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Chapter 7: Problem 11

When is an estimator said to be consistent? Is the sample mean, \(\bar{x}\), a consistent estimator of \(\mu\) ? Explain.

Short Answer

Expert verified
An estimator is considered consistent if it approaches the true parameter value as the sample size increases. The sample mean, \(\bar{x}\), is a consistent estimator of the population mean \(\mu\) because as the number of observations increases, the sample mean tends towards the population mean.

Step by step solution

01

Definition of a Consistent Estimator

An estimator is said to be consistent if, as the sample size approaches infinity, the estimator tends to the true value of the parameter being estimated. In terms of mathematical probability, if an estimator \(\hat{胃}_n\) is trying to estimate the true value \(胃\); as \(n鈫掆垶\), the probability \(P[ | \hat{胃}_n 鈭� 胃 | < 蔚]\) must go to 1 for all \(蔚 > 0\). Simply put, as we have more and more samples, the estimated values should get closer and closer to the actual value.
02

Properties of sample mean

In statistics, the sample mean \(\bar{x}\) is an estimator of the population mean \(\mu\). As an estimator, it is unbiased and efficient. This means that on average, if we were to compute the sample mean from many separate samples, the average would be the population mean.
03

Is the sample mean a consistent estimator

As the number of observations increases, the sample mean \(\bar{x}\) tends towards the true population mean \(\mu\), satisfying the given condition of consistency. Therefore, the sample mean is a consistent estimator of the population mean.

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Most popular questions from this chapter

Mong Corporation makes auto batteries. The company claims that \(80 \%\) of its LL 70 batteries are good for 70 months or longer. Assume that this claim is true. Iet \(\hat{p}\) be the proportion in a sample of 100 such batteries that are good for 70 months or longer. a. What is the probability that this sample proportion is within \(.05\) of the population proportion? b. What is the probability that this sample proportion is less than the population proportion by .06 or more? c. What is the probability that this sample proportion is greater than the population proportion by \(.07\) or more?

According to a National Center for Education Statistics survey released in \(2007,41.5 \%\) of Utah households used a public library or bookmobile within the past 1 month (http:/harvestercensus.gov/imls/ pubs/Publications/2007327.pdf). Suppose that this percentage is true for the current population of households in Utah. a. Suppose that \(51.4 \%\) in a sample of 70 Utah households have used a public library or bookmobile within the past 1 month. How likely is it for the sample proportion in a sample of 70 to be \(.514\) or more when the population proportion is \(.415\) ? b. Refer to part a. How likely is it for the sample proportion to be \(.514\) or more when the sample size is 250 and the population proportion is \(.415\) ? c. What is the smallest sample size that will produce a sample proportion of \(.514\) or more in no more than \(1 \%\) of all sample surveys of that size?

Johnson Electronics Corporation makes electric tubes. It is known that the standard deviation of the lives of these tubes is 150 hours. The company's research department takes a sample of 100 such tubes and finds that the mean life of these tubes is 2250 hours. What is the probability that this sample mean is within 25 hours of the mean life of all tubes produced by this company?

Consider a large population with \(p=.21 .\) Assuming \(n / N \leq .05\), find the mean and standard deviation of the sample proportion \(\hat{p}\) for a sample size of a. 400 b. 750

Investigation of all five major fires in a western desert during one of the recent summers found the following causes. \(\begin{array}{lll}\text { Accident } & \text { Arson } & \text { Accident }\end{array}\) \(\begin{array}{lll}\text { Arson } & \text { Accident } & \text { Accident } & \text { Arson }\end{array}\) a. What proportion of those fires were due to arson? b. How many total samples (without replacement) of size three can be selected from this population? c. List all the possible samples of size three that can be selected from this population and calculate the sample proportion \(\hat{p}\) of the fires due to arson for each sample. Prepare the table that gives the sampling distribution of \(\hat{p}\). d. For each sample listed in part c, calculate the sampling error.
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