Download Summary of Process for Constructing Confidence Intervals for

MAT140: Applied Statistical Methods
Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters
Inferences about a population parameter can be made using sample statistics for the following:
1. Proportion (percentage of population that possesses some characteristic or trait)
2. Mean (average value of whatever quantity is being measured for the population)
3. Variance (NOT standard deviation, although variance is used to work with standard deviation)
A confidence interval (CI) for one of the parameters listed above is generally constructed through the
following process:
1. Determine the corresponding statistic from a sample
2. Determine the range of values (upper and lower bounds) assumed to contain the true value of the
population parameter at the stated level of confidence
It should be noted that although following the above procedure is the general process for making
inferences about a population parameter based on a sample statistic, each of the three parameters for
which valid inferences can be made has its own unique formulas to use. Furthermore, obtaining the
range of values that is hoped to contain the true value of the actual population parameter does not
guarantee that the true value of the population parameter actually falls in that range.
Related to confidence intervals is the concept of determining the minimum sample size necessary to
ensure valid results, which is done simply by solving the pertinent formula for
where applicable.
The following variables and abbreviations are used at various times when performing calculations
related to confidence intervals and, later on, hypothesis testing (which is covered in a future chapter):
=
=
=
=
=
=
=
=
=
=
=
=
̂
⁄
⁄
⁄
=
=
=
=
=
=
population proportion ( is the complement of ;
and
)
population mean
population variance
population standard deviation
sample proportion ( ̂ is the complement of ̂ ; ̂
̂ and ̂
̂)
sample mean
sample variance
sample standard deviation
sample size
confidence interval
margin of error (used for calculating
for
or )
complement of confidence level associated with
(for example, a
confidence level means
, which means
area in each tail for a confidence interval (
represents the middle range of values)
critical value in normal distribution for
for
or
with known value of
critical value in distribution for
for
with unknown value of
degrees of freedom (
)
critical value in
distribution for lower bound of
for
(or )
critical value in
distribution for upper bound of
for
(or )
)
Prof. Fowler
Procedure for Estimating a Population Proportion
1. Verify that all requirements are satisfied:
a. The sample is a simple random sample (this is either given or assumed in this course)
b. The conditions for binomial distribution are satisfied:
i. Fixed number of trials ( is set)
ii. Trials are independent (the outcome of any one trial has no effect on any other trial)
iii. Exactly two categories of outcomes for each trial (success and failure)
iv. Probabilities remain constant for each trial ( and
do not change)
c. There are at least
successes and
failures (equivalent to
and
)
2. If not provided, calculate the value of the sample proportion
̂:
(Note: ̂
̂
̂
3. Determine the value of
4. Look up the value of
based on the stated confidence level:
in the table of values for the normal distribution
⁄
5. Calculate the margin of error
:
⁄
6. Calculate the limits and state the
a. Preferred method:
b. ̂
c. ( ̂
̂
̂
√
̂̂
for the population proportion
in one of the following ways:
̂
To calculate the minimum sample size necessary to guarantee valid results for a stated confidence level
of (
and margin of error , simply solve the equation for
above for :
[
⁄
] ̂̂
Notes: 1. It is highly unlikely that there will be prior knowledge to provide a value for ̂ . In such
instances, it is customary to use ̂
(and, therefore, ̂
) in the calculation for
(If prior knowledge provides a value of ̂ to be used, then that should be used instead.)
2. Always round the value of
up to the next highest integer if the value is not an integer.
.
Prof. Fowler
Procedure for Estimating a Population Mean When
Is Known
Note: In general, this procedure is not used since it would be extremely unlikely that a value for
would be known without also knowing the true value of
since the calculation of depends
on the value of , although rare cases may exist; the more practical approach for estimating
is outlined in the next section of this packet.
1. Verify that all requirements are satisfied:
a. The sample is a simple random sample (this is either given or assumed in this course)
b. The value of the population standard deviation
is known
c. Either or both of the following conditions are met:
i. The population is normally distributed
ii. The sample size
2. If not provided, calculate the value of the sample mean
3. Determine the value of
4. Look up the value of
based on the stated confidence level:
⁄
in the table of values for the normal distribution
5. Calculate the margin of error
:
⁄
6. Calculate the limits and state the
√
for the population mean
in one of the following ways:
a. Preferred method:
b.
c. (
To calculate the minimum sample size necessary to guarantee valid results for a stated confidence level
of (
and margin of error , simply solve the equation for
above for :
[
Note: Always round the value of
⁄
]
up to the next highest integer if the value is not an integer.
Prof. Fowler
Procedure for Estimating a Population Mean When
Is Not Known
1. Verify that all requirements are satisfied:
a. The sample is a simple random sample (this is either given or assumed in this course)
b. Either or both of the following conditions are met:
i. The population is normally distributed
ii. The sample size
2. If not provided, calculate the value of the sample mean
3. If not provided, calculate the value of the sample standard deviation
4. Determine the value of
based on the stated confidence level:
5. Determine the number of degrees of freedom,
6. Look up the value of
⁄
:
in the table of values for the
7. Calculate the margin of error
distribution
:
⁄
8. Calculate the limits and state the
√
for the population mean
in one of the following ways:
a. Preferred method:
b.
c. (
To calculate the minimum sample size necessary to guarantee valid results for a stated confidence level
of (
and margin of error , it is generally acceptable to simply solve the equation for
above
for
and use the value in the row labeled “Large” for degrees of freedom as the value of ⁄ (this is
equivalent to the value of ⁄ used in the formula from the previous section in which the population
standard deviation is known):
[
⁄
]
Notes: 1. If statistics from a sample of known size are provided, the value of ⁄ from the row for
the corresponding number of degrees of freedom can be used, resulting in a slightly larger
minimum necessary sample size which will theoretically provide more accurate results.
2. To obtain the greatest minimum necessary sample size using this approach, the preceding
formula can be used with
degree of freedom, although the resulting sample size will be
significantly larger and may not be practical when formally gathering data.
3. If the population cannot be assumed to be at least reasonably close to normally distributed,
calculating a minimum sample size becomes complex beyond the scope of this course.
4. Always round the value of
up to the next highest integer if the value is not an integer.
Prof. Fowler
Procedure for Estimating a Population Variance
1. Verify that all requirements are satisfied:
a. The sample is a simple random sample (this is either given or assumed in this course)
b. The population must be normally distributed regardless of sample size
2. If not provided, calculate the value of the sample standard deviation
3. Determine the value of
based on the stated confidence level:
4. Determine the number of degrees of freedom,
5. Look up the values of
and
in the table of values for the
6. Calculate the limits and state the
for the population variance
(
7. If the
:
for the population standard deviation
(
√
distribution
:
(
is desired, simply take square roots:
(
√
Because the
distribution depends on the number of degrees of freedom, and the number of degrees
of freedom depends on the sample size, it is not possible to determine the minimum sample size
necessary to guarantee valid results for a stated confidence level of (
and margin of error
by
simply solving the equation for
above for . However, complex procedures do exist to determine
such sample sizes. Since those procedures are beyond the scope of this course, it is sufficient to simply
note that Table 7-2 on page 376 of the textbook provides required minimum sample sizes for many of
the most common confidence levels and margins of error. When necessary throughout this course, it is
sufficient to simply look up a required minimum sample size in that table.
Prof. Fowler