Question

Will \(\hat{p}\) from a random sample of size 400 tend to be closer to the actual value of the population proportion when \(p=0.4\) or when \(p=0.7 ?\) Provide an explanation for your choice.

Short answer

The problem doesn't provide enough data to make a definitive judgement. Therefore, without more information, we can't definitively say whether \(\hat{p}\) will be closer to p = 0.4 or p = 0.7. The closeness depends on the actual value of \(\hat{p}\) which we don't have.

Step-by-step solution

Understand the Law of large numbers

The Law of Large Numbers is a theory in probability and statistics that suggests that as the size of a sample increases, the mean value of the sample will get closer to the mean value of the whole population. In other words, larger samples will provide estimates that are closer to the actual population parameters.

Apply the Law of large numbers to our problem

In the context of this exercise, the \(\hat{p}\) (sample proportion) should get closer to the p (population proportion) as the sample size increases. Here, our sample size is already large (400), hence, according to the Law of Large Numbers, our sample proportion \(\hat{p}\) should be closer to our population proportion p. As we have two different population proportions (0.4 and 0.7), the question becomes: which one will \(\hat{p}\) be closer to?

Compare \(\hat{p}\) with different values of p

The law of large numbers doesn't tell us that \(\hat{p}\) will necessarily be closer to p = 0.4 or p = 0.7. It simply tells us that as sample size increases, \(\hat{p}\) gets closer to p. The comparison of how close \(\hat{p}\) is to either 0.4 or 0.7 doesn't depend on the law of large numbers but depends on the difference between \(\hat{p}\) and p. Neither value is necessarily better or worse for the sample proportion to be closer to, it depends on the specific value of \(\hat{p}\). For a given sample, we calculate \(\hat{p}\) and then compare it to the two values of p to find out which one it is closer to.