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6.1 Estimating with
Confidence
How can we be sure?
AGAIN, we look at this question about the
mean of a sample estimating the mean of
a population. This time, however, we will
be able to say how “good” our mean
sample is compared to the mean
population, even though we don’t know
what that is.
We define a sample mean x to be a point
estimate for the population μ.
But rather than giving a point estimate for the
mean of the population, we seek to give an
interval estimate; that is, an interval in which we
think the population mean lies (obviously the
smaller the better).
We will also be able to give a level of confidence
which answers the question ‘how sure are you?’
e.g. 95% sure, 98% sure etc.
Let’s show how we might go about this in
general on the board.
We have shown the
following:
For a sample size
n>29, a 1-α
confidence interval for
a population mean μ
is given by
x  z / 2


   x  z / 2
n
n
Margin of error
Notice in the above inequality that we
have taken our point estimate and simply
added zα/2(σ/n1/2) on the right side of the
inequality and subtracted zα/2(σ/n1/2) on the
left side of the inequality. We define the
margin of error of the estimate with
confidence 1-α to be m=zα/2(σ/n1/2).
Levels of Confidence
The following table is
helpful to have in front
of you when
computing confidence
intervals:
These and any other
z values are easily
found by looking at
the table way in the
back.
Level of
zα/2
Confidence
1-α
90%
1.65
95%
1.96
98%
2.33
99%
2.58
A physician wanted to estimate the mean
length of time μ that a patient had to wait
to see him after arriving at the office. A
random sample of 50 patients showed a
mean waiting time of 23.4 minutes and a
standard deviation of 7.1 minutes. Find a
95% confidence interval for μ and identify
the maximum error of estimate.
The mean monthly rent of an SRS of 44
unfurnished one bedroom apartments in
Boston is $1400. Assume that the
standard deviation is $220. Find a 95%
confidence interval for the margin of error.
Practice Problem
Suppose a student measuring the boiling
temperature of a certain liquid observes
readings (in degrees Celsius). He
calculates the sample mean of 124
readings to be 101.82. If he knows that the
standard deviation for this procedure is 1.2
degrees, what is the confidence interval
for the population mean at a 98%
confidence level?
Determining the Sample Size
Sometimes we may have a
certain confidence interval in
mind that we would like to be
able to say our mean is within.
In this case, we may ask what
sample size we would need in
order to fall within this interval.
Let’s do this on the board.
Hence, if we know the
standard deviation and would
like the mean to fall within a
certain confidence interval, we
must choose n at least
n
z
2

 /2
m
2
2
How many households in a large town
should be randomly sampled to estimate
the mean number of dollars spent per
household per week on food supplies to
within $3 with 80% confidence? Use
σ=15.