Ka-fu Wong © 2003 Chap 9- 1 Dr. Ka-fu Wong ECON1003 Analysis of Economic Data.

Ka-fu Wong © 2003 Chap 9- 1

Dr. Ka-fu Wong

ECON1003Analysis of Economic Data

Ka-fu Wong © 2003 Chap 9- 2l

GOALS

1. Define a what is meant by a point estimate.2. Define the term level of confidence.3. Construct a confidence interval for the

population mean when the population standard deviation is known.

4. Construct a confidence interval for the population mean when the population standard deviation is unknown.

5. Construct a confidence interval for the population proportion.

6. Determine the sample size for attribute and variable sampling.

Chapter NineEstimation and Confidence IntervalsEstimation and Confidence Intervals


Point and Interval Estimates

A point estimate is a single value (statistic) used to estimate a population value (parameter).

A confidence interval is a range of values within which the population parameter is expected to occur.


Confidence Intervals

The degree to which we can rely on the statistic is as important as the initial calculation. Remember, most of the time we are working from samples. And samples are really estimates. Ultimately, we are concerned with the accuracy of the estimate.

1. Confidence interval provides Range of Values Based on Observations from 1 Sample

2. Confidence interval gives Information about Closeness to Unknown Population Parameter Stated in terms of Probability Knowing Exact Closeness Requires Knowing

Unknown Population Parameter


Areas Under the Normal Curve

Between:± 1 - 68.26%± 2 - 95.44%± 3 - 99.74%

µµ-1σµ+1σ

µ-2σ µ+2σµ+3σµ-3σ

If we draw an observation from the normal distributed population, the drawn value is likely (a chance of 68.26%) to lie inside the interval of (µ-1σ, µ+1σ).

P((µ-1σ <x<µ+1σ) =0.6826.


P(µ-1σ <x<µ+1σ) vsP(x-1σ <µ <x+1σ)

P(µ-1σ <x<µ+1σ) is the probability that a drawn observation will lie between (µ-1σ, µ+1σ).

P(µ-1σ <x<µ+1σ) = P(µ-1σ -µ-x <x-µ-x<µ +1σ -µ-x) = P(-1σ -x <-µ<1σ -x)= P(-(-1σ -x )>-(-µ)>-(1σ -x))= P(1σ +x >µ>-1σ +x)= P(x-1σ <µ <x+1σ)

P(x-1σ <µ <x+1σ) is the probability that the population mean will lie between (x-1σ, x+1σ).


Elements of Confidence Interval Estimation

Confidence Interval

Sample Statistic

Confidence Limit (Lower)

Confidence Limit (Upper)

A probability that the population parameter falls somewhere within the interval.


Confidence Intervals

90% Samples

95% Samples

99% Samples

x_

nZX

XZX

X

XXXX 58.2645.1645.158.2

XX 96.196.1


Level of Confidence

1. Probability that the unknown population parameter falls within the interval

2. Denoted (1 - level of confidence Is the Probability That the

Parameter Is Not Within the Interval3. Typical Values Are 99%, 95%, 90%


Interpreting Confidence Intervals

Once a confidence interval has been constructed, it will either contain the population mean or it will not.

For a 95% confidence interval, if you were to produce all the possible confidence intervals using each possible sample mean from the population, 95% of these intervals would contain the population mean.


Intervals & Level of Confidence

Sampling Distribution of Mean

Large Number of Intervals

Intervals Extend from

(1 - ) % of Intervals Contain .

% Do Not.

x =

1 - /2/2

X_

x_

XZX

XZX

to


Point Estimates and Interval Estimates

The factors that determine the width of a confidence interval are:

1. The size of the sample (n) from which the statistic is calculated.

2. The variability in the population, usually estimated by s.

3. The desired level of confidence.

nZX

XZX

)2/

()2/

(



If the population standard deviation is known or the sample is greater than 30 we use the z distribution.

n

szX



If the population standard deviation is unknown and the sample is less than 30 we use the t distribution.

n

stX


Student’s t-Distribution

The t-distribution is a family of distributions that is bell-shaped and symmetric like the standard normal distribution but with greater area in the tails. Each distribution in the t-family is defined by its degrees of freedom. As the degrees of freedom increase, the t-distribution approaches the normal distribution.


About Student

Student is a pen name for a statistician named William S. Gosset who was not allowed to publish under his real name. Gosset assumed the pseudonym Student for this purpose. Student’s t distribution is not meant to reference anything regarding college students.


Zt

0

t (df = 5)

Standard Normal

t (df = 13)Bell-Shaped

Symmetric

‘Fatter’ Tails

Student’s t-Distribution


Upper Tail Area

df .25 .10 .05

1 1.000 3.078 6.314

2 0.817 1.886 2.920

3 0.765 1.638 2.353

t0

Student’s t Table

Assume:n = 3df = n - 1 = 2 = .10/2 =.05

2.920t Values

/ 2

.05


Degrees of freedom

Degrees of freedom refers to the number of independent data values available to estimate the population’s standard deviation. If k parameters must be estimated before the population’s standard deviation can be calculated from a sample of size n, the degrees of freedom are equal to n - k.


