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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 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 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 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 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.

1 answer

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.

This is because larger sample sizes tend to yield more accurate estimates of population parameters, as they typically exhibit less variability and produce a sampling distribution that is closer to the true population proportion—reflecting the Central Limit Theorem. With a larger sample size (like 400), the sampling distribution is expected to be more normal (and therefore symmetrical), resulting in a mean closer to the population proportion.