Question
You want to demonstrate that the mean of the sampling distribution is approximately equal to the population proportion. You have two sets of data to use: one is a sample size of 75, and the other is a sample size of 400. Which of the following statements is true?(1 point)
Responses
The sample size of 75 will give a better approximation of population proportion, as it has more variability in the data and is less symmetric.
The sample size of 75 will give a better approximation of population proportion, as it has more variability in the data and is less symmetric.
The sample size of 75 will not give a better approximation of population proportion, as it has less variability in the data and is more symmetrical.
The sample size of 75 will not give a better approximation of population proportion, as it has less variability in the data and is more symmetrical.
The sample size of 400 will give a better approximation of population proportion, as it is symmetrical and the mean falls close to the middle of the distribution.
The sample size of 400 will give a better approximation of population proportion, as it is symmetrical and the mean falls close to the middle of the distribution.
The sample size of 400 will not give a better approximation of population proportion, as it has more variability and the mean falls close to the middle of the distribution.
Responses
The sample size of 75 will give a better approximation of population proportion, as it has more variability in the data and is less symmetric.
The sample size of 75 will give a better approximation of population proportion, as it has more variability in the data and is less symmetric.
The sample size of 75 will not give a better approximation of population proportion, as it has less variability in the data and is more symmetrical.
The sample size of 75 will not give a better approximation of population proportion, as it has less variability in the data and is more symmetrical.
The sample size of 400 will give a better approximation of population proportion, as it is symmetrical and the mean falls close to the middle of the distribution.
The sample size of 400 will give a better approximation of population proportion, as it is symmetrical and the mean falls close to the middle of the distribution.
The sample size of 400 will not give a better approximation of population proportion, as it has more variability and the mean falls close to the middle of the distribution.
Answers
GPT-4o mini
The correct statement is:
**The sample size of 400 will give a better approximation of population proportion, as it is symmetrical and the mean falls close to the middle of the distribution.**
### Explanation:
1. **Central Limit Theorem**: According to the Central Limit Theorem, as the sample size increases, the sampling distribution of the sample proportion becomes more normally distributed, regardless of the shape of the population distribution, and it approaches the true population proportion. A larger sample size (in this case, 400) is likely to produce a more reliable estimate of the population proportion.
2. **Variability**: Larger samples generally have less variability and their means are more stable and closer to the true population mean (or proportion). In contrast, smaller samples (like 75) may exhibit greater variability and thus provide less accurate estimates.
3. **Shape of the Distribution**: A larger sample size leads to a more symmetrical distribution, which improves the approximation of the mean of the sampling distribution to the population proportion.
Thus, the larger sample size of 400 will provide a better approximation of the population proportion compared to the smaller sample size of 75.
**The sample size of 400 will give a better approximation of population proportion, as it is symmetrical and the mean falls close to the middle of the distribution.**
### Explanation:
1. **Central Limit Theorem**: According to the Central Limit Theorem, as the sample size increases, the sampling distribution of the sample proportion becomes more normally distributed, regardless of the shape of the population distribution, and it approaches the true population proportion. A larger sample size (in this case, 400) is likely to produce a more reliable estimate of the population proportion.
2. **Variability**: Larger samples generally have less variability and their means are more stable and closer to the true population mean (or proportion). In contrast, smaller samples (like 75) may exhibit greater variability and thus provide less accurate estimates.
3. **Shape of the Distribution**: A larger sample size leads to a more symmetrical distribution, which improves the approximation of the mean of the sampling distribution to the population proportion.
Thus, the larger sample size of 400 will provide a better approximation of the population proportion compared to the smaller sample size of 75.