Lecture notes - Estimation, estimation PDF

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Estimation, Estimation...

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Economics 261 Principles of Statistics I Lecture Notes Topic 7: Estimation 1. 2. 3. 4.

The Process of Estimation Confidence Intervals Sample Size Estimation Alternative Levels of Confidence

1.

The Process of Estimation. 1.

Estimation is the process of assigning a value to a population parameter from a sample statistic. 1.

Estimator:

A statistic which is used to estimate a parameter.

2.

Parameter:

A population summary measure.

3.

Estimate:

The numerical value of an estimator.

Estimator _ X

Parameter µ

Continuous example

p

π

Percentage example

S

2.

2

Estimate

2

σ

σ

S _ _ X1 - X2

µ1 - µ2

p1 - p2

π1 - π2

Two types of estimates. 1.

2.

Point estimate:

A single number estimate of a parameter; example.

Point estimate for a mean:

_ ΣX X = CCCCC n

→

µ

Point estimate for a proportion:

X p = CCCCC n

→

π

Interval estimate:

A range estimate of a parameter; confidence interval; example. 7.1

7.2 2.

Confidence Intervals. Note: 1.

The following are confidence intervals for large samples (n ≥ 30).

Confidence interval for a population mean, µ. 1.

The case of known σ: σ

_ µ = X ± Z σㄡ

where:

σㄡ = CCC √n

where:

µ _ X

= population mean.

Z

= standard normal variable for a selected level of confidence; from Z-table.

= sample mean.

σㄡ = standard error of the mean; defined above. σ

= population standard deviation.

n

= sample size. _

_

X – Z σㄡ ≤ µ ≤ X + Z σㄡ (LCL) (UCL)

CI:

2.

The case of unknown σ: Note:

Sample standard deviation (S) is substituted for the population standard deviation (σ).

_

S

µ = X ± Z Sㄡ

where:

Sㄡ = CCC √n

where:

µ _ X

= population mean.

Z

= standard normal variable for a selected level of confidence; from Z-table.

= sample mean.

Sㄡ = standard error of the mean; defined above.

_ CI:

S

= sample standard deviation.

n

= sample size. _

X – Z Sㄡ ≤ µ ≤ X + Z Sㄡ (LCL) (UCL)

7.3 2.

Confidence interval for a population proportion, π. Note:

Sample proportion (p) is substituted for the population proportion (π) since the unknown π is the parameter to be estimated.

π = p ± Z Sp

where:

where:

p (1-p) Sp = √ CCCCCCC n

←

π (1-π) σp = √ CCCCCCC n

π

= population proportion.

p

= sample proportion.

Z

= standard normal variable for a selected level of confidence; from Z-table.

Sp = standard error of the proportion; defined above. n

CI:

= sample size.

p – Z Sp ≤ π ≤ p + Z Sp (LCL) (UCL)

7.4 3.

Sample Size Estimation. 1.

The case of mean (µ) analyses.

n = (Zσ/E) where:

2

n = sample size. Z = standard normal variable for a selected level of confidence; from Z-table. σ = population standard deviation. _ E = error of estimation; sampling error, X-µ. (1) The value for σ can be determined from previous studies, or estimated as σ ≈ 1/4 Range. (2) E is the allowable sampling error determined by the researcher.

Notes:

2.

The case of proportion (π) analyses. 2

Z (π)(1-π) n = CCCCCCCCCCCC 2 E where:

n = sample size. Z = standard normal variable for a selected level of confidence; from Z-table. π = population proportion. E = error of estimation; sampling error, p-π.

Notes:

(1) The value for π can be determined from previous studies, or set at .5 to maximize the sample size. (2) E is the allowable sampling error determined by the researcher.

7.5 4.

Alternative Levels of Confidence. Confidence Coefficient

Level of Significance*

(Level) α α/2 .5-(α/2) Zα/2 _______________________________________________________________ .90 (90%)

.10

.05

.4500

1.645

.95 (95%)

.05

.025

.4750

1.96

.99 (99%) .01 .005 .4950 2.575 __________________________________________________________________ *Level of significance (α) = 1-Confidence coefficient.

4.

Special meaning of a confidence interval. 1.

A given confidence interval expresses the percentage (given by the Confidence Coefficient) of sample intervals that contain the estimated parameter.

2.

Some examples....