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Statistical Inference
Confidence Interval for the Population Mean
EC103 Lecture 19 Yuji Tamura
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We have so far looked at
point estimates
such as
the sample mean for the population mean
and
the sample standard deviation
for the population standard deviation.
A point estimate is a statistic, computed from sample
information, which is used to estimate a population
parameter of interest.
EC103 Lecture 19 Yuji Tamura
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The confidence interval
… is a range of values
constructed from sample data
so that the population parameter is likely
to lie within that range
at a specified probability
(at a level of confidence).
EC103 Lecture 19 Yuji Tamura
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Revisiting
the sampling distribution of the sample mean
For a sufficiently large sample (greater than 30),
the central limit theorem applies.
As a consequence, …
EC103 Lecture 19 Yuji Tamura
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95% confidence interval
• 95% of the sample means selected from a
population will lie within 1.96 standard errors of the
mean
99% confidence interval
• 99% of the sample means selected from a
population will lie within 2.58 standard errors of the
mean
EC103 Lecture 19 Yuji Tamura
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The percentage refers to
the middle 95(or 99)% of the observations.
From the table for the standard normal distribution,
we can see that 1.96 and 2.58 are the z values
to get the area .95 and .99 respectively.
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A confidence interval for a given sample mean is
obtained by
x ± zσ / n
if the population standard error is known.
But we often come across the case where it is
unknown. If we have a sufficiently large sample, we
can replace this with the sample counterpart, ie,
x ± zs / n
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σ / n is the standard error of the mean.
s / n can replace it if s is based on a sufficiently
large sample, ie, more than 30 observations.
We have z = 1.96 if we wanted a 95% confidence
interval.
We have z = 2.58 if we wanted a confidence interval
at a higher level, i.e., 99%.
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Example
Take a random sample of 49 observations from a
normally distributed population.
If the sample mean is 55 and the standard error of the
mean is 10, …
… what's the 99% confidence interval for the
population mean?
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55 ± 2.58(10 / 49 )
1/ 2
gives us 51.314 and 58.686
So we are 99%-confident that the population mean
takes a value between these two.
EC103 Lecture 19 Yuji Tamura
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