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Section 8.1
Estimating Population Means
( Known)
And some added content
by D.R.S., University of Cordele
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Estimating Population Means
A point estimate is a single-number estimate of a
population parameter.
An unbiased estimator is a point estimate that does
not consistently underestimate or overestimate the
population parameter.
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𝑥 is the best point estimate of 𝜇
Usually, we can’t find the exact population mean 𝜇 by
surveying the entire population (time, expense, or
sheer impossibility).
We can take a sample and found the 𝑥.
That’s as good as we can do.
But we want to express some uncertainty.
Being uncertain in a formal mathematical way.
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Confidence Intervals
math courseware specialists
8.1 Introduction to Estimating
Population Means
Confidence Interval for Population Means:
E is the
Margin of Error
We claim that μ is between these values.
We claim that μ is in this (low,high) interval
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Example 8.1: Finding a Point Estimate for a
Population Mean
Find the best point estimate for the population mean of
test scores on a standardized biology final exam. The
following is a simple random sample taken from the
population of test scores.
45
68
72
91
100 71
69
83
86
55
89
97
76
68
92
75
84
70
81
90
85
74
88
99
76
91
93
85
96
100
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Example 8.1: Finding a Point Estimate for a
Population Mean (cont.)
Solution
The best point estimate for the population mean is a
sample mean because it is an unbiased estimator. The
sample mean for the given sample of test scores is
xi

x
 81.6. Thus, the best point estimate for the
n
population mean of test scores on this standardized
exam is 81.6.
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Estimating Population Means
An interval estimate is a range of possible values for a
population parameter.
The level of confidence is the probability that the
interval estimate contains the population parameter.
A confidence interval is an interval estimate associated
with a certain level of confidence.
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Estimating Population Means
The margin of error, or maximum error of estimate, E,
is the largest possible distance from the point estimate
that a confidence interval will cover.
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Example 8.2: Constructing a Confidence Interval
with a Given Margin of Error
A college student researching study habits collects data
from a random sample of 250 college students on her
campus and calculates that the sample mean is
hours per week. If the margin of error for her data
using a 95% level of confidence is E = 0.6 hours, x  15.7
construct a 95% confidence interval for her data.
Interpret your results.
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Example 8.2: Constructing a Confidence Interval
with a Given Margin of Error (cont.)
Solution
The best point estimate for the population mean is a
sample mean, so use x  15.7 as the point estimate for
this population parameter. We are given the value of
the margin of error for our confidence interval, E = 0.6.
As indicated in Figure 8.1, the endpoints of a
confidence interval are found by subtracting E from and
adding E to the point estimate.
To find the lower endpoint, subtract the margin of error
from the sample mean; that is, calculate x  E.
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Example 8.2: Constructing a Confidence Interval
with a Given Margin of Error (cont.)
Thus, the lower endpoint is calculated as follows.
Lower endpoint: x  E  15.7  0.6
 15.1 hours per week
To find the upper endpoint, add the margin of error to
the sample mean; that is, calculate x  E. Thus, the
upper endpoint is calculated as follows.
Upper endpoint: x  E  15.7  0.6
 16.3 hours per week
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Example 8.2: Constructing a Confidence Interval
with a Given Margin of Error (cont.)
Therefore, the confidence interval ranges from 15.1 to
16.3. The confidence interval can be written
mathematically using either inequality symbols or
interval notation, as shown below.
15.1    16.3
or
15.1, 16.3
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Example 8.2: Constructing a Confidence Interval
with a Given Margin of Error (cont.)
The interpretation of our confidence interval is that we
are 95% confident that the true population mean for
the number of hours per week that students on this
campus spend studying is between 15.1 and 16.3
hours.
But this was too easy. They handed us the value of E.
We did minus on one side of the mean.
We did plus on the other side of the mean.
We really want to know “Where does that E value
come from?”
As the old saying goes, “Give a man the value of E and he will calculate one confidence
interval; teach a man how to find E and he will enjoy statistics for a lifetime.”
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Using the Standard Normal Distribution to
Estimate a Population Mean
Margin of Error of a Confidence Interval for a
Population Mean ( Known)
When the population standard deviation is known, the
sample taken is a simple random sample, and either
the sample size is at least 30 or the population
distribution is approximately normal, the margin of
error of a confidence interval for a population mean is
given by
  
