Lesson7-1 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Lesson 7:
Estimation and Confidence Estimation and Confidence IntervalsIntervals
Lesson7-2 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Outline
Point and interval estimates
Confidence intervals
Student’s t-distribution
Degree of freedom
Confidence interval for population mean
Confidence interval for a population proportion
Selecting a sample size
Summary
Lesson7-3 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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.
Lesson7-4 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Confidence Intervals
The degree to which we can rely on the statistic is as important as the initial calculation.
Samples give us estimates of the population parameter – only 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 Exact closeness not known because knowing exact
closeness requires knowing unknown population parameter
Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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.
Lesson7-6 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
P(µ-1σ <x<µ+1σ) vsP(x-1σ <µ <x+1σ)
P(µ-1σ <x<µ+1σ) is the probability that a randomly 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).
Lesson7-7 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
P(µ-1σm <x<µ+1 σm) vsP(m-1 σm <µ <m+1 σm) (m=sample mean)
P(µ-1 σm <m<µ+1 σm) is the probability that a randomly drawn sample will have a sample mean between (µ-1σ, µ+1σ).
P(µ-1 σm <m<µ+1 σm)
= P(µ-1 σm -µ-m <x -µ-m<µ +1 σm -µ-m)
= P(-1 σm -m <-µ<1 σm -m)
= P(-(-1 σm -m )>-(-µ)>-(1 σm -m))
= P(1 σm +m>µ>-1 σm +m)
= P(m - 1 σm <µ <m+1 σm)
P(m-1 σm <µ <m+1 σm) is the probability that the population mean will lie between (m-1 σm , m+1 σm).
Lesson7-8 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
P(µ-a <x<µ+b) vs P(x-a<µ <x+b)
P(µ-a <x<µ+b) is the probability that a drawn observation will lie between (µ-a, µ+b).
P(x-a <µ <x+b) is the probability that the population mean will lie between (x - a, x+ b).
Generally, P(µ-a <x<µ+b) = P(x-a <µ <x+b)
Generally, P(µ-a <x<µ+b) and P(x-a <µ <x+b) are not equal. They are equal only if a = b. That is, when the confidence interval is symmetric.
Generally, P(µ-a <x<µ+b) and P(x-a <µ <x+b) are not equal. They are equal only if a = b. That is, when the confidence interval is symmetric.
NO!!!!
Lesson7-9 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
P(µ-a <x<µ+b) = P(x-b <µ <x+a)
P(µ-a <x<µ+b) is the probability that a drawn observation will lie between (µ-a, µ+b).
P(µ-a <x<µ+b) = P(µ-a -µ-x <x -µ-x<µ +b -µ-x) = P(-a -x <-µ<b -x)= P(-(-a -x )>-(-µ)>-(b -x))= P(a +x >µ>-b +x)
= P(x - b <µ <x+a)
P(x-b <µ <x+a) is the probability that the population mean will lie between (x - b, x+ a).
Lesson7-10 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Elements of Confidence Interval Estimation
Confidence Interval
Sample Statistic
Confidence Limit (Lower)
Confidence Limit (Upper)
We are concerned about the probability that the population parameter falls somewhere within the interval around the sample statistic.
XZX
XX
ZX
Generally, we consider symmetric confidence intervals only.Generally, we consider symmetric confidence intervals only.
Lesson7-11 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Confidence Intervals
90% Samples
95% Samples
99% Samples
x_
nZ
XZ
X
XXXX 58.2645.1645.158.2
XX 96.196.1
The likelihood (probability) that the sample mean of a randomly drawn sample will fall within the interval:
Lesson7-12 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Confidence Intervals
)(X
ZXX
ZP
The likelihood (or probability) that the sample mean will fall within “1 standard deviation” of the population mean is the same as the likelihood (or probability) that the population mean will fall within “1 standard deviation” of the sample mean.
Z
1.645
1.96
2.58
0.90
0.95
0.99
0.90
0.95
0.99
)(X
ZXX
ZXP
Lesson7-13 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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%
Lesson7-14 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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 we were to draw 1000 samples and construct
the 95% confidence interval for the population mean for each of the 1000 samples.
Some of the intervals contain the population mean, some not.
