Download Point Estimate (Critical Value)(Standard Error)

Chapter 7
Estimating Population
Values
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Chapter 7 - Chapter Outcomes
After studying the material in this chapter,
you should be able to:
•Distinguish between
a point estimate
and a confidence interval estimate.
•Construct and interpret a confidence
interval estimate for a single population
mean using both the z and t
distributions.
Chapter 7 - Chapter Outcomes
(continued)
After studying the material in this chapter, you
should be able to:
•Determine
the required sample size for
an estimation application involving a
single population mean.
•Establish and interpret a confidence
interval estimate for a single population
proportion.
Point Estimates
A point estimate is a single number
determined from a sample that is
used to estimate the corresponding
population parameter.
Sampling Error
Sampling error refers to the difference
between a value (a statistic) computed
from a sample and the corresponding
value (a parameter) computed from a
population.
Confidence Intervals
A confidence interval refers to an interval
developed from randomly sample values
such that if all possible intervals of a given
width were constructed, a percentage of
these intervals, known as the confidence
level, would include the true population
parameter.
Confidence Intervals
Lower Confidence
Limit
Upper Confidence
Limit
Point Estimate
95% Confidence Intervals
(Figure 7-3)
0.95
z.025= -1.96
z.025= 1.96
Confidence Interval
- General Format -
Point Estimate  (Critical Value)(Standard Error)
Confidence Intervals
The confidence level refers to a percentage
greater than 50 and less than 100 that
corresponds to the percentage of all possible
confidence intervals, based on a given size
sample, that will contain the true population
value.
Confidence Intervals
The confidence coefficient refers to the
confidence level divided by 100% -- i.e.,
the decimal equivalent of a confidence
level.
Confidence Interval
- General Format:  known -
Point Estimate  z (Standard Error)
Confidence Interval
Estimates
CONFIDENCE INTERVAL
ESTIMATE FOR  ( KNOWN)
xz

n
where:
z = Critical value from standard
normal table
 = Population standard deviation
n = Sample size
Example of a Confidence
Interval Estimate for 
A random sample of 100 cans, from a
population with  = 0.20, produced a sample
mean equal to 12.09. A 95% confidence
interval would be:
xz

n
0.20
12.09  1.96
100
12.09  0.039
12.051 ounces
12.129 ounces
Special Message about
Interpreting Confidence Intervals
Once a confidence interval has been
constructed, it will either contain the
population mean or it will not. For a 95%
confidence interval, if you were to produce
all the possible confidence intervals using
each possible sample mean from the
population, 95% of these intervals would
contain the population mean.
Margin of Error
The margin of error is the largest
possible sampling error at the
specified level of confidence.
Margin of Error
MARGIN OF ERROR (ESTIMATE FOR  WITH
 KNOWN)
ez
where:

n
e = Margin of error
z = Critical value
 = Standard error of the sampling
distribution
n
Example of Impact of Sample
Size on Confidence Intervals
If instead of random sample of 100 cans, suppose a
random sample of 400 cans, from a population with  =
0.20, produced a sample mean equal to 12.09. A 95%
confidence interval would be:
xz
12.0704 ounces
12.051 ounces

n
0.20
12.09  1.96
400
12.09  0.0196
n=400
n=100
12.1096 ounces
12.129 ounces
Student’s t-Distribution
The t-distribution is a family of distributions that
is bell-shaped and symmetric like the standard
normal distribution but with greater area in the
tails. Each distribution in the t-family is defined
by its degrees of freedom. As the degrees of
freedom increase, the t-distribution approaches
the standard normal distribution.
Degrees of freedom
Degrees of freedom refers to the number of
independent data values available to estimate
the population’s standard deviation. If k
parameters must be estimated before the
population’s standard deviation can be
calculated from a sample of size n, the degrees
of freedom are equal to n - k.
t-Values
t-VALUE
where:
x
t
s
n
x = Sample mean
 = Population mean
s = Sample standard deviation
n = Sample size
Confidence Interval Estimates
CONFIDENCE INTERVAL
( UNKNOWN)
s
x t
n
where:
t = Critical value from t-distribution
with n-1 degrees of freedom
x = Sample mean
s = Sample standard deviation
n = Sample size
Confidence Interval Estimates
CONFIDENCE INTERVAL-LARGE
SAMPLE WITH  UNKNOWN
s
xz
n
where:
z =Value from the standard normal
distribution
x = Sample mean
s = Sample standard deviation
n = Sample size
Determining the Appropriate
Sample Size
SAMPLE SIZE REQUIREMENT ESTIMATING  WITH  KNOWN
z
 z 
n 2 

e
 e 
2
where:
2
2
z = Critical value for the specified
confidence interval
e = Desired margin of error
 = Population standard deviation
Pilot Samples
A pilot sample is a random sample taken
from the population of interest of a size
smaller than the anticipated sample size
that is used to provide and estimate for
the population standard deviation.
Example of Determining
Required Sample Size
(Example 7-7)
The manager of the Georgia Timber Mill wishes
to construct a 90% confidence interval with a
margin of error of 0.50 inches in estimating the
mean diameter of logs. A pilot sample of 100 logs
yields a sample standard deviation of 4.8 inches.
2
2
1.645 (4.8)
n
 249.38  250
2
0
.
50
Note, the manager needs only 150 more logs since
the 100 in the pilot sample can be used.
Estimating A Population
Proportion
SAMPLE PROPORTION
x
p
n
where:
x = Number of occurrences
n = Sample size
Estimating a Population
Proportion
STANDARD ERROR FOR p
p 
p(1  p)
n
where:
 =Population proportion
n = Sample size
Confidence Interval
Estimates for Proportions
CONFIDENCE INTERVAL FOR 
p(1  p)
pz
n
where:
p = Sample proportion
n = Sample size
z = Critical value from the standard
normal distribution
Example of Confidence
Interval for Proportion
(Example 7-8)
62 out of a sample of 100 individuals who were
surveyed by Quick-Lube returned within one
month to have their oil changed. To find a 90%
confidence interval for the true proportion of
customers who actually returned:
x 62
p 
 0.62
n 100
0.54
(0.62)(1  0.62)
0.62  1.645
100
0.70
Determining the Required
Sample Size
MARGIN OF ERROR FOR ESTIMATING 
ez
where:
 (1   )
n
 = Population proportion
z = Critical value from standard
normal distribution
n = Sample size
Determining the Required
Sample Size
SAMPLE SIZE FOR ESTIMATING 
z  (1   )
n
2
e
2
where:
 = Value used to represent the
population proportion
e = Desired margin of error
z = Critical value from the standard
normal table
Key Terms
•
•
•
•
•
Confidence Coefficient
Confidence Interval
Confidence Level
Degrees of Freedom
Margin of Error
•
•
•
•
Pilot Sample
Point Estimate
Sampling Error
Student’s tdistribution