E  z 2   x   z 2 
 n 
 
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 
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Using the Standard Normal Distribution to
Estimate a Population Mean
Margin of Error of a Confidence Interval for a
Population Mean ( Known) (cont.)
Where z 2 is the critical value for the level of
confidence, c = 1 − , such that the area under the
standard normal distribution to the right of z 2 is

equal to ,
2
 is the population standard deviation, and
n is the sample size.
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A convenient table for common critical values
Level of Confidence
0.80 (or 80%)
0.85 (or 85%)
0.90 (or 90%)
0.95 (or 95%)
0.98 (or 98%)
0.99 (or 99%)
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zα/2
1.28
1.44
1.645
1.96
2.33
2.575
You already know
how to get these
values. For example:
1. For an 80% level of
confidence,
2. α = 1 – 80% = 1 –
0.8000 = 0.2000
3. α / 2 = 0.1000
4. What z (and –z)
value causes area
0.8000 in the
middle and 0.1000
in each tail?
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Example 8.3: Finding the Margin of Error of a Confidence
Interval for a Population Mean ( Known)
Researchers want to estimate the mean monthly
electricity bill in a large urban area using a simple
random sample of 100 households. Assume that the
population standard deviation is known to be $15.50.
Find the margin of error for a 99% confidence interval.
Round your answer to two decimal places.
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Example 8.3: Finding the Margin of Error of a Confidence
Interval for a Population Mean ( Known) (cont.)
Solution
First, refer to the table of critical z-values to find the
critical value for c = 0.99. In the table, we see that for
c = 0.99, the critical value is z 2  z0.01 2  z0.005  2.575.
The population standard deviation is  = 15.50, and the
sample size is n = 100.
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Example 8.3: Finding the Margin of Error of a Confidence
Interval for a Population Mean ( Known) (cont.)
Substituting these values into the formula for the
margin of error, we get the following.
 
E  z 2
  


n
 15.50 
  2.575 
 100 
 3.99125
So the margin of error for this 99% confidence interval
is approximately $3.99.
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Example 8.4: Constructing a Confidence Interval
for a Population Mean ( Known)
In order to estimate the number of calls to expect at a
new suicide hotline, volunteers contact a random
sample of 35 similar hotlines across the nation and find
that the sample mean is 42.0 calls per month.
Construct a 95% confidence interval for the mean
number of calls per month. Assume that the population
standard deviation is known to be 6.5 calls per month.
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Example 8.4: Constructing a Confidence Interval
for a Population Mean ( Known) (cont.)
Step 1: Find the point estimate.
To find a confidence interval, you need to know a point
estimate and a margin of error. The point estimate for a
population mean is the sample mean. In this example,
the sample mean is given to us as 42.0 calls per month.
The margin of error must be calculated using the
formula for population means with  known.
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Example 8.4: Constructing a Confidence Interval
for a Population Mean ( Known) (cont.)
Step 2: Find the margin of error.
To calculate the margin of error, refer to the table of
critical z-values to find the critical value for c = 0.95.
Note that z 2  z0.05 2  z0.025  1.96 is the critical value.
The population standard deviation is  = 6.5, and the
sample size is n = 35.
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Example 8.4: Constructing a Confidence Interval
for a Population Mean ( Known) (cont.)
Substituting these values into the formula for the
margin of error, we get the following.
 
