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Chapter 9: Problem 76

Consider taking a random sample from a population with \(p=0.25\) a. What is the standard error of \(\hat{p}\) for random samples of size \(400 ?\) b. Would the standard error of \(\hat{p}\) be smaller for random samples of size 200 or samples of size \(400 ?\) c. Does cutting the sample size in half from 400 to 200 double the standard error of \(\hat{p} ?\)

Short Answer

Expert verified
a) The standard error of \(\hat{p}\) for random samples of size 400 is obtained by plugging \(p=0.25\) and \(n=400\) into the formula \[ SE =\sqrt{\frac{{p(1-p)}}{{n}} } \] ; (b) The standard error of \(\hat{p}\) would be smaller for samples of size 400 than for samples of size 200; (c) Cutting the sample size in half from 400 to 200 does not exactly double the standard error of \(\hat{p}\), it increases it by the factor of \(\sqrt{2}\), which is about 1.41 - smaller than doubling.

Step by step solution

01

Calculate the standard error for a sample size of 400

We need to provide the standard error (SE) for a sample of size 400. By using the formula above, SE can be calculated as: \[ SE_1 = \sqrt{\frac{{0.25*(1-0.25)}}{{400}} } \]
02

Compare the standard error for samples of size 200 and 400

The question now inquires if the standard error would be smaller for random samples of size 200 or size 400. Let's calculate the standard error for a sample size of 200 following the same formula: \[ SE_2 = \sqrt{\frac{{0.25*(1-0.25)}}{{200}} } \]Then, compare the value obtained with the SE computed for the sample of size 400.
03

Determine relationship of cutting sample size in half with the standard error

The exercise asks whether cutting the sample size in half from 400 to 200 doubles the standard error or not. For this, we need to compare the two standard errors obtained in steps 1 and 2.

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Most popular questions from this chapter

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 \(p\) ? $$ \begin{array}{ll} \text { i. } & n=500, p=0.6 \\ \text { ii. } & n=450, p=0.7 \\ \text { iii. } & n=400, p=0.8 \end{array} $$

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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 do not own a car. For which of the following situations will the estimate tend to be closest to the actual value of \(p ?\) i. \(\quad n=500, p=0.1\) $$ \text { ii. } \quad n=1,000, p=0.2 $$ iii. \(n=1,200, p=0.3\)

A car manufacturer is interested in learning about the proportion of people purchasing one of its cars who plan to purchase another car of this brand in the future. A random sample of 400 of these people included 267 who said they would purchase this brand again. For each of the three statements below, indicate if the statement is correct or incorrect. If the statement is incorrect, explain what makes it incorrect. Statement 1 : The estimate \(\hat{p}=0.668\) will never differ from the value of the actual population proportion by more than \(0.0462 .\) Statement 2 : It is unlikely that the estimate \(\hat{p}=0.668\) differs from the value of the actual population proportion by more than 0.0235 . Statement 3: It is unlikely that the estimate \(\hat{p}=0.668\) differs from the value of the actual population proportion by more than 0.0462 .
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