Transcript STA 291-021 Summer 2007

Lecture 14
Dustin Lueker
x  Z / 2

s
n
This interval will contain μ with a 100(1-α)%
confidence
◦ If we are estimating µ, then why it is unreasonable
for us to know σ?
 Thus we replace σ by s (sample standard deviation)
 This formula is used for large sample size (n≥30)
 If we have a sample size less than 30 a different
distribution is used, the t-distribution, we will get to this
later
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
Incorrect statement
◦ With 95% probability, the population mean will fall
in the interval from 3.5 to 5.2

To avoid the misleading word “probability” we
say
◦ We are 95% confident that the true population mean
will fall between 3.5 and 5.2
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
Changing our confidence level will change
our confidence interval
◦ Increasing our confidence level will increase the
length of the confidence interval
 A confidence level of 100% would require a confidence
interval of infinite length
 Not informative

There is a tradeoff between length and
accuracy
◦ Ideally we would like a short interval with high
accuracy (high confidence level)
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
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
 Why are each of these true?
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
Start with the confidence interval formula
assuming that the population standard
deviation is known

x  Z / 2

n
 xE
Mathematically we need to solve the above
equation for n
2
 Z / 2 
n  

 E 
2
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
About how large a sample would have been
adequate if we merely needed to estimate the mean
to within 0.5, with 95% confidence? Assume   5
Note: We will always round the sample size up to ensure that we get within the desired
error bound.
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
To account for the extra variability of using a
sample size of less than 30 the student’s tdistribution is used instead of the normal
distribution
x  t / 2
s
n
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



t-distributions are
bell-shaped and
symmetric around
zero
The smaller the
degrees of freedom
the more spread out
the distribution is
t-distribution look
much like normal
distributions
In face, the limit of the
t-distribution is a
normal distribution as
n gets larger
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
Need to know α and degrees of freedom (df)
◦ df = n-1

α=.05, n=23
◦ tα/2=

α=.01, n=17
◦ tα/2=

α=.1, n=20
◦ tα/2=
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
A sample of 12 individuals yields a mean of
5.4 and a variance of 16. Estimate the
population mean with 98% confidence.
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