT-4o mini
Let's evaluate each scenario based on the two criteria mentioned: whether the sample is random and whether the sample size is large enough.
1. **Middle School Principal:**
- **Sample Randomness:** The principal uses a random number generator to select 100 students from a total of 500. Since each student has an equal chance of being selected, the randomness criterion is **valid**.
- **Sample Size:** A sample size of 100 from a population of 500 is reasonably large, providing a good representation. Thus, this criterion is also **valid**.
- **Conclusion Validity:** Given the valid sample randomness and size, the conclusion about the average number of clubs students participate in is **valid**.
2. **George Asking Friends:**
- **Sample Randomness:** George asks only five of his friends, who may not be representative of the entire student population. This sample is not random or representative, making the randomness criterion **invalid**.
- **Sample Size:** While the sample size of five is small, the more critical issue is the lack of randomness. Thus, the sample size is **insufficient** (invalid) due to lack of randomness.
- **Conclusion Validity:** Given these points, George's conclusion that all students will attend the basketball game is **invalid**.
3. **Person Determining Most Popular Sport:**
- **Sample Randomness:** The person puts all students' names in a box and randomly draws 100 of them. Since every student had an equal chance to be selected, the randomness criterion is **valid**.
- **Sample Size:** A sample size of 100 from a population of 500 is adequate for determining popularity, so this criterion is also **valid**.
- **Conclusion Validity:** The conclusion about soccer being the most popular sport is therefore **valid** based on the valid sample randomness and size.
### Summary:
1. **Middle School Principal's Study:** Valid
2. **George's Conclusion:** Invalid
3. **Sport Popularity Study:** Valid
1. **Middle School Principal:**
- **Sample Randomness:** The principal uses a random number generator to select 100 students from a total of 500. Since each student has an equal chance of being selected, the randomness criterion is **valid**.
- **Sample Size:** A sample size of 100 from a population of 500 is reasonably large, providing a good representation. Thus, this criterion is also **valid**.
- **Conclusion Validity:** Given the valid sample randomness and size, the conclusion about the average number of clubs students participate in is **valid**.
2. **George Asking Friends:**
- **Sample Randomness:** George asks only five of his friends, who may not be representative of the entire student population. This sample is not random or representative, making the randomness criterion **invalid**.
- **Sample Size:** While the sample size of five is small, the more critical issue is the lack of randomness. Thus, the sample size is **insufficient** (invalid) due to lack of randomness.
- **Conclusion Validity:** Given these points, George's conclusion that all students will attend the basketball game is **invalid**.
3. **Person Determining Most Popular Sport:**
- **Sample Randomness:** The person puts all students' names in a box and randomly draws 100 of them. Since every student had an equal chance to be selected, the randomness criterion is **valid**.
- **Sample Size:** A sample size of 100 from a population of 500 is adequate for determining popularity, so this criterion is also **valid**.
- **Conclusion Validity:** The conclusion about soccer being the most popular sport is therefore **valid** based on the valid sample randomness and size.
### Summary:
1. **Middle School Principal's Study:** Valid
2. **George's Conclusion:** Invalid
3. **Sport Popularity Study:** Valid