Transcript Document

Chapter 13
Confidence intervals: the basics
BPS - 3rd Ed.
Chapter 13
1
Statistical Inference
 Two
general types of statistical inference
– Confidence Intervals (introduced this chapter)
– Tests of Significance (introduced next chapter)
BPS - 3rd Ed.
Chapter 13
2
Starting Conditions
1.
2.
3.
SRS from population
Normal distribution X~N(m, s) in the
population
Although the value of m is unknown,
the value of the population standard
deviation s is known
BPS - 3rd Ed.
Chapter 13
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Case Study
NAEP Quantitative Scores
(National Assessment of Educational Progress)
Rivera-Batiz, F. L. (1992). Quantitative literacy and the
likelihood of employment among young adults. Journal of
Human Resources, 27, 313-328.
The NAEP survey includes a short test of quantitative
skills, covering mainly basic arithmetic and the ability to
apply it to realistic problems. Young people have a better
chance of good jobs and wages if they are good with
numbers.
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Chapter 13
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Case Study
NAEP Quantitative Scores

Given:
– Scores on the test range from 0 to 500
– Higher scores indicate greater numerical ability
– It is known NAEP scores have standard deviation s = 60.

In a recent year, 840 men 21 to 25 years of age were
in the NAEP sample
– Their mean quantitative score was 272 (x-bar).
– On the basis of this sample, estimate the mean score µ in
the population of 9.5 million young men in this age range
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Case Study
NAEP Quantitative Scores
1.
2.
3.
To estimate the unknown population mean m,
use the sample mean x = 272.
The law of large numbers suggests that x
will be close to m, but there will be some error in
the estimate.
The sampling
 distribution of x has a Normal
distribution with unknown mean
 m and standard
deviation:
s
60

 2.1
n
840

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Chapter 13
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Case Study
NAEP Quantitative Scores
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Case Study
NAEP Quantitative Scores
4.
The 68-95-99.7 rule
indicates that
x and m are within
two standard
deviations (4.2) of
each other in about
95% of all samples.
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Case Study
NAEP Quantitative Scores
So, if we estimate that m lies within 4.2 of
we’ll be right about 95% of the time.
x,
x  4.2 is a 95% confidence interval for µ
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NAEP Illustration (cont.)
x  4.2 is a 95% confidence interval for µ

The confidence interval has the form
estimate ± margin of error
estimate (x-bar in this case) is our guess for
unknown µ
 margin of error (± 4.2 in this case) shows
accuracy of estimate

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Level of Confidence (C)


Probability that interval will capture the true
parameter in repeated samples; the “success rate”
for the method
You can choose any level of confidence, but the most
common levels are:
– 90%
– 95%
– 99%

e.g., If we use 95% confidence, we are saying “we
got this interval by a method that gives correct results
95% of the time” (next slide)
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Fig 13.4


BPS - 3rd Ed.
Chapter 13
Twenty-five
samples from the
same population
gave 25 95%
confidence intervals
In the long run, 95%
of samples give an
interval that capture
the true population
mean µ
12
Confidence Interval
Mean of a Normal Population
Take an SRS of size n from a Normal
population with unknown mean m and
known standard deviation s. A “level C”
confidence interval for m is:
σ
x z
n

Confidence level C
90%
95%
Critical value z*
1.645
1.960 2.576
BPS - 3rd Ed.
Chapter 13
99%
13
Confidence Interval
Mean of a Normal Population
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Case Study
NAEP Quantitative Scores
Using the 68-95-99.7 rule gave an approximate 95%
confidence interval. A more precise 95% confidence
interval can be found using the appropriate value of z*
(1.960) with the previous formula
x  (1.960)(2. 1) = 272  4.116 = 267.884
x  (1.960)(2. 1) = 272  4.116 = 276.116
We are 95% confident that the average NAEP
quantitative score for all adult males is between
267.884 and 276.116.
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How Confidence Intervals Behave
 The
margin of error is:
margin of error = z

s
n
 The
margin of error gets smaller, resulting in
more accurate inference,
– when n gets larger
– when z* gets smaller (confidence level gets
smaller)
– when s gets smaller (less variation)
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Case Study
NAEP Quantitative Scores
95% Confidence Interval
x  (1.960)(2. 1) = 272  4.116 = 267.884
x  (1.960)(2. 1) = 272  4.116 = 276.116
90% Confidence Interval
x  (1.645)(2. 1) = 272  3.4545 = 268.5455
x  (1.645)(2. 1) = 272  3.4545 = 275.4545
The 90% CI is narrower than the 95% CI.
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Choosing the Sample Size
The confidence interval for the mean of
a Normal population will have a
specified margin of error m when the
sample size is:
z σ 

n  

m



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18
Case Study
NAEP Quantitative Scores
Suppose that we want to estimate the
population mean NAEP scores using a 90%
confidence interval, and we are instructed to do
so such that the margin of error does not
exceed 3 points.
What sample size will be required to enable us
to create such an interval?
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Case Study
NAEP Quantitative Scores

2
 z σ   (1.645)(60 ) 
 
n
 1082.41

 m  
3



2
Thus, we will need to sample at least 1082.41 men
aged 21 to 25 years to ensure a margin of error not to
exceed 3 points.
Note that since we can’t sample a fraction of an
individual and using 1082 men will yield a margin of
error slightly more than 3 points, our sample size
should be n = 1083 men.
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