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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

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Sampling Distribution of the Sample Variance

a. Find the value of the population variance 蟽2.

b. Table 6-2 describes the sampling distribution of the sample mean. Construct a similar table representing the sampling distribution of the sample variance s2. Then combine values of s2that are the same, as in Table 6-3 (Hint: See Example 2 on page 258 for Tables 6-2 and 6-3, which describe the sampling distribution of the sample mean.)

c. Find the mean of the sampling distribution of the sample variance.

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Durations of PregnanciesThe lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days.

a. In a letter to 鈥淒ear Abby,鈥� a wife claimed to have given birth 308 days after a brief visit fromher husband, who was working in another country. Find the probability of a pregnancy lasting308 days or longer. What does the result suggest?

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