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Chapter 9: Problem 67

Data from a representative sample were used to estimate that \(32 \%\) of all computer users in 2011 had tried to get on a Wi-Fi network that was not their own in order to save money (USA TODAY, May 16,2011 ). You decide to conduct a survey to estimate this proportion for the current year. 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 0.32 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?

Short Answer

Expert verified
The required sample size using the preliminary estimate of 0.32 is 335, and the required sample size using the conservative estimate of 0.5 is 385. Since the difference between the two sample sizes is not very large, it is recommended to use the conservative sample size of 385, as it provides a better level of reliability for the study.

Step by step solution

01

Determine desired confidence level of the study

Since the confidence level is not provided, we will assume a standard 95% confidence level. For a 95% confidence level, the z-score can be found using a z-table or standard calculator, yielding a z-score of approximately 1.96. #Step 2: Calculate sample size using preliminary estimate of 0.32#
02

Plug the values into the margin of error formula and solve for n

We are given a margin of error E = 0.05, and the preliminary estimate of p = 0.32. We will now use the margin of error formula to calculate the required sample size: \(0.05 = 1.96 \sqrt{\frac{0.32(1-0.32)}{n}}\) Next, we will square both sides of the equation to remove the square root: \(0.0025 = 1.96^2 * \frac{0.32(1-0.32)}{n}\) Now, isolate n: \(n = \frac{1.96^2 * 0.32(1-0.32)}{0.0025}\) Finally, calculate n: \(n \approx 334.41\) Since we cannot have a fraction of a person in our sample size, we round up to the nearest whole number, which is 335. #Step 3: Calculate sample size using conservative estimate of 0.5#
03

Plug the values into the margin of error formula and solve for n

To calculate the required sample size using the conservative estimate of p = 0.5, we follow the same procedure as in Step 2: \(0.05 = 1.96 \sqrt{\frac{0.5(1-0.5)}{n}}\) Square both sides of the equation: \(0.0025 = 1.96^2 * \frac{0.5(1-0.5)}{n}\) Isolate n: \(n = \frac{1.96^2 * 0.5(1-0.5)}{0.0025}\) Calculate n: \(n \approx 384.16\) Again, we round up to the nearest whole number to get a sample size of 385. #Step 4: Compare sample sizes and make recommendation#
04

Analyze the two sample sizes and choose the best one for the study

The two sample sizes are: - 335 using the preliminary estimate of 0.32 - 385 using the conservative estimate of 0.5 The conservative estimate yields a larger sample size, which would result in a more accurate estimation of the proportion. However, a larger sample size is also more time-consuming and expensive to gather. Since the difference between the two sample sizes (50) is not very large, it is recommended to use the conservative sample size of 385, as it provides a better level of reliability for the study.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Margin of Error
The margin of error is a statistic that represents the maximum amount by which the sample estimate of a population parameter is expected to differ from the actual population parameter. For example, in the context of the exercise, a margin of error of 0.05 means that the true population proportion could be as much as 0.05 higher or lower than the estimate obtained from the sample.

This concept is crucial in the field of statistics as it affects how the results of a survey or study are interpreted. It provides a way to quantify uncertainty in sampling, allowing for a range of possible values rather than a single, possibly misleading estimate. To calculate it, you need to know the confidence level and the variance within the sample. The formula to calculate the margin of error includes the standard error of the estimated proportion and the critical value from the z-distribution corresponding to the desired confidence level. The formula can be expressed as:
\[ E = z \times \sqrt{\frac{p(1-p)}{n}} \]
where E is the margin of error, z is the z-score corresponding to the confidence level, p is the estimated proportion, and n is the sample size. Smaller margins of error require larger sample sizes and vice versa, given a fixed confidence level.
Confidence Level
The confidence level signifies the degree of certainty that the calculated confidence interval contains the true population parameter. Common confidence levels include 90%, 95%, and 99%. A 95% confidence level, which was assumed in the exercise for the sake of calculation, indicates that if the study was repeated multiple times, the true population parameter would lie within the calculated interval 95% of the time.

