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### Subject 4. Point Estimates of the Population Mean

Very often, there are a number of different estimators that can be used to estimate unknown population parameters. When faced with such a choice, it is desirable to know that the estimator chosen is the "best" under the circumstances, that is, it has more desirable properties than any of the other options available to us. There are three desirable properties of estimators:

• unbiasedness

An estimator's expected value (the mean of its sampling distribution) equals the parameter it is intended to estimate. For example, the sample mean is an unbiased estimator of the population mean because the expected value of the sample mean is equal to the population mean.

• efficiency

An estimator is efficient if no other unbiased estimator of the sample parameter has a sampling distribution with smaller variance. That is, in repeated samples, analysts expect the estimates from an efficient estimator to be more tightly grouped around the mean than estimates from other unbiased estimators. For example, the sample mean is an efficient estimator of the population mean, and the sample variance is an efficient estimator of the population variance.

• consistency

A consistent estimator is one for which the probability of accurate estimates (estimates close to the value of the population parameter) increases as sample size increases. In other words, a consistent estimator's sampling distribution becomes concentrated on the value of the parameter it is intended to estimate as the sample size approaches infinity. As the sample size increases to infinity, the standard error of the sample mean declines to 0 and the sampling distribution concentrates around the population mean. Therefore, the sample mean is a consistent estimator of the population mean.

The single estimate of an unknown population parameter calculated as a sample mean is called a point estimate of the mean. The formula used to compute the point estimate is called an estimator. The specific value calculated from sample observations using an estimator is called an estimate. For example, the sample mean is a point estimate of the population mean. Suppose two samples are taken from a population and the sample means are 16 and 21 respectively. Therefore, 16 and 21 are two estimates of the population mean. Note that an estimator will yield different estimates as repeated samples are taken from the sample population.

A confidence interval is an interval for which one can assert with a given probability 1 - α, called the degree of confidence, that it will contain the parameter it is intended to estimate. This interval is often referred to as the (1 - α)% confidence interval for the parameter, where α is referred to as the level of significance. The end points of a confidence interval are called the lower and upper confidence limits.

For example, suppose that a 95% confidence interval for the population mean is 20 to 40. This means that:

• There is a 95% probability that the population mean lies in the range of 20 to 40.
• "95%" is the degree of confidence.
• "5%" is the level of significance.
• 20 and 40 are the lower and higher confidence limits, respectively.

#### Practice Question 1

A random sample of 85 students in Chicago city high schools take a course designed to improve SAT scores. Based on these students, a 90% confidence interval for the mean improvement in SAT scores from this course for all Chicago city high school students is computed as (72.3, 91.4) points. The correct interpretation of this interval is that ______

I. 90% of the students in the sample had their scores improved by between 72.3 and 91.4 points.
II. 90% of the students in the population should have their scores improved by between 72.3 and 91.4.

Correct Answer: Neither of these statements

The confidence level states the probability that the method will give a correct answer with repeated use. In other words, if you use a 90% confidence interval often, in the long run, 90% of your intervals will contain the true parameter value. This is the proper interpretation of the confidence level.

#### Practice Question 2

I collect a random sample of size n from a population with standard deviation σ, and from the data collected I compute a 95% confidence interval for the mean of the population. Which of the following would produce a new confidence interval with smaller width (smaller margin of error) based on these sample data?

I. Increase σ.
II. Use a lower confidence level.
III. Use a smaller sample size.

The margin of error of a confidence interval is reduced by using a lower level of confidence or increasing the sample size.

#### Practice Question 3

For a 90% confidence interval, the alpha value is ______.

A. 90%
B. 10%
C. 5%

A 90% confidence interval, x-bar - E < m < x-bar + E, has a alpha value of 1 - 90% = 10%. That is, P(x-bar - E < m < x-bar + E) = 1 - a.

#### Practice Question 4

A(n) ______ of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used to estimate the population parameter.

A. statistic
B. average
C. point estimator

#### Practice Question 5

Which of the following are desirable properties of estimators?