If the interval is a 95% confidence interval, about 950 of the confidence intervals will contain the population mean.
That is, 95% of the samples will contain the population mean.
(1-)
(1-)
(1-)
(1-)
Lesson7-15 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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
Lesson7-16 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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/
(
=
1 - /2/2
X_
x_
x
Lesson7-17 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Point and Interval Estimates
We may use the z distribution if one of the following conditions hold: The population is normal and its standard deviation
is known The sample has more than 30 observations (The
population standard deviation can be known or unknown).
n
szX
Technical note: If the random variables A and B are normally distributed,
Y = A+B and X=(A+B)/2 will be normally distributed. If the population is normal, the sample mean of a
random sample of n observations (for any integer n) will be normally distributed.
Lesson7-18 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Point and Interval Estimates
Use the t distribution if all of the following conditions are fulfilled: The population is normal The population standard deviation is unknown
and the sample has less than 30 observations.
n
stX
Note that the t distribution does not cover those non-normal populations.
Lesson7-19 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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.
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.
Lesson7-20 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Zt
0
t (df = 5)
Standard Normal
t (df = 13)Bell-Shaped
Symmetric
‘Fatter’ Tails
Student’s t-Distribution
Lesson7-21 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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
Lesson7-22 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Degrees of freedom (df)
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.
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
Lesson7-23 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
t-Values
where:= Sample mean= Population mean
s = Sample standard deviationn = Sample size
n
sx
t
x
Lesson7-24 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Confidence interval for mean ( unknown in small sample)
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
Lesson7-25 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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
Lesson7-26 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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
Lesson7-27 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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
Lesson7-28 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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
Lesson7-29 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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
Lesson7-30 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Constructing General Confidence Intervals for µ
In general, a confidence interval for the mean is computed by:
n
szX
Lesson7-31 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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.
Lesson7-32 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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.
Lesson7-33 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Confidence Interval for a Population Proportion
The confidence interval for a population proportion is estimated by:
1)ˆ1(ˆ
ˆ 2/
npp
Zp
Lesson7-34 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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.
0456.35. 1500
)65)(.35(.33.235.
33.2
02.098.0)1(
01.02/
ZZ
Lesson7-35 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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 when is
known:
1
N
nN
nx
Standard error of the sample means when is NOT known and need to be estimated by s:
NnN
ns
x
Lesson7-36 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Finite-Population Correction Factor
Standard error of the sample proportions:
NnN
npp
p
1)ˆ1(ˆ
ˆ ˆ
This adjustment is called the finite-population correction factor.
If n/N < .05, the finite-population correction factor is ignored.
Lesson7-37 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Finite-Population Correction Factor
Given the information in EXAMPLE 3, 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.
0102.100.24)500
49500)(
494
(96.124
0648.100.24)1500
49500)(
49
4(96.124
Lesson7-38 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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.
Lesson7-39 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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/
(
Lesson7-40 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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
Lesson7-41 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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
Lesson7-42 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
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
Lesson7-43 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Summary: Confidence interval for sample mean
General confidence interval:
ˆ),(ˆ nr
( = population mean; = confidence level; = standard deviation)
Sample Size (n)
<30
≥30
known unknown
Normal
Population distribution Unknow
nNormal
Population distribution Unknow
n
nZ
2/ˆnn
t
ˆ1,2/ˆ
nZ
ˆ
2/ˆ n
Z
2/ˆ
2/1
)1/(2)ˆ(ˆ
n
ix
? ?
Lesson7-44 Ka-fu Wong © 2004 ECON1003: Analysis of Economic Data
Summary: Confidence Interval for sample proportion
General confidence interval: p
nrp ˆ),(ˆ (p= population mean; = confidence level; = standard deviation)
Sample Size (n)
<30
≥30
known unknown
Normal
Population distribution Unknow
nNormal
Population distribution Unknow
n
nZp
2/ˆ
nntp
ˆ1,2/
ˆ
nZp
ˆ2/
ˆ n
Zp
2/
ˆ
2/1)ˆ1(ˆˆ pp
Because = p(1-p), we know if only if we know p. If we know p, there is no need to estimate p or to construct the confidence interval for p.
1ˆn
1ˆn
2/1)1( pp
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