Question

A company gives an anonymous survey to its employees to see what percent of its employees are happy. The company is too large to check each response, so samples of 50 are taken, and the tendency is that three-fourths of the employees are happy. For the mean of the sampling distribution of sample proportions, answer the following questions if the sample size is doubled.

a) Increases
b) Decreases
c) Remains the same

Answer 1

The mean of the sampling distribution of sample proportions remains the same when the sample size is doubled, while the standard deviation decreases, resulting in a tighter distribution around the true population proportion.

Step-by-step explanation:

If the sample size is doubled in a survey to determine what percent of a company's employees are happy, the mean of the sampling distribution of sample proportions remains the same. This mean is the population proportion of happy employees, which is given as three-fourths or 75%. When we double the sample size, we do not change this population proportion; rather, we are likely to get a more accurate estimate of that proportion.

However, what does change when you increase the sample size is the standard deviation of the sampling distribution; it decreases. This is because the standard deviation of the sampling distribution of the sample proportion is given by the formula \(√{{pq}/{n}}\), where p is the population proportion, q is the complement of the population proportion (1-p), and n is the sample size. So as n increases, the standard deviation of the sampling distribution becomes smaller, leading to a tighter distribution around the true proportion.

In summary, doubling the sample size results in the same mean but a smaller standard deviation for the sampling distribution.