Question

In the 2004 Utah election, 45% of all Utah voters voted for Initiative 1. Suppose a student decided to predict the theoretical sampling distribution of p^ for samples of size 1000. Another student decides to predict the theoretical sampling distribution of p^ only she wants to predict for samples of size 1500. How will the means of these two sampling distributions for p^ compare?

Answer 1

The means of the theoretical sampling distributions for p^ with sample sizes of 1000 and 1500 will be the same, both being 45%. The standard deviations, however, will differ, with the larger sample size of 1500 having a smaller standard deviation.

Step-by-step explanation:

The means of the theoretical sampling distributions of î²ê for samples sizes of 1000 and 1500 when predicting the percentage of voters who voted for Initiative 1 in the 2004 Utah election will be the same. This is because the mean of a sampling distribution of the sample proportion (î²ê) is equal to the population proportion (p). Since the population proportion is 45% for both sample sizes, the means of the sampling distributions will be 0.45 or 45% for both samples of size 1000 and 1500.

The difference between the two distributions lies in their standard deviations. The standard deviation of the sampling distribution is given by the square root of (pq)/n, where p is the proportion of successes, q is the proportion of failures (q=1-p), and n is the sample size. For the sample size of 1500, the standard deviation will be smaller than that for the sample size of 1000, indicating that the sampling distribution for the larger sample size will be more tightly clustered around the mean.