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Chapter 9: Problem 11

Use the formula for the standard error of \(\hat{p}\) to explain why increasing the sample size decreases the standard error.

Short Answer

Expert verified
Increasing the sample size decreases the standard error because the sample size (\(n\)) is in the denominator of the standard error formula. Thus, increasing \(n\) makes the whole fraction smaller, hence reducing the standard error.

Step by step solution

01

Understanding the concept of Standard Error and Sample Size

The standard error of an estimate is a measure of how much we expect it to vary if we take multiple random samples from the population. In our case, the estimate is \(\hat{p}\) - the sample proportion. So, the standard error of \(\hat{p}\) tells us how much we expect \(\hat{p}\) to vary from sample to sample. Sample size (\(n\)) refers to the number of observations or replicates in a sample.
02

Analysis of the formula for Standard Error of \(\hat{p}\)

The formula for standard error of \(\hat{p}\) is \(\sqrt{(\hat{p}(1-\hat{p})/n)}\). In this formula, the denominator under the square root symbol is the sample size \(n\). Since standard error measures the expected variability of \(\hat{p}\) and \(n\) appears in the denominator of this formula, it tells that standard error is inversely related to the square root of the sample size.
03

Reasoning why increasing the sample size decreases the Standard Error

Since sample size (\(n\)) is in the denominator of the standard error formula, as we increase the sample size, the value of the whole denominator increases. For any fraction, if the denominator gets larger while the numerator stays the same, the overall value of the fraction gets smaller. The square root doesn't change this relationship. Thus, increasing the sample size (\(n\)) will decrease the standard error of \(\hat{p}\).

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Most popular questions from this chapter

USA Today (October 14,2002 ) reported that \(36 \%\) of adult drivers admit that they often or sometimes talk on a cell phone when driving. This estimate was based on data from a representative sample of 1,004 adult drivers. A margin of error of \(3.1 \%\) was also reported. Is this margin of error correct? Explain.

An article in the Chicago Tribune (August 29, 1999) reported that in a poll of residents of the Chicago suburbs, \(43 \%\) felt that their financial situation had improved during the past year. The following statement is from the article: "The findings of this Tribune poll are based on interviews with 930 randomly selected suburban residents. The sample included suburban Cook County plus DuPage, Kane, Lake, McHenry, and Will Counties. In a sample of this size, one can say with \(95 \%\) certainty that results will differ by no more than \(3 \%\) from results obtained if all residents had been included in the poll." Give a statistical argument to justify the claim that the estimate of \(43 \%\) is within \(3 \%\) of the actual percentage of all residents who feel that their financial situation has improved.

Suppose that county planners are interested in learning about the proportion of county residents who would pay a fee for a curbside recycling service if the county were to offer this service. Two different people independently selected random samples of county residents and used their sample data to construct the following confidence intervals for the proportion who would pay for curbside recycling: Interval 1:(0.68,0.74) Interval 2:(0.68,0.72) a. Explain how it is possible that the two confidence intervals are not centered in the same place. b. Which of the two intervals conveys more precise information about the value of the population proportion? c. If both confidence intervals are associated with a \(95 \%\) confidence level, which confidence interval was based on the smaller sample size? How can you tell? d. If both confidence intervals were based on the same sample size, which interval has the higher confidence level? How can you tell?

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 \(p=0.8\)

Suppose that 935 smokers each received a nicotine patch, which delivers nicotine to the bloodstream at a much slower rate than cigarettes do. Dosage was decreased to 0 over a 12 -week period. Of these 935 people, 245 were still not smoking 6 months after treatment. Assume this sample is representative of all smokers. a. Use the given information to estimate the proportion of all smokers who, when given this treatment, would refrain from smoking for at least 6 months. 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.
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