Question
Two random samples were taken to determine how often people in a community listen to the local radio station each month.
The first sample surveyed 15 people as they exited the bank on Main Street. The first sample found that the mean number of hours they listened to the radio each month was 20.
The second sample surveyed every sixth person as they exited the only local grocery store until 100 people were surveyed. The second sample found that the mean number of hours they listened to the radio each month was 12.
Which statements are true? Check all that apply.
The second sample is random.
The second sample is likely to be more representative of the population.
The first sample is likely to be more representative of the population.
The second sample will give a better representation because is it larger.
The first sample will give a better representation because it is smaller.
Community members are more likely to listen to the radio station an average of 12 hours a month than 20 hours a month.
The first sample surveyed 15 people as they exited the bank on Main Street. The first sample found that the mean number of hours they listened to the radio each month was 20.
The second sample surveyed every sixth person as they exited the only local grocery store until 100 people were surveyed. The second sample found that the mean number of hours they listened to the radio each month was 12.
Which statements are true? Check all that apply.
The second sample is random.
The second sample is likely to be more representative of the population.
The first sample is likely to be more representative of the population.
The second sample will give a better representation because is it larger.
The first sample will give a better representation because it is smaller.
Community members are more likely to listen to the radio station an average of 12 hours a month than 20 hours a month.
Answers
GPT-4o mini
To evaluate the statements regarding the two samples, let's review some core concepts about sampling and representation.
1. **The second sample is random.**
- **False.** The second sample was taken by surveying every sixth person exiting a grocery store, which might introduce some bias based on who shops at that time or location. While systematic sampling can be random, it does not guarantee randomness in the same way a true random sample does.
2. **The second sample is likely to be more representative of the population.**
- **False.** Although the second sample has a larger number of participants (100 vs. 15), it may not be more representative if there are inherent biases in the sample selection (e.g., people shopping at a grocery store may not represent the entire community).
3. **The first sample is likely to be more representative of the population.**
- **False.** The first sample, while smaller, was also taken in a specific location (the bank) and may not be representative of the entire community, similar to the second sample.
4. **The second sample will give a better representation because it is larger.**
- **False.** While larger samples can provide more reliable estimates generally, a larger sample does not guarantee better representativeness if it is taken from a biased group.
5. **The first sample will give a better representation because it is smaller.**
- **False.** Size alone doesn’t determine representativeness. A small sample can be biased just as easily as a large one.
6. **Community members are more likely to listen to the radio station an average of 12 hours a month than 20 hours a month.**
- **Not necessarily true.** This statement assumes that the means (12 hours vs. 20 hours) reflect true listening behavior in the community; however, without evidence or additional context regarding the sampling methods and population, we cannot conclude that people are "more likely" to listen for fewer hours based on these two samples alone.
Overall, none of the statements are clearly true based on the information provided. The representativity of samples is influenced by how they are obtained, not merely by their sizes or means.
1. **The second sample is random.**
- **False.** The second sample was taken by surveying every sixth person exiting a grocery store, which might introduce some bias based on who shops at that time or location. While systematic sampling can be random, it does not guarantee randomness in the same way a true random sample does.
2. **The second sample is likely to be more representative of the population.**
- **False.** Although the second sample has a larger number of participants (100 vs. 15), it may not be more representative if there are inherent biases in the sample selection (e.g., people shopping at a grocery store may not represent the entire community).
3. **The first sample is likely to be more representative of the population.**
- **False.** The first sample, while smaller, was also taken in a specific location (the bank) and may not be representative of the entire community, similar to the second sample.
4. **The second sample will give a better representation because it is larger.**
- **False.** While larger samples can provide more reliable estimates generally, a larger sample does not guarantee better representativeness if it is taken from a biased group.
5. **The first sample will give a better representation because it is smaller.**
- **False.** Size alone doesn’t determine representativeness. A small sample can be biased just as easily as a large one.
6. **Community members are more likely to listen to the radio station an average of 12 hours a month than 20 hours a month.**
- **Not necessarily true.** This statement assumes that the means (12 hours vs. 20 hours) reflect true listening behavior in the community; however, without evidence or additional context regarding the sampling methods and population, we cannot conclude that people are "more likely" to listen for fewer hours based on these two samples alone.
Overall, none of the statements are clearly true based on the information provided. The representativity of samples is influenced by how they are obtained, not merely by their sizes or means.