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Chapter 7: Problem 51

Consider a large population with \(p=.21\). Assuming \(n / N \leq .05\), find the mean and standard deviation of the sample proportion \(\hat{p}\) for a sample size of a. 400 b. 750

Short Answer

Expert verified
a. For a sample size of 400, the mean is .21 and the standard deviation is computed as \(\sqrt{.21(.79) / 400}\). b. For a sample size of 750, the mean is .21 and the standard deviation is \(\sqrt{.21(.79) / 750}\).

Step by step solution

01

Compute the Mean

First, let's calculate the mean of the sample proportions (\(\mu_{\hat{p}}\)). Since the mean of the sample proportion equals the population proportion, we have \(\mu_{\hat{p}}=p\). Therefore, \(\mu_{\hat{p}}=.21\). There is no need to differentiate between different sample sizes.
02

Compute the Standard Deviation for a Sample Size of 400

Now, we calculate the standard deviation of the sample proportion (\(\sigma_{\hat{p}}\)) for a sample size \(n=400\). We use the formula \(\sigma_{\hat{p}}=\sqrt{p(1-p) / n}\), so the computation would be \(\sigma_{\hat{p}}=\sqrt{.21(.79) / 400}\).
03

Compute the Standard Deviation for a Sample Size of 750

Similarly, for \(n=750\), we calculate \(\sigma_{\hat{p}}\) by plugging \(n=750\) into the formula, i.e., \(\sigma_{\hat{p}}=\sqrt{.21(.79) / 750}\).

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Most popular questions from this chapter

A certain elevator has a maximum legal carrying capacity of 6000 pounds. Suppose that the population of all people who ride this elevator have a mean weight of 160 pounds with a standard deviation of 25 pounds. If 35 of these people board the elevator, what is the probability that their combined weight will exceed 6000 pounds? Assume that the 35 people constitute a random sample from the population.

For a population, \(N=10,000, \mu=124\), and \(\sigma=18\). Find the \(z\) value for each of the following for \(n=36\). a. \(\bar{x}=128.60\) b. \(\bar{x}=119.30\) c. \(\bar{x}=116.88\) d. \(\bar{x}=132.05\)

How does the value of \(\sigma_{\bar{x}}\) change as the sample size increases? Explain.

In a January 2014 survey conducted by the Associated PressWe TV, \(68 \%\) of American adults said that owning a home is the most important thing or \(a\) very important but not the most important thing (opportunityagenda.org). Assume that this percentage is true for the current population of American adults. Let \(\hat{p}\) be the proportion in a random sample of 1000 American adults who hold the above opinion. Find the mean and standard deviation of the sampling distribution of \(\hat{p}\) and describe its shape.

In a large city, \(88 \%\) of the cases of car burglar alarms that go off are false. Let \(\hat{p}\) be the proportion of false alarms in a random sample of 80 cases of car burglar alarms that go off in this city. Calculate the mean and standard deviation of \(\hat{p}\), and describe the shape of its sampling distribution.
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