Suppose you are trying to determine the mean rent of a twobedroom apartment
in your town. You might look in the classified section of the newspaper, write
down several rents listed, and average them together. You would have obtained a
point estimate of the true mean. If you are trying to determine the percent of times
you make a basket when shooting a basketball, you might count the number of
shots you make and divide that by the number of shots you attempted. In this
case, you would have obtained a point estimate for the true proportion.
We use sample data to make generalizations about an unknown population. This
part of statistics is called inferential statistics. The sample data help us to
make an estimate of a population parameter. We realize that the point estimate is
most likely not the exact value of the population parameter, but close to it. After
calculating point estimates, we construct confidence intervals in which we believe
the parameter lies.
In this chapter, you will learn to construct and interpret confidence intervals. You
will also learn a new distribution, the Student'st, and how it is used with these
intervals. Throughout the chapter, it is important to keep in mind that the
confidence interval is a random variable. It is the parameter that is
fixed.
If you worked in the marketing department of an entertainment company, you
might be interested in the mean number of compact discs (CD's) a consumer
buys per month. If so, you could conduct a survey and calculate the sample
mean, x¯
x
, and the sample standard deviation, ss. You would use x¯
x
to estimate
the population mean and ss to estimate the population standard deviation. The
sample mean, x¯
x
, is the point estimate for the population mean, μμ. The sample
standard deviation, ss, is the point estimate for the population standard deviation,
σσ.
Each of x¯
x
and
ss is also called a statistic.
A confidence interval is another type of estimate but, instead of being just one number, it is an interval of numbers. The interval of numbers is a range of values calculated from a given set of sample data. The confidence interval is likely to include an unknown population parameter.
Suppose for the CD example we do not
know the population mean μμ but we do know that the population standard
deviation is σ=1σ=1 and our sample size is 100. Then by the Central Limit
Theorem, the standard deviation for the sample mean is
σ
n
=
1
100
=
0.1
σ
n
=
1
100
=0.1.
The Empirical Rule, which applies to bellshaped distributions, says that in
approximately 95% of the samples, the sample mean,
x¯
x
, will be within two standard
deviations of the population mean μμ. For our CD example, two standard deviations
is (2)(0.1) = 0.2(2)(0.1) = 0.2. The sample mean
x¯
x
is likely to be within 0.2 units of μμ.
Because
x¯
x
is within 0.2 units of μμ, which is unknown, then μμ is likely to be within 0.2 units
of
x¯
x
in 95% of the samples. The population mean μμ is contained in an interval
whose lower number is calculated by taking the sample mean and subtracting
two standard deviations ((2)(0.1)(2)(0.1)) and whose upper number is calculated by
taking the sample mean and adding two standard deviations. In other words, μμ
is between
x¯

0.2
x
0.2
and
x¯
+
0.2
x
+0.2
in 95% of all the samples.
For the CD example, suppose that a sample produced a sample mean
x¯
=
2
x
=2. Then the
unknown population mean μμ is between
x¯

0.2
=
2

0.2
=
1.8
x
0.2=20.2=1.8
and
x¯
+
0.2
=
2
+
0.2
=
2.2
x
+0.2=2+0.2=2.2
We say that we are 95% confident that the unknown population mean number of CDs
is between 1.8 and 2.2. The 95% confidence interval is (1.8, 2.2).
The 95% confidence interval implies two possibilities. Either the interval (1.8, 2.2)
contains the true mean μμ or our sample produced an
x¯
x
that is not within 0.2 units of
the true mean μμ. The second possibility happens for only 5% of all the samples
(100%  95%).
Remember that a confidence interval is created for an unknown population parameter
like the population mean, μμ. Confidence intervals for some parameters have the form
(point estimate  margin of error, point estimate + margin of error)
The margin of error depends on the confidence level or percentage of confidence.
When you read newspapers and journals, some reports will use the phrase
"margin of error." Other reports will not use that phrase, but include a confidence interval as the point estimate + or  the margin of
error. These are two ways of expressing the same concept.
Although the text only covers symmetric confidence intervals, there are nonsymmetric confidence intervals (for example, a confidence interval for the standard deviation).
"Reviewer's Comments: 'I recommend this book. Overall, the chapters are very readable and the material presented is consistent and appropriate for the course. A wide range of exercises introduces […]"