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Chapter 9: Problem 13

If two statistics are available for estimating a population characteristic, under what circumstances might you choose a biased statistic over an unbiased statistic?

Short Answer

Expert verified
A biased statistic might be chosen over an unbiased one if it results in a lower Mean Squared Error (MSE), which takes into account both bias and variance. Also, it might be preferred due to computational simplicity, especially if the bias can be corrected.

Step by step solution

01

Define Biased and Unbiased Statistics

An unbiased statistic or estimator is one for which the mean of the statistic's sampling distribution is equal to the population parameter being estimated. That is, the statistic doesn't overestimate or underestimate the parameter on average. On the other hand, a biased statistic is one that systematically overestimates or underestimates the parameter.
02

Discuss scenarios for choosing biased statistic

A biased estimator might be used in preference to an unbiased one if it leads to a lower Mean Squared Error (MSE). The MSE of an estimate is the variance of the estimator plus the square of the bias. Thus, a biased estimator with low variance could have a lower MSE than an unbiased one with high variance. Also, in some cases, the computational simplicity of biased estimators might make them more attractive. This could be especially true in situations where the bias is known, and adjustments can be made to counteract it.
03

Illustrate with examples

For example, the sample variance (dividing by n-1 instead of n) is a biased estimator of the population variance. Despite being biased, it is preferable to the unbiased estimator (dividing by n) because it has a lower MSE.

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Most popular questions from this chapter

The article "Viewers Speak Out Against Reality TV" (Associated Press, September 12,2005\()\) included the following statement: "Few people believe there's much reality in reality TV: a total of \(82 \%\) said the shows are either 'totally made up' or 'mostly distorted."" This statement was based on a survey of 1,002 randomly selected adults. Calculate and interpret the margin of error for the reported percentage.

In a survey of 1,000 randomly selected adults in the United States, participants were asked what their most favorite and least favorite subjects were when they were in school (Associated Press, August 17,2005\()\). In what might seem like a contradiction, math was chosen more often than any other subject in both categories. Math was chosen by 230 of the 1,000 as their most favorite subject and chosen by 370 of the 1,000 as their least favorite subject. a. Construct and interpret a \(95 \%\) confidence interval for the proportion of U.S. adults for whom math was their most favorite subject. b. Construct and interpret a \(95 \%\) confidence interval for the proportion of U.S. adults for whom math was their least favorite subject.

Appropriate use of the interval $$ \hat{p} \pm(z \text { critial value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$ 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. $$ \begin{array}{l} \text { a. } n=100 \text { and } \hat{p}=0.70 \\ \text { b. } n=40 \text { and } \hat{p}=0.25 \\ \text { c. } n=60 \text { and } \hat{p}=0.25 \end{array} $$ d. \(n=80\) and \(\hat{p}=0.10\)

In a survey on supernatural experiences, 722 of 4,013 adult Americans reported that they had seen a ghost (鈥淲hat Supernatural Experiences We've Had," USA Today, February 8,2010 ). Assume that this sample is representative of the population of adult Americans. a. Use the given information to estimate the proportion of adult Americans who would say they have seen a ghost. b. Verify that the conditions for use of the margin of error formula to be appropriate are met. c. Calculate the margin of error. d. Interpret the margin of error in context. e. Construct and interpret a \(90 \%\) confidence interval for the proportion of all adult Americans who would say they have seen a ghost. f. Would a \(99 \%\) confidence interval be narrower or wider than the interval calculated in Part (e)? Justify your answer.

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?
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