I. unbiased
II. consistent
III. complete
IV. efficient

A. All but II
B. All but I
C. None of these choices

An estimator is a formula used to compute sample statistics. An estimator should be unbiased, efficient, and consistent.

#### Practice Question 6

True or False? If false, correct the statement.

For a given situation, the longer your confidence interval is, the lower your confidence in it is.

For a given situation, the longer your confidence interval is, the higher your confidence in it is.

#### Practice Question 7

Which of the following statements is correct?

A. An efficient estimator is one that maximizes variance.
B. Consistency can be referred to as probability in convergence.
C. Unbiasedness is a desirable property of estimators because it means that the expected value of the estimators is the parameter it is intended to estimate.

Statements A and B are incorrect. Efficient estimators minimize variance. Consistency is also called convergence in probability.

#### Practice Question 8

The single estimate of the population parameter calculated as a sample mean is called ______.

A. a point estimate
B. an unbiased estimator
C. a consistent estimator

The single estimate of the population parameter calculated as a sample mean is called a point estimate.

#### Practice Question 9

An unbiased estimator is ______.

A. an estimator whose expected value equals the parameter it is intended to estimate
B. an estimator whose expected value is less than the parameter it is intended to estimate
C. a consistent and efficient estimator

An unbiased estimator is an estimator whose expected value equals the parameter it is intended to estimate.

#### Practice Question 10

Select the correct statement(s).

I. An estimator is efficient if the sample size exceeds 30.
II. A consistent estimator is an estimator where the sample size exceeds 30.

Correct Answer: Neither of these statements are correct.

#### Practice Question 11

Seventy-five employees from a large company were randomly surveyed with regard to the length of time they have been working for the company. The sample revealed a mean of 8.2 years and a standard deviation of 2.3 years.

What is the value of the point estimate of the population mean?

A. 75 years
B. 8.2 years
C. 2.3 years

#### Practice Question 12

The wildlife department has been feeding a special food to rainbow trout fingerlings in a pond. A sample of the weights of 40 trout revealed that the mean weight was 402.7 grams and the standard deviation 8.8 grams. What is the point estimate?

A. 10.0675
B. 40
C. 402.7

The sample mean is a good estimate for the population mean.

#### Practice Question 13

A 95% confidence interval for a population parameter signifies which of the following?

I. 95% of similarly constructed intervals will contain the population parameter.
II. For a given sample size, 95% of the samples that have the sample statistic for the population parameter lie within the specified confidence interval around the actual population parameter.
III. The confidence interval will correctly estimate the population parameter with a probability of 95%.

A. I and II
B. II and III
C. I, II and III

Note that III is incorrect because the confidence interval is a range estimate and hence cannot represent a point estimate of the population parameter itself.

#### Practice Question 14

An estimator whose limit (as the sample size approaches infinity) is the parameter it is intended to estimate is called ______.

A. reliable
B. unbiased
C. consistent

An estimator is called consistent if the accuracy (of measuring the parameter) increases as the sample size increases. As the sample size goes toward infinity, the resulting estimate is the population parameter.

#### Practice Question 15

An estimator is efficient if ______

A. the expected value of the estimate equals the population parameter.
B. no other unbiased estimator of the same parameter has a sampling distribution with smaller variance.
C. the estimator gets better as we use more data.

An estimator is efficient if no other estimator of the same parameter has a sampling distribution with a smaller variance.

#### Practice Question 16

A consistent estimator is an estimator ______.

A. where the expected value of the estimate equals the population parameter
B. where no other unbiased estimator of the same parameter has a sampling distribution with smaller variance
C. for which the probability of accurate estimates increases as sample size increases

A consistent estimator is an estimator for which the probability of accurate estimates increases as sample size increases.

#### Practice Question 17

A sample mean possesses all of the following properties EXCEPT ______.

A. consistency and efficiency
B. unbiasedness
C. symmetry

The distribution of sample means may not be symmetrical, especially for small sample sizes.

#### Practice Question 18

Which of the following is least likely to be a desirable property of an estimator?

A. consistency
B. efficiency
C. reliability