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Chapter 6: Q5 (page 293)

Unbiased Estimators

a. What is an unbiased estimator?

b. For the following statistics, identify those that are unbiased estimators: mean, median, range, variance, proportion.

c. Determine whether the following statement is true or false: 鈥淭he sample standard deviation is a biased estimator, but the bias is relatively small in large samples, so is often used to estimate 蟽鈥�

Short Answer

Expert verified

a. An unbiased estimator is a statistic whose sampling distribution has a mean value equal to the corresponding population parameter.

b. Out of the given list of estimators, mean, variance, and proportion are unbiased estimators.

c. The sample standard deviation is often used to estimate the population standard deviation 蟽because the bias is small in large samples. Thus, this statement is true.

Step by step solution

01

Given Information

The definition of an unbiased estimator is considered. A set of estimators is provided.

02

Definition of unbiased estimator

a.

A sample statistic is an unbiased estimator of the corresponding population parameter if the sample statistic's mean value (or expected value) over all possible samples of size is equal to the corresponding population parameter.

03

Identification of unbiased estimators

b.

The following statistics are considered to be unbiased estimators of their respective population parameters:

  • Proportion
  • Mean
  • Variance

Thus, out of the given list of estimators, mean, variance, and proportion are unbiased estimators. On the other hand, median and range are biased estimators.

04

Sample standard deviation as an estimator of population standard deviation

c.

The sample standard deviation s is often used to estimate the population standard deviation 蟽because the bias is small in large samples. Thus, this statement is true.

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Most popular questions from this chapter

Sampling Distribution Data Set 4 鈥淏irths鈥� in Appendix B includes a sample of birth weights. If we explore this sample of 400 birth weights by constructing a histogram and finding the mean and standard deviation, do those results describe the sampling distribution of the mean? Why or why not?

Standard normal distribution, assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, Draw a graph, then find the probability of the given bone density test score. If using technology instead of Table A-2, round answers to four decimal places.

Less than 4.55

Continuous Uniform Distribution. In Exercises 5鈥�8, refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.

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Births: Sampling Distribution of Sample Proportion For three births, assume that the genders are equally likely. Construct a table that describes the sampling distribution of the sample proportion of girls from three births. Does the mean of the sample proportions equal the proportion of girls in three births? (Hint: See Exercise 15 for two births.)

In Exercises 11鈥�14, use the population of {34, 36, 41, 51} of the amounts of caffeine (mg/12oz) in鈥侰oca-Cola鈥俍ero,鈥侱iet鈥侾epsi,鈥侱r鈥侾epper,鈥俛nd鈥侻ellow鈥俌ello鈥俍ero.

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