Degrees of Freedom (df )

1. Number of Observations that Are Free to Vary After Sample Statistic Has Been Calculated

2. Example

Sum of 3 Numbers Is 6X1 = 1 (or Any Number)X2 = 2 (or Any Number)X3 = 3 (Cannot Vary)Sum = 6

degrees of freedom = n -1 = 3 -1= 2


t-Values

where:= Sample mean= Population mean

s = Sample standard deviationn = Sample size

n

sx

t

x


Estimation Example Mean ( Unknown)

A random sample of n = 25 has = 50 and S = 8. Set up a 95% confidence interval estimate for .

X tS

nX t

S

nn n

/ , / ,

. .

. .

2 1 2 1

50 2 06398

2550 2 0639

8

2546 69 53 30

X


Central Limit Theorem

For a population with a mean and a variance 2 the sampling distribution of the means of all possible samples of size n generated from the population will be approximately normally distributed.

The mean of the sampling distribution equal to and the variance equal to 2/n.

),?(~ 2X

)/,(~ 2 nNXn The sample mean of n observation

The population distribution


Standard Error of the Sample Means

The standard error of the sample mean is the standard deviation of the sampling distribution of the sample means.

It is computed by

is the symbol for the standard error of the sample mean.

σ is the standard deviation of the population.

n is the size of the sample.

nx

x


Standard Error of the Sample Means

If is not known and n 30, the standard deviation of the sample, designated s, is used to approximate the population standard deviation. The formula for the standard error is:

n

ssx


95% and 99% Confidence Intervals for the sample mean

The 95% and 99% confidence intervals are constructed as follows:95% CI for the sample mean is given by

n

s96.1

n

s58.2

99% CI for the sample mean is given by


95% and 99% Confidence Intervals for µ

The 95% and 99% confidence intervals are constructed as follows:95% CI for the population mean is given by

n

sX 96.1

n

sX 58.2

99% CI for the population mean is given by


Constructing General Confidence Intervals for µ

In general, a confidence interval for the mean is computed by:

n

szX


EXAMPLE 3

The Dean of the Business School wants to estimate the mean number of hours worked per week by students. A sample of 49 students showed a mean of 24 hours with a standard deviation of 4 hours. What is the population mean?

The value of the population mean is not known. Our best estimate of this value is the sample mean of 24.0 hours. This value is called a point estimate.


Example 3 continued

Find the 95 percent confidence interval for the population mean.

12.100.2449

496.100.2496.1

n

sX

The confidence limits range from 22.88 to 25.12.About 95 percent of the similarly constructed intervals include the population parameter.


Confidence Interval for a Population Proportion

The confidence interval for a population proportion is estimated by:

n

ppzp

)1(


EXAMPLE 4

A sample of 500 executives who own their own home revealed 175 planned to sell their homes and retire to Arizona. Develop a 98% confidence interval for the proportion of executives that plan to sell and move to Arizona.

0497.35. 500

)65)(.35(.33.235.


Finite-Population Correction Factor

A population that has a fixed upper bound is said to be finite.

For a finite population, where the total number of objects is N and the size of the sample is n, the following adjustment is made to the standard errors of the sample means and the proportion:

Standard error of the sample means:

1

N

nN

nx


Finite-Population Correction Factor

Standard error of the sample proportions:

1

)1(

N

nN

n

ppp

This adjustment is called the finite-population correction factor.

If n/N < .05, the finite-population correction factor is ignored.


EXAMPLE 5

Given the information in EXAMPLE 4, construct a 95% confidence interval for the mean number of hours worked per week by the students if there are only 500 students on campus.

Because n/N = 49/500 = .098 which is greater than 05, we use the finite population correction factor.

0648.100.24)1500

49500)(

49

4(96.124


Selecting a Sample Size

There are 3 factors that determine the size of a sample, none of which has any direct relationship to the size of the population. They are:The degree of confidence selected. The maximum allowable error.The variation in the population.


Selecting a Sample Size

To find the sample size for a variable:

where : E is the allowable error, z is the z- value corresponding to the selected level of confidence, and s is the sample deviation of the pilot survey.

2*

*

E

sznE

n

sz

nZX

XZX

)2/

()2/

(


EXAMPLE 6

A consumer group would like to estimate the mean monthly electricity charge for a single family house in July within $5 using a 99 percent level of confidence. Based on similar studies the standard deviation is estimated to be $20.00. How large a sample is required?

1075

)20)(58.2(2

n


Sample Size for Proportions

The formula for determining the sample size in the case of a proportion is:

where p is the estimated proportion, based on past experience or a pilot survey; z is the z value associated with the degree of confidence selected; E is the maximum allowable error the researcher will tolerate.

2

)1(

E

Zppn


EXAMPLE 7

The American Kennel Club wanted to estimate the proportion of children that have a dog as a pet. If the club wanted the estimate to be within 3% of the population proportion, how many children would they need to contact? Assume a 95% level of confidence and that the club estimated that 30% of the children have a dog as a pet.

89703.

96.1)70)(.30(.

2

n


- END -

Chapter NineEstimation and Confidence IntervalsEstimation and Confidence Intervals

Ka-fu Wong © 2003 Chap 9- 1 Dr. Ka-fu Wong ECON1003 Analysis of Economic Data.

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