ECON 4630
ECON 5630
TOPIC #8: ESTIMATION AND CONFIDENCE INTERVALS
I.
Introduction
A.
Motivation
B.
Definitions
1.
A point estimate is calculated from sample data in order to estimate an
unknown population parameter. For example,
2.
A confidence interval is a range of values between which the true mean
will probably lie. The specific probability is called the level of
confidence. For example,
1
II.
Population standard deviation (σ) is known
A.
General
If we know the true population standard deviation:
90% of the sample means will lie within _______ standard deviations of μ
95% of the sample means will lie within _______ standard deviations of μ 99% of
the sample means will lie within _______ standard deviations of μ
In other words, our 90%, 95%, and 99% confidence intervals can be specified as follows:
A 90% confidence interval for a mean will be:
A 95% confidence interval for a mean will be:
A 99% confidence interval for a mean will be:
More generally, a confidence interval for the mean will be:
What about an 86% confidence interval?
2
3
B.
Example
Suppose we are interested in the lifetimes of light bulbs produced by a particular
manufacturer. Suppose we draw a sample of 49 light bulbs, and discover that the sample
mean is 1200 hours with the sample standard deviation equal to 400 hours. Let us construct
and interpret 90%, 95%, and 99% confidence intervals.
4
III.
σ is unknown
A. General
In this case, we need to make two revisions to our earlier confidence interval formula:
Use S rather than σ
Use Student’s t rather than standard normal
5
Student’s t Distribution
df
0.100
1
2
3
4
5
0.020
3.078
1.886
1.638
1.533
1.476
Confidence Intervals
90%
95%
98%
99%
Level of Significance for One-Tailed Test
0.050
0.025
0.010
0.005
Level of Significance for Two-Tailed Test
0.10
0.05
0.02
0.01
6.314
12.706
31.821
63.657
2.920
4.303
6.965
9.925
2.353
3.182
4.541
5.841
2.132
2.776
3.747
4.604
2.015
2.571
3.365
4.032
6
7
8
9
10
1.440
1.415
1.397
1.383
1.372
1.943
1.895
1.860
1.833
1.812
2.447
2.365
2.306
2.262
2.228
3.143
2.998
2.896
2.821
2.764
3.707
3.499
3.355
3.250
3.169
5.959
5.408
5.041
4.781
4.587
11
12
13
14
15
1.363
1.356
1.350
1.345
1.341
1.796
1.782
1.771
1.761
1.753
2.201
2.179
2.160
2.145
2.131
2.718
2.681
2.650
2.624
2.602
3.106
3.055
3.012
2.977
2.947
4.437
4.318
4.221
4.140
4.073
16
17
18
19
20
1.337
1.333
1.330
1.328
1.325
1.746
1.740
1.734
1.729
1.725
2.120
2.110
2.101
2.093
2.086
2.853
2.567
2.552
2.539
2.528
2.921
2.898
2.878
2.861
2.845
4.015
3.965
3.922
3.883
3.850
21
22
23
24
25
1.323
1.321
1.319
1.318
1.316
1.721
1.717
1.714
1.711
1.708
2.080
2.074
2.069
2.064
2.060
2.518
2.508
2.500
2.492
2.485
2.831
2.819
2.807
2.797
2.787
3.819
3.792
3.768
3.745
3.725
26
27
28
29
30
1.315
1.314
1.313
1.311
1.310
1.706
1.703
1.701
1.699
1.697
2.056
2.052
2.048
2.045
2.042
2.479
2.473
2.467
2.462
2.457
2.779
2.771
2.763
2.756
2.750
3.707
3.690
3.674
3.659
3.646
40
60
120

1.303
1.296
1.289
1.282
1.684
1.671
1.658
1.645
2.021
2.000
1.980
1.960
2.423
2.390
2.358
2.326
2.704
2.660
2.617
2.576
3.551
3.460
3.373
3.291
80%
99.9%
0.0005
0.001
636.619
31.599
12.924
8.610
6.869
6
A confidence interval for the mean in this situation is:
X t
S
,
n
where t is the critical value from the t-table.
B.
Example
The owner of Britten’s Egg Farm wants to estimate the mean number of eggs laid
per chicken. A sample of 20 chickens shows they laid an average of 20 eggs per
month with a standard deviation of 2 eggs per month.
a.
What is the value of the population mean? What is the best estimate of this
value?
b.
Explain why we need to use the t-distribution. What assumption do you
need to make?
c.
For a 95% confidence interval, what is the value of t?
d.
Develop the 95% confidence interval for the population mean.
e.
Would it be reasonable to conclude that the population mean is 21 eggs?
What about 25 eggs?
7
IV.
Confidence Intervals for Proportions
A.
General
What if our interest is in the proportion (π) of the population with some
characteristic?
We can estimate π with the sample proportion, p:
The confidence interval for a population proportion looks like this:
pz
B.
p (1  p )
n
Example
In a recent of 332 high school students conducted by the online polling website
whatthenationthinks.com, 222 said that if they found $1,000 behind a bush at
school they would keep the money. Develop a 90% confidence interval for the
population proportion.
8
V.
Choosing the Appropriate Sample Size
A.
General
How large should the sample size be?
Above we discussed the “margin of error,” E  z
S
n
With a little algebra,
B.
Example
In a recent survey of small and microenterprises the sample standard deviation for a
continuous variable (number of employees per firm) was found to be 2.15.
Suppose the researcher desires a 95% confidence level, and the tolerable margin of
error is 0.05 workers. What should the sample size be?
9
NOTES:
10