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Chapter 9: Problem 2

Why is an unbiased statistic generally preferred over a biased statistic for estimating a population characteristic? Does unbiasedness alone guarantee that the estimate will be close to the true value? Explain. Under what circumstances might you choose a biased statistic over an unbiased statistic if two statistics are available for estimating a population characteristic?

Short Answer

Expert verified
An unbiased statistic is preferred because it gives the correct average value of the population characteristic, minimising systematic error. However, unbiasedness alone doesn't guarantee accuracy, as high variance might spread estimates. A biased statistic could be preferred if the unbiased one has larger variance or higher obtaining cost, as per bias-variance trade-off.

Step by step solution

01

Understanding Statistical Bias

A statistic is biased if it is systematically over- or under-estimating the true parameter. This means that on average, the statistic is not equal to the parameter it is intended to estimate.
02

Preference of Unbiased Estimators

An unbiased statistic is generally preferred because on average, it gives the correct value of the population characteristic, which is not guaranteed by a biased statstic. Thus, by preferring unbiased statistics, one minimises systematic errors.
03

Unbiasedness and Accuracy

Unbiasedness alone does not guarantee that the estimate will be close to the true value. The estimator could be unbiased but have a high variance, implying that the estimates are spread out over a large range. Therefore, both bias and variance need to be considered when assessing the quality of an estimator.
04

Circumstances to Prefer Biased Estimators

Under certain circumstances, if the unbiased estimator has a much larger variance than a slightly biased alternative or if the cost of obtaining an unbiased estimator is too high, a biased statistic might be preferred. This is due to the trade-off between bias and variance, also known as the bias-variance trade-off.

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Most popular questions from this chapter

The formula used to compute a confidence interval for the mean of a normal population when \(n\) is small is $$\bar{x} \pm(t \text { critical value }) \frac{s}{\sqrt{n}}$$ What is the appropriate \(t\) critical value for each of the following confidence levels and sample sizes? a. \(95 \%\) confidence, \(n=17\) b. \(90 \%\) confidence, \(n=12\) c. \(99 \%\) confidence, \(n=24\) d. \(90 \%\) confidence, \(n=25\) e. \(90 \%\) confidence, \(n=13\) f. \(95 \%\) confidence, \(n=10\)

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 sample of 1004 adult drivers, and a bound on the error of estimation of \(3.1 \%\) was reported. Assuming a \(95 \%\) confidence level, do you agree with the reported bound on the error? Explain.

In a survey on supernatural experiences, 722 of 4013 adult Americans surveyed reported that they had seen or been with a ghost ("What Supernatural Experiences We've Had," USA Today, February 8,2010 ). a. What assumption must be made in order for it to be appropriate to use the formula of this section to construct a confidence interval to estimate the proportion of all adult Americans who have seen or been with a ghost? b. Construct and interpret a \(90 \%\) confidence interval for the proportion of all adult Americans who have seen or been with a ghost. c. Would a \(99 \%\) confidence interval be narrower or wider than the interval computed in Part (b)? Justify your answer.

In a study of academic procrastination, the authors of the paper "Correlates and Consequences of Behavioral Procrastination" (Procrastination, Current Issues and New Directions [2000]) reported that for a sample of 411 undergraduate students at a midsize public university preparing for a final exam in an introductory psychology course, the mean time spent studying for the exam was \(7.74\) hours and the standard deviation of study times was \(3.40\) hours. For purposes of this exercise, assume that it is reasonable to regard this sample as representative of students taking introductory psychology at this university. a. Construct a \(95 \%\) confidence interval to estimate \(\mu\), the mean time spent studying for the final exam for students taking introductory psychology at this university. b. The paper also gave the following sample statistics for the percentage of study time that occurred in the 24 hours prior to the exam: \(n=411 \quad \bar{x}=43.18 \quad s=21.46\) Construct and interpret a \(90 \%\) confidence interval for the mean percentage of study time that occurs in the 24 hours prior to the exam.

Acrylic bone cement is sometimes used in hip and knee replacements to fix an artificial joint in place. The force required to break an acrylic bone cement bond was measured for six specimens under specified conditions, and the resulting mean and standard deviation were \(306.09\) Newtons and \(41.97\) Newtons, respectively. Assuming that it is reasonable to believe that breaking force under these conditions has a distribution that is approximately normal, estimate the mean breaking force for acrylic bone cement under the specified conditions using a \(95 \%\) confidence interval.
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