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Chapter 9: Problem 73

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

Expert verified
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

01

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.
02

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?
03

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.

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Most popular questions from this chapter

An Associated Press article on potential violent behavior reported the results of a survey of 750 workers who were employed full time (San Luis Obispo Tribune, September 7 , 1999). Of those surveyed, 125 indicated that they were so angered by a coworker during the past year that they felt like hitting the coworker (but didn't). Assuming that it is reasonable to regard this sample as representative of the population of full-time workers, use this information to construct and interpret a \(90 \%\) confidence interval estimate of \(p,\) the proportion of all full-time workers so angered in the last year that they wanted to hit a coworker.

A random sample will be selected from the population of all adult residents of a particular city. The sample proportion \(\hat{p}\) will be used to estimate \(p,\) the proportion of all adult residents who are employed full time. For which of the following situations will the estimate tend to be closest to the actual value of \(p\) ? $$ \begin{array}{ll} \text { i. } & n=500, p=0.6 \\ \text { ii. } & n=450, p=0.7 \\ \text { iii. } & n=400, p=0.8 \end{array} $$

A researcher wants to estimate the proportion of students enrolled at a university who are registered to vote. Would the standard error of the sample proportion \(\hat{p}\) be larger if the actual population proportion was \(p=0.4\) or \(p=0.8\) ?

The article "Consumers Show Increased Liking for Diesel Autos" (USA Today, January 29,2003 ) reported that \(27 \%\) of U.S. consumers would opt for a diesel car if it ran as cleanly and performed as well as a car with a gas engine. Suppose that you suspect that the proportion might be different in your area. You decide to conduct a survey to estimate this proportion for the adult residents of your city. What is the required sample size if you want to estimate this proportion with a margin of error of 0.05 ? Calculate the required sample size first using 0.27 as a preliminary estimate of \(p\) and then using the conservative value of \(0.5 .\) How do the two sample sizes compare? What sample size would you recommend for this study?

Use the formula for the standard error of \(\hat{p}\) to explain why a. The standard error is greater when the value of the population proportion \(p\) is near 0.5 than when it is near \(1 .\) b. The standard error of \(\hat{p}\) is the same when the value of the population proportion is \(p=0.2\) as it is when \(p=0.8\)
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