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You want to demonstrate that the mean of the sampling distribution is approximately equal to the population proportion. You hav...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 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 not give a better approximation of population proportion, as it has more variability 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. 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 400 will give a better approximation of population proportion, as it is symmetrical and the mean falls close to the middle of the distribution.
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GPT-4o mini
The correct statement among the options provided 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."
### Explanation:
1. **Sample Size and Variability**: A larger sample size, such as 400, tends to have less variability in the sample proportion, leading to a more accurate estimate of the population proportion compared to a smaller sample size, such as 75.
2. **Sampling Distribution**: The sampling distribution of the sample proportion approaches a normal distribution as the sample size increases, according to the Central Limit Theorem. A sample size of 400 is more likely to yield a distribution that is symmetrical and centered around the true population proportion.
3. **Accuracy**: Larger sample sizes generally provide better approximations of population parameters due to reduced sampling error.
In summary, with a larger sample size of 400, both the variability is reduced and the estimate is more likely to be symmetric, making it a better approximation of the population proportion than a smaller sample size of 75.
"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."
### Explanation:
1. **Sample Size and Variability**: A larger sample size, such as 400, tends to have less variability in the sample proportion, leading to a more accurate estimate of the population proportion compared to a smaller sample size, such as 75.
2. **Sampling Distribution**: The sampling distribution of the sample proportion approaches a normal distribution as the sample size increases, according to the Central Limit Theorem. A sample size of 400 is more likely to yield a distribution that is symmetrical and centered around the true population proportion.
3. **Accuracy**: Larger sample sizes generally provide better approximations of population parameters due to reduced sampling error.
In summary, with a larger sample size of 400, both the variability is reduced and the estimate is more likely to be symmetric, making it a better approximation of the population proportion than a smaller sample size of 75.