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Confidence Intervals

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Title: Confidence Intervals


1
Confidence Intervals
Chapter 6
2
6.1
  • Confidence Intervals for the Mean (Large
    Samples)

3
Point Estimate for Population µ
A point estimate is a single value estimate for a
population parameter. The most unbiased point
estimate of the population mean, ?, is the sample
mean, ?.
Example A random sample of 32 textbook prices
(rounded to the nearest dollar) is taken from a
local college bookstore. Find a point estimate
for the population mean, ?.
34 34 38 45 45 45 45 54
56 65 65 66 67 67 68 74
79 86 87 87 87 88 90 90
94 95 96 98 98 101 110 121
The point estimate for the population mean of
textbooks in the bookstore is 74.22.
4
Interval Estimate
An interval estimate is an interval, or range of
values, used to estimate a population parameter.
How confident do we want to be that the interval
estimate contains the population mean, µ?
5
Level of Confidence
The level of confidence c is the probability that
the interval estimate contains the population
parameter.
c is the area beneath the normal curve between
the critical values.
Use the Standard Normal Table to find the
corresponding z-scores.
The remaining area in the tails is 1 c .
6
Common Levels of Confidence
If the level of confidence is 90, this means
that we are 90 confident that the interval
contains the population mean, µ.
zc 1.645
?zc ? 1.645
The corresponding z-scores are 1.645.
7
Common Levels of Confidence
If the level of confidence is 95, this means
that we are 95 confident that the interval
contains the population mean, µ.
zc 1.96
?zc ? 1.96
The corresponding z-scores are 1.96.
8
Common Levels of Confidence
If the level of confidence is 99, this means
that we are 99 confident that the interval
contains the population mean, µ.
zc 2.575
?zc ? 2.575
The corresponding z-scores are 2.575.
9
Margin of Error
The difference between the point estimate and the
actual population parameter value is called the
sampling error.
When µ is estimated, the sampling error is the
difference µ ?. Since µ is usually unknown,
the maximum value for the error can be calculated
using the level of confidence.
Given a level of confidence, the margin of error
(sometimes called the maximum error of estimate
or error tolerance) E is the greatest possible
distance between the point estimate and the value
of the parameter it is estimating.
10
Margin of Error
Example A random sample of 32 textbook prices is
taken from a local college bookstore. The mean
of the sample is ? 74.22, and the sample
standard deviation is s 23.44. Use a 95
confidence level and find the margin of error for
the mean price of all textbooks in the bookstore.

We are 95 confident that the margin of error for
the population mean (all the textbooks in the
bookstore) is about 8.12.
11
Confidence Intervals for µ
A c-confidence interval for the population mean µ
is ? ? E lt µ lt ? E. The probability that
the confidence interval contains µ is c.
Example A random sample of 32 textbook prices is
taken from a local college bookstore. The mean
of the sample is ? 74.22, the sample standard
deviation is s 23.44, and the margin of error
is E 8.12. Construct a 95 confidence interval
for the mean price of all textbooks in the
bookstore.
Continued.
12
Confidence Intervals for µ
Example continued Construct a 95 confidence
interval for the mean price of all textbooks in
the bookstore.
? 74.22
s 23.44
E 8.12
? ? E 74.22 8.12
? E 74.22 8.12
66.1
82.34
With 95 confidence we can say that the cost for
all textbooks in the bookstore is between 66.10
and 82.34.
13
Finding Confidence Intervals for µ
Finding a Confidence Interval for a Population
Mean (n ? 30 or s known with a normally
distributed population)
In Words In Symbols
  1. Find the sample statistics n and ?.
  2. Specify ?, if known. Otherwise, if n ? 30, find
    the sample standard deviation s and use it as an
    estimate for ?.
  3. Find the critical value zc that corresponds to
    the given level of confidence.
  4. Find the margin of error E.
  5. Find the left and right endpoints and form the
    confidence interval.

Use the Standard Normal Table.
Left endpoint ??E Right endpoint ? E Interval
??E lt µ lt ? E
14
Confidence Intervals for µ (? Known)
Example A random sample of 25 students had a
grade point average with a mean of 2.86. Past
studies have shown that the standard deviation is
0.15 and the population is normally distributed.
Construct a 90 confidence interval for the
population mean grade point average.
n 25
? 2.86
? 0.15
zc 1.645
2.81 lt µ lt 2.91
? E 2.86 0.05
With 90 confidence we can say that the mean
grade point average for all students in the
population is between 2.81 and 2.91.
15
Sample Size
Given a c-confidence level and a maximum error of
estimate, E, the minimum sample size n, needed to
estimate ?, the population mean, is If ? is
unknown, you can estimate it using s provided you
have a preliminary sample with at least 30
members.
Example You want to estimate the mean price of
all the textbooks in the college bookstore. How
many books must be included in your sample if you
want to be 99 confident that the sample mean is
within 5 of the population mean?
Continued.
16
Sample Size
Example continued You want to estimate the mean
price of all the textbooks in the college
bookstore. How many books must be included in
your sample if you want to be 99 confident that
the sample mean is within 5 of the population
mean?
? 74.22
? ? s 23.44
zc 2.575
(Always round up.)
You should include at least 146 books in your
sample.
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