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Introduction to EstimationIntroduction to Estimation

Chapter 10

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Concepts of Estimation

The objective of estimation is to determine the value of a population parameter on the basis of a sample statistic. There are two types of estimators:Point EstimatorInterval estimator

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Point Estimator

A point estimator draws inference about a population by estimating the value of an unknown parameter using a single value or point.

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Population distribution

Point Estimator

Parameter

?

Sampling distribution

A point estimator draws inference about a population by estimating the value of an unknown parameter using a single value or point.

Point estimator

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An interval estimator draws inferences about a population by estimating the value of an unknown parameter using an interval.

Interval estimator

Population distribution

Sample distribution

Parameter

Interval Estimator

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Selecting the right sample statistic to estimate a parameter value depends on the characteristics of the statistic.

Estimator’s Characteristics

Estimator’s desirable characteristics: Unbiasedness: An unbiased estimator is one whose

expected value is equal to the parameter it estimates.Consistency: An unbiased estimator is said to be

consistent if the difference between the estimator and the parameter grows smaller as the sample size increases.

Relative efficiency: For two unbiased estimators, the one with a smaller variance is said to be relatively efficient.

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Unbiased Estimate

)(XE

ppE )ˆ(

22

22

)(

1

)(

sE

n

xxs

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Consistency…An unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size grows larger.E.g. X is a consistent estimator of because: V(X) is That is, as n grows larger, the variance of X grows smaller.

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Relative Efficiency…If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to be relatively efficient.E.g. both the the sample median and sample mean are unbiased estimators of the population mean, however, the sample median has a greater variance than the sample mean, so we choose since it is relatively efficient when compared to the sample median.

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Estimating the Population Mean when the Population Variance is Known

How is an interval estimator produced from a sampling distribution?A sample of size n is drawn from the population, and

its mean is calculated.By the central limit theorem is normally distributed

(or approximately normally distributed.), thus…x

x

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nx

Z

We have established before that

1)n

zxn

z(P 22

Estimating the Population Mean when the Population Variance is Known

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1)n

zxn

zx(P 22

This leads to the following equivalent statement

The Confidence Interval for ( is known)

The confidence interval

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Interpreting the Confidence Interval for

1 – of all the values of obtained in repeatedsampling from a given distribution, construct an interval

that includes (covers) the expected value of thepopulation.

1 – of all the values of obtained in repeatedsampling from a given distribution, construct an interval

that includes (covers) the expected value of thepopulation.

x

nzx,

nzx 22

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x

nz2 2

nzx 2

n

zx 2

Lower confidence limit Upper confidence limit

1 -

Confidence level

Graphical Demonstration of the Confidence Interval for

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The Confidence Interval for ( is known)

Four commonly used confidence levels

Confidence level 0.90 0.10 0.05 1.6450.95 0.05 0.025 1.960.98 0.02 0.01 2.330.99 0.01 0.005 2.575

Confidence level 0.90 0.10 0.05 1.6450.95 0.05 0.025 1.960.98 0.02 0.01 2.330.99 0.01 0.005 2.575

z

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The width of the confidence interval is affected by

• the population standard deviation ()• the confidence level (1-)• the sample size (n).

The width of the confidence interval is affected by

• the population standard deviation ()• the confidence level (1-)• the sample size (n).

The Width of the Confidence Interval

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90%

Confidence level

To maintain a certain level of confidence, a larger standard deviation requires a larger confidence interval.To maintain a certain level of confidence, a larger standard deviation requires a larger confidence interval.

n)645.1(2

nz2 05.

/2 = .05/2 = .05

n

5.1)645.1(2

n

5.1z2 05.

Suppose the standard deviation has increased by 50%.

The Affects of on the interval width

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n)96.1(2

nz2 025.

/2 = 2.5%/2 = 2.5%

/2 = 5%/2 = 5%

n)645.1(2

nz2 05.

Confidence level 90%95%

Let us increase the confidence level from 90% to 95%.

Larger confidence level produces a wider confidence intervalLarger confidence level produces a wider confidence interval

The Affects of Changing the Confidence Level

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90%

Confidence level

n)645.1(2

nz2 05.

Increasing the sample size decreases the width of the confidence interval while the confidence level can remain unchanged.

Increasing the sample size decreases the width of the confidence interval while the confidence level can remain unchanged.

The Affects of Changing the Sample Size

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Example 1

a. The mean of a random sample of 25 observations from a normal population whose standard deviation is 40 is 200. Estimate the population mean with 95% confidence.

b. Repeat part a changing the population standard deviation to 25.

c. Repeat part a changing the population standard deviation to 10

d. Describe what happens to the confidence interval estimate when the standard deviation is decreased

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Solution

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Example 2

The following data represent a random sample of 9 marks (out of 10) on a statistics quiz. The marks are normally distributed with a standard deviation of 2. Estimate the population mean with 90% confidence.

4 6 8 5 4 8 3 10 9

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Solution

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Selecting the Sample size

We can control the width of the confidence interval by changing the sample size.

Thus, we determine the interval width first, and derive the required sample size.

The phrase “estimate the mean to within W units”, translates to an interval estimate of the form

wx wx

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The required sample size to estimate the mean is

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w

zn

22

w

zn

Selecting the Sample size

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Example 3

a. A statistics practitioner would like to estimate a population mean to within 10 units. The confidence level has been set at 95% and =200. Determine the sample size

b. Suppose that the sample mean was calculated as 500. Estimate the population mean with 95% confidence

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Solution

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Example 4

A statistics professor wants to compare today’s students with those 25 years ago. All of his current students’ marks are stored on a computer so that he can easily determine the population mean. However, the marks 25 years ago reside only in his musty files. He does not want to retrieve all the marks and will be satisfied with a 95% confidence interval estimate of the mean mark 25 years ago. If he assumes that the population standard deviation is 10, how large a sample should he take to estimate the mean to within 3 marks?

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Solutions