E  z 2
  


n
 6.5 
 1.96 
 35 
 2.153453
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Example 8.4: Constructing a Confidence Interval
for a Population Mean ( Known) (cont.)
Step 3: Subtract the margin of error from and add the
margin of error to the point estimate.
To find the lower endpoint, subtract the margin of error
from the sample mean. Thus, the lower endpoint is
calculated as follows.
Lower endpoint: x  E  42.0  2.153453
 39.8 calls per month
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Example 8.4: Constructing a Confidence Interval
for a Population Mean ( Known) (cont.)
To find the upper endpoint, add the margin of error to
the sample mean. Thus, the upper endpoint is
calculated as follows.
Upper endpoint: x  E  42.0  2.153453
 44.2 calls per month
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Example 8.4: Constructing a Confidence Interval
for a Population Mean ( Known) (cont.)
Therefore, the 95% confidence interval ranges from
39.8 to 44.2 calls per month. The confidence interval
can be written mathematically using either inequality
symbols or interval notation, as shown below.
39.8    44.2
or
 39.8, 44.2
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Example 8.4: Constructing a Confidence Interval
for a Population Mean ( Known) (cont.)
The interpretation of our confidence interval is that we
are 95% confident that the true population mean for
the numbers of calls to suicide hotlines across the
nation is between 39.8 and 44.2 calls per month.
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Example 8.5: Constructing a Confidence Interval
for a Population Mean ( Known)
A toy company wants to know the mean number of
new toys per child bought each year. Marketing
strategists at the toy company collect data from the
parents of 1842 randomly selected children. The
sample mean is found to be 4.7 toys per child.
Construct a 90% confidence interval for the mean
number of new toys per child purchased each year.
Assume that the population standard deviation is
known to be 1.9 toys per child per year.
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Example 8.5: Constructing a Confidence Interval
for a Population Mean ( Known)
Solution
Step 1: Find the point estimate.
The point estimate for the population mean is the
sample mean, which we are told is 4.7 toys per child.
Step 2: Find the margin of error.
Next, we need to calculate the margin of error. Since
we want a 90% confidence interval, we can use the
table to find the critical z-value, which is
z 2  z0.10 2  z0.05  1.645.
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Example 8.5: Constructing a Confidence Interval
for a Population Mean ( Known)
Substituting the values we have into the formula for
margin of error, we get the following.
 
