Question
wo 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
Let's analyze the statements based on the information provided about the two samples:
1. **The second sample is random.**
- This statement is **partially true**. The second sample, which surveyed every sixth person exiting the grocery store, could provide a systematic sample, but it is not necessarily random as it is based on a specific selection criterion (every sixth person).
2. **The second sample is likely to be more representative of the population.**
- This statement could be **true** if the grocery store customers are more representative of the entire community than bank customers. However, without additional context about whether the people at the grocery store differ significantly from those exiting the bank, we cannot conclusively say this is true. It may or may not be representative.
3. **The first sample is likely to be more representative of the population.**
- This is **unlikely**. Since the first sample is based on people exiting a bank, which may not represent the listening habits of the broader community, it’s reasonable to conclude that it may not be as representative as the second sample.
4. **The second sample will give a better representation because it is larger.**
- This statement is **not necessarily true**. While a larger sample size can lead to a more accurate estimate, it depends on whether the sample is representative of the population. If the larger sample (second sample) is not as representative, it could yield misleading results despite its size.
5. **The first sample will give a better representation because it is smaller.**
- This statement is **false**. A smaller sample does not guarantee better representation. In many cases, a larger sample can provide better results if it is also representative of the entire population.
6. **Community members are more likely to listen to the radio station an average of 12 hours a month than 20 hours a month.**
- This statement is an interpretation of the results from the samples. Based on the mean hours reported (12 in the second sample and 20 in the first), one might conclude that, according to these samples, community members listen to the radio more on average for the second sample, but we cannot definitively state this as a fact without knowing the true distribution in the population.
In summary, the most applicable statements are:
- The second sample is random (partially true).
- Community members are more likely to listen to the radio station an average of 12 hours a month than 20 hours a month (debatable without further context but suggested by the data).
The other statements are either misleading, likely false, or not definitively true.
1. **The second sample is random.**
- This statement is **partially true**. The second sample, which surveyed every sixth person exiting the grocery store, could provide a systematic sample, but it is not necessarily random as it is based on a specific selection criterion (every sixth person).
2. **The second sample is likely to be more representative of the population.**
- This statement could be **true** if the grocery store customers are more representative of the entire community than bank customers. However, without additional context about whether the people at the grocery store differ significantly from those exiting the bank, we cannot conclusively say this is true. It may or may not be representative.
3. **The first sample is likely to be more representative of the population.**
- This is **unlikely**. Since the first sample is based on people exiting a bank, which may not represent the listening habits of the broader community, it’s reasonable to conclude that it may not be as representative as the second sample.
4. **The second sample will give a better representation because it is larger.**
- This statement is **not necessarily true**. While a larger sample size can lead to a more accurate estimate, it depends on whether the sample is representative of the population. If the larger sample (second sample) is not as representative, it could yield misleading results despite its size.
5. **The first sample will give a better representation because it is smaller.**
- This statement is **false**. A smaller sample does not guarantee better representation. In many cases, a larger sample can provide better results if it is also representative of the entire population.
6. **Community members are more likely to listen to the radio station an average of 12 hours a month than 20 hours a month.**
- This statement is an interpretation of the results from the samples. Based on the mean hours reported (12 in the second sample and 20 in the first), one might conclude that, according to these samples, community members listen to the radio more on average for the second sample, but we cannot definitively state this as a fact without knowing the true distribution in the population.
In summary, the most applicable statements are:
- The second sample is random (partially true).
- Community members are more likely to listen to the radio station an average of 12 hours a month than 20 hours a month (debatable without further context but suggested by the data).
The other statements are either misleading, likely false, or not definitively true.