The selection of a confidence level depends on how precise the researcher wants the estimate to be. Higher confidence levels lead to wider confidence intervals, which means that the estimate is less precise but that you are more sure that the interval contains the true population proportion. To express this confidence level numerically when calculating the margin of error, a z-score is used. This z-score corresponds to the chosen confidence level and can be found in z-tables or calculated by statistical software. In practical terms, a higher confidence level requires a larger sample size to maintain the same margin of error.
Sample Size Calculation
Calculating the sample size for a survey or experiment is a step to ensure the reliability of the results. This process involves determining the number of participants needed to estimate the population parameter with a certain level of confidence and margin of error.

The formula for calculating the sample size when estimating a proportion is derived from setting up the margin of error equation and solving for n. It is a function of the desired margin of error E, the confidence level-specific z-score, and the estimated proportion p as shown in the provided solution. In scenarios where there's no preliminary estimate, a conservative estimate using a proportion of 0.5 is often used as it maximizes the product p(1-p), leading to the largest required sample size for a given margin of error and confidence level. This ensures that the sample is large enough to account for maximum variability in the population.
Population Proportion
The population proportion, denoted by p, is a measure of frequency. It indicates the fraction of members in a population that possess a particular attribute or characteristic. For instance, in the exercise provided, the number of computer users in 2011 who tried to access a non-private Wi-Fi network to save money is the characteristic of interest.

To estimate the current year's population proportion for the same behavior, we would sample some computer users and calculate the sample proportion, which serves as an estimate of the true population proportion. The accuracy of this estimation will depend on the sample size and the true variability in the population. A preliminary estimate of the population proportion, if known from previous studies or pilot tests, can be used in the sample size calculation to increase the efficiency of the sampling process. If such preliminary data is unavailable, a conservative approach assumes a population proportion of \( p = 0.5 \), which safeguards against underestimation of the required sample size.

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Most popular questions from this chapter

Appropriate use of the interval $$ \hat{p} \pm(z \text { critical value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$ requires a large sample. For each of the following combinations of \(n\) and \(\hat{p}\), indicate whether the sample size is large enough for this interval to be appropriate. a. \(n=100\) and \(\hat{p}=0.70\) b. \(n=40\) and \(\hat{p}=0.25\) c. \(n=60\) and \(\hat{p}=0.25\) d. \(n=80\) and \(\hat{p}=0.10\)

The report "The 2016 Consumer Financial Literacy Survey" (The National Foundation for Credit Counseling, www.nfcc.org, retrieved October 28,2016 ) summarized data from a representative sample of 1668 adult Americans. Based on data from this sample, it was reported that over half of U.S. adults would give themselves a grade of \(\mathrm{A}\) or \(\mathrm{B}\) on their knowledge of personal finance. This statement was based on observing that 934 people in the sample would have given themselves a grade of \(\mathrm{A}\) or \(\mathrm{B}\). a. Construct and interpret a \(95 \%\) confidence interval for the proportion of all adult Americans who would give themselves a grade of \(\mathrm{A}\) or \(\mathrm{B}\) on their financial knowledge of personal finance. b. Is the confidence interval from Part (a) consistent with the statement that a majority of adult Americans would give themselves a grade of \(\mathrm{A}\) or \(\mathrm{B}\) ? Explain why or why not.

The paper "Sleeping with Technology: Cognitive, Affective and Technology Usage Predictors of Sleep Problems Among College Students" (Sleep Health [2016]: 49-56) summarized data from a survey of a sample of college students. Of the 734 students surveyed, 125 reported that they sleep with their cell phones near the bed and check their phones for something other than the time at least twice during the night. For purposes of this exercise, assume that this sample is representative of college students in the United States. a. Use the given information to estimate the proportion of college students who check their cell phones for something other than the time at least twice during the night. b. Verify that the conditions needed in order for the margin of error formula to be appropriate are met. c. Calculate the margin of error. d. Interpret the margin of error in the context of this problem.

Describe how each of the following factors affects the width of the large- sample confidence interval for \(p\) : a. The confidence level b. The sample size c. The value of \(\hat{p}\)

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 registered to vote. For which of the following situations will the estimate tend to be closest to the actual value of \(p ?\) I. \(\quad n=1000, p=0.5\) II. \(\quad n=200, p=0.6\) III. \(n=100, p=0.7\)
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