Question

If we used a much smaller sample size of n = 50, would you guess that the standard error for ˆp would be larger or smaller than when we used n = 1000? (Intuitively, it seems like more data is better than less data, and generally that is correct! The typical error when p = 0.88 and n = 50 would be larger than the error we would expect when n = 1000.)

Answer 1

The standard error for ˆp would be larger with a smaller sample size of n = 50 compared to a larger sample size of n = 1000, due to the inverse relationship between standard error and the square root of sample size.

Step-by-step explanation:

The question relates to the concept of standard error in statistics, particularly how the standard error of the proportion (often denoted as ˆp) changes with varying sample sizes. When the size of a sample decreases, the standard error increases. Specifically, if we have a small sample size of n = 50, the standard error for ˆp would be larger compared to a larger sample size of n = 1000. This is because the standard error is inversely proportional to the square root of the sample size; that is, standard error = σ / √n, where σ is the standard deviation of the population and n is the sample size. This principle can be observed in other sample sizes too. For example, increasing the sample size to n = 100 decreases the margin of error, which is tied to the standard error. More data tends to provide a more accurate estimate of the population parameter, hence reducing the standard error and margin of error.