Transcript STA 291-021 Summer 2007

Lecture 15
Dustin Lueker

The width of a confidence interval
◦
◦
◦
◦
Increases
Increases
Increases
Increases
as
as
as
as
the
the
the
the
confidence level increases
error probability decreases
standard error increases
sample size n decreases
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

n≥30
x  Z / 2
s
n
x  t / 2
s
n
n<30
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
Start with the confidence interval formula
assuming that the population standard
deviation is known
x  Z / 2

s
 x  ME
n
Mathematically we need to solve the above
equation for n
2
 Z / 2 
ns 

 ME 
2
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
The sample proportion is an unbiased and
efficient estimator of the population
proportion
◦ The proportion is a special case of the mean
pˆ  Z / 2
pˆ (1  pˆ )
n
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
To calculate the confidence interval, we use
the Central Limit Theorem (np and nq ≥ 5)
◦ What if this isn’t satisfied?

Instead of the typical p̂ estimator, we will
use
x2
~
p
n4

Then the formula for confidence interval
becomes
~
~
p (1  p )
~
p  Z 2
n4
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
As with a confidence interval for the sample
mean a desired sample size for a given
margin of error (E) and confidence level can
be computed for a confidence interval about
the sample proportion
2
 Z / 2 
n  pˆ (1  pˆ )

 E 
◦ This formula requires guessing p̂ before taking the
sample, or taking the safe but conservative
approach of letting p̂ = .5
 Why is this the worst case scenario? Or the
conservative approach?
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
Two independent samples
◦ Different subjects in the different samples
◦ Two subpopulations
 Ex: Male/Female
◦ The two samples constitute independent samples from two
subpopulations

Two dependent samples
◦ Natural matching between an observation in one sample
and an observation in the other sample
 Ex: Two measurements of the same subject
 Left/right hand
 Performance before/after training
◦ Important: Data sets with dependent samples require
different statistical methods than data sets with
independent samples
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
Is the proportion who favor national health
insurance different for Democrats and
Republicans?
◦ Democrats and Republicans would be your two samples
◦ Yes and No would be your responses, how you’d find
your proportions

Is the proportion of people who experience pain
different for the two treatment groups?
◦ Those taking the drug and placebo would be your two
samples
 Could also have them take different drugs
◦ No pain or pain would be your responses, how you’d
find your proportions
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

Take independent samples from both groups
Sample sizes are denoted by n1 and n2
◦ To use the large sample approach both samples
should be greater than 30
 Subscript notation is same for sample means
( x1  x2 )  Z / 2
2
1
2
2
s
s

n1 n2
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
In the 1982 General Social Survey, 350
subjects reported the time spent every day
watching television. The sample yielded a
mean of 4.1 and a standard deviation of 3.3.
In the 1994 survey, 1965 subjects yielded a
sample mean of 2.8 hours with a standard
deviation of 2.
◦ Construct a 95% confidence interval for the
difference between the means in 1982 and 1994.
 Is it plausible that the mean was the same in both
years?
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
For large samples
◦ For this we will consider a large sample to be those
with at least five observations for each choice
(success, failure)
 All we will deal with in this class

Large sample confidence interval for p1-p2
pˆ1  pˆ 2  Z / 2
pˆ1 (1  pˆ1 ) pˆ 2 (1  pˆ 2 )

n1
n2
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
Two year Italian study on the effect of condoms
on the spread of HIV
◦ Heterosexual couples where one partner was infected
with HIV virus
 171 couples who always used condoms, 3 partners became
infected with HIV
 55 couples who did not always use a condom, 8 partners
became infected with HIV
◦ Estimate the infection rates for the two groups
◦ Construct a 95% confidence interval to compare them
 What can you conclude about the effect of condom use on
being infected with HIV from the confidence interval?
 Was your Sex Ed teacher lying to you?
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