E  z 2
  


n
 1.9 
 1.645 
 1842 
 0.072824
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Example 8.5: Constructing a Confidence Interval
for a Population Mean ( Known)
Step 3: Subtract the margin of error from and add the
margin of error to the point estimate.
To find the lower endpoint, subtract the margin of error
from the best point estimate, that is, the sample mean.
Lower endpoint: x  E  4.7  0.072824
 4.6 toys per child
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Example 8.5: Constructing a Confidence Interval
for a Population Mean ( Known)
To find the upper endpoint, add the margin of error to
the point estimate.
Upper endpoint: x  E  4.7  0.072824
 4.8 toys per child
Thus, the 90% confidence interval ranges from 4.6 to
4.8 new toys per child per year.
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Example 8.5: Constructing a Confidence Interval
for a Population Mean ( Known)
The confidence interval can be written mathematically
using either inequality symbols or interval notation, as
shown below.
4.6    4.8
or
 4.6, 4.8
Therefore, we are 90% confident that the true
population mean for the number of new toys per child
bought each year is between 4.6 and 4.8 toys per child.
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Example 8.6: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Mean ( Known)
The owners of a local company that produces handknitted socks want to know, for women in their area,
the average length of a woman’s foot, from toe to heel.
They collect data from 431 randomly selected women.
The sample mean is found to be 8.72 inches. Construct
a 95% confidence interval for the mean length of a
woman’s foot in the company's area. Assume the
owners know that the population standard deviation is
0.36 inches.
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Example 8.6: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Mean ( Known) (cont.)
Solution
Using a TI-83/84 Plus calculator, press
, scroll to
TESTS, and then select option 7:ZInterval, since
the population standard deviation is known. We are
given the sample statistics, so we highlight the Stats
option, and enter the values for Ç, Ë, and n. For this
example,  = 0.36, x  8.72, and n = 431. C-Level is
the confidence level, which should be entered as a
decimal. The confidence level for this example is 95%,
so we enter 0.95.
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Example 8.6: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Mean ( Known) (cont.)
Highlight Calculate and press
. The
calculator screen then gives the confidence interval in
interval form and also reiterates the sample mean and
sample size.
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Example 8.6: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Mean ( Known) (cont.)
Thus, the 95% confidence interval ranges from 8.69 to
8.75 inches. The confidence interval can be written
mathematically using either inequality symbols or
interval notation, as shown below.
8.69    8.75
or
 8.69, 8.75
Therefore, we are 95% confident that the true
population mean for the length of a woman’s foot is
between 8.69 and 8.75 inches.
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Example 8.6: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Mean ( Known)
With Excel =CONFIDENCE.NORM( α, σ, n)
• α = level of significance = 1 – level of confidence
• σ = population standard deviation
• n = sample size
Further words about ZInterval
•
•
•
If you’re asked for a confidence interval,
• Use ZInterval for a normal distrib. situation.
• It’s easier than using the primitive formula
• The calculator keeps more decimal precision
If the problem asks for a critical value of z, too,
• Then you have to use invNorm( or the printed
table to answer that question.
Make the right choice between
• Stats, if you’re given the mean, etc.
• Data, if you’re given a list of raw data
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Minimum Sample Size for Estimating a
Population Mean
Minimum Sample Size for Estimating a Population
Mean
The minimum sample size required for estimating a
population mean at a given level of confidence with a
particular margin of error is given by
 z 2   
n
 E 
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Minimum Sample Size for Estimating a
Population Mean
Minimum Sample Size for Estimating a Population
Mean (cont.)
Where z 2 is the critical value for the level of
confidence, c = 1 − , such that the area under the
standard normal distribution to the right of z 2 is equal
to  ,
2
 is the population standard deviation, and
E is the desired maximum margin of error.
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Example 8.8: Finding the Minimum Sample Size Needed for
a Confidence Interval for a Population Mean
Determine the minimum sample size needed if we wish
to be 90% confident that the sample mean is within
two units of the population mean. An estimate for the
population standard deviation of 8.4 is available from a
previous study.
Solution
From the information that we are given, we know the
following values: c = 0.90 and  ≈ 8.4. The phrase
“within two units” indicates that the desired maximum
margin of error is two, so E = 2.
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Example 8.8: Finding the Minimum Sample Size Needed for
a Confidence Interval for a Population Mean (cont.)
Since we desire a 90% level of confidence, we can use
the table of critical z-values to determine that
z 2  z0.10 2  z0.05  1.645.
Substituting these values into our formula for minimum
sample size, we obtain the following.
2
2
z


  2   1.645  8.4 
n

 47.734281  48



2
 E  
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Example 8.8 with TI-84
Since we desire a 90% level of confidence, we can use
the table of critical z-values to determine that
z 2  z0.10 2  z0.05  1.645.
2
 z 2     1.645  8.4 
n

 47.734281  48



2
 E  
2
TI-84 in one line: (1.645*8.4/2)2 (used X2 key)
Or even more slick: : (-invNorm(0.05)*8.4/2)2
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Example 8.8 with Excel
• Line 1: typing the formula with all numbers.
• Line 2: wrap it in the CEILING(value, multiple of)
function to bump up to next highest integer.
• Lines 3 and 4: same, but we use
–NORM.S.INV(0.05) instead of the table lookup.
Example 8.8: Finding the Minimum Sample Size Needed for
a Confidence Interval for a Population Mean (cont.)
So the size of the sample that we need to construct a
90% confidence interval for the population mean with
the desired margin of error is at least 48.
What’s good about this?
• We have some assurance about the results we get,
knowing how many we need in our sample.
• We know not to over-sample
• Saves time
• Saves effort
• Saves money!
Drawback? Need to know σ. Or rely on some claim about σ.
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