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 with a sample size of 75, and the other with a sample size of 400. Which of the following statements is true?
The sample size of 75 will not give a better approximation of the population proportion, as it has less variability in the data and is more symmetrical.
The sample size of 75 will give a better approximation of the population proportion, as it has more variability in the data and is less symmetric.
The sample size of 400 will give a better approximation of the 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 the population proportion, as it has more variability and the mean falls close to the middle of the distribution.
The sample size of 75 will not give a better approximation of the population proportion, as it has less variability in the data and is more symmetrical.
The sample size of 75 will give a better approximation of the population proportion, as it has more variability in the data and is less symmetric.
The sample size of 400 will give a better approximation of the 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 the 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 the population proportion, as it is symmetrical and the mean falls close to the middle of the distribution.**
Here's why this statement is true:
1. **Larger Sample Size**: A larger sample size (in this case, 400) leads to a more reliable estimate of the population proportion. This is due to the Central Limit Theorem, which states that as the sample size increases, the sampling distribution of the sample proportion becomes approximately normally distributed, centered at the population proportion.
2. **Less Variability**: Larger samples typically have less variability in their estimates. This means that the mean of the sampling distribution will be closer to the true population proportion.
3. **Symmetry**: Larger sample sizes result in a distribution that is more symmetrical and less affected by outliers or skewness, which helps ensure that the mean of the sample proportion is a good estimate of the population proportion.
In contrast, smaller sample sizes (like 75) can exhibit more variability and may not provide as accurate or reliable an estimate of the population proportion.
**The sample size of 400 will give a better approximation of the population proportion, as it is symmetrical and the mean falls close to the middle of the distribution.**
Here's why this statement is true:
1. **Larger Sample Size**: A larger sample size (in this case, 400) leads to a more reliable estimate of the population proportion. This is due to the Central Limit Theorem, which states that as the sample size increases, the sampling distribution of the sample proportion becomes approximately normally distributed, centered at the population proportion.
2. **Less Variability**: Larger samples typically have less variability in their estimates. This means that the mean of the sampling distribution will be closer to the true population proportion.
3. **Symmetry**: Larger sample sizes result in a distribution that is more symmetrical and less affected by outliers or skewness, which helps ensure that the mean of the sample proportion is a good estimate of the population proportion.
In contrast, smaller sample sizes (like 75) can exhibit more variability and may not provide as accurate or reliable an estimate of the population proportion.