Transcript Understandable Statistics Eighth Edition By Brase and

Understandable Statistics
Eighth Edition
By Brase and Brase
Prepared by: Lynn Smith
Gloucester County College
Edited by: Jeff, Yann, Julie, and Olivia
Chapter 8: Estimation
Section 8.1
Estimating μ When σ Is Known
Focus Points
• Explain the meaning of confidence level,
error of estimate, and critical value.
• Find the critical value corresponding to a
given confidence level.
• Compute confidence intervals for μ when
σ is known. Interpret the results.
Statistics Quote
“There are three kinds of lies—lies, damned
lies, and statistics.” —Benjamin Disraeli
Assumptions About the Random
Variable x
1. We have a simple random sample of size n
drawn from a population of x values.
2. The value of sigma, the population standard
deviation of x, is known.
3. If the x distribution is normal, then our methods
work for any sample size n.
4. If x has an unknown distribution, then we
require a sample size n ≥ 30. However, if the x
distribution is distinctly skewed and definitely
not mound-shaped, a sample size of 50 or
even 100 or higher may be necessary.
Point Estimate
an estimate of a population
parameter given by a single
number
Examples of Point Estimates
Examples of Point Estimates
• x is used as a point estimate for .
Examples of Point Estimates
•
x is used as a point estimate for .
•
s is used as a point estimate for .
Margin of Error
the magnitude of the difference
between the point estimate and
the true parameter value
The margin of error using
x as a point estimate for 
is
x
Confidence Level
• A confidence level, c, is a measure of the
degree of assurance we have in our
results.
• The value of c may be any number
between zero and one.
• Typical values for c include 0.90, 0.95, and
0.99.
Critical Value for a Confidence
Level, c
the value zc such that the area
under the standard normal curve
falling between – zc and zc is
equal to c.
Critical Value for a Confidence
Level, c
P(– zc < z < zc ) = c
Area Between –z.99 and z.99 is .99
Find z0.90 such that 90% of the
area under the normal curve lies
between z-0.90 and z0.90.
P(-z0.90 < z < z0.90 ) = 0.90
.90
– z.90
0
z.90
Find z0.90 such that 90% of the
area under the normal curve
lies between z-0.90 and z0.90.
P(0< z < z0.90 ) = 0.90/2 = 0.4500
.4500
– z.90
0
z.90
Find z0.90 such that 90% of the
area under the normal curve
lies between z-0.90 and z0.90.
P( z < z0.90 ) = .5 + 0.4500 = .9500
.9500
– z.90
0
z.90
Find z0.90 such that 90% of the
area under the normal curve
lies between z-0.90 and z0.90.
• According to Table 5a in Appendix II,
0.9500 lies exactly halfway between two area
values in the table (.9495 and .9505).
• Averaging the z values associated with these
areas gives z0.90 = 1.645.
Common Levels of Confidence
and Their Corresponding Critical
Values
Level of Confidence, c
Critical Value zc
0.90, or 90%
1.645
0.95, or 95%
1.96
0.98, or 98%
2.33
0.99, or 99%
2.58
More Confidence Values
C Confidence Interval for μ
An interval computed
from sample data in
such a way that c is
the probability of
generating an interval
containing the actual
value of μ
Confidence Interval for the
Mean of Large Samples (n 
30)
xE
where
   xE
x  Sample Mean
Confidence Interval for the
Mean of Large Samples (n 
30)
xE
where
   xE
x  Sample Mean
s
Ez
n
c
Confidence Interval for the
Mean of Large Samples (n 
30)
xE
where



xE
x  Sample Mean
s
Ez
n
s  sample standard deviation
c
Confidence Interval for the
Mean of Large Samples (n 
30)
xE



xE
x  Sample Mean
s
Ez
n
s  sample standard deviation
c  confidence level (0  c  1)
where
c
Confidence Interval for the
Mean of Large Samples (n 
30)
xE
where



xE
x  Sample Mean
s
Ez
n
s  sample standard deviation
c
c  confidence level (0  c  1)
z  critical value for confidence level c
c
Confidence Interval for the
Mean of Large Samples (n 
30)
   xE
x  Sample Mean
s
Ez
n
s  sample standard deviation
c  confidence level (0  c  1)
xE
where
c
z  critical value for confidence level c
c
n  sample size
Create a 95% confidence
interval for the mean driving
time between Philadelphia
and Boston.
Assume that the mean driving
time of 64 trips was 6.4 hours
with a standard deviation of
0.9 hours.
x
= 6.4 hours
s = 0.9 hours
c = 95%,
so zc = 1.96
n = 64
x = 6.4 hours
s = 0.9 hours
Approximate  as s = 0.9 hours.
95% Confidence interval will be from
x  E to x  E
x
= 6.4 hours
s = 0.9 hours
c = 95%, so zc = 1.96
n = 64
E  zc
s
0.9
 1.96
 .2205
n
64
95% Confidence Interval:
6.4 – .2205 <  < 6.4 + .2205
6.1795 <  < 6.6205
We are 95% sure that the true time is
between 6.18 and 6.62 hours.
Calculator Instructions
CONFIDENCE INTERVALS FOR A POPULATION MEAN
The TI-83 Plus and TI-84 Plus fully support confidence
intervals. To access the confidence interval choices,
press Stat and select TESTS. The confidence interval
choices are found in items 7 through B.
Example (σ is known with a
Large Sample)
Suppose a random sample of 250 credit
card bills showed an average balance of
$1200. Also assume that the population
standard deviation is $350. Find a 95%
confidence interval for the population
mean credit card balance.
Example (σ is known)
• Since σ is known and we
have a large sample, we
use the normal
distribution. Select
7:ZInterval.
• In this example, we have
summary statistics, so we
will select the STATS
option for input. We enter
the value of σ, the value
of x and the sample size
n. Use 0.95 for the CLevel.
Example (σ is known)
• Highlight Calculate
and press Enter to
get the results. Notice
that the interval is
given using standard
mathematical notation
for an interval. The
interval for μ goes
from $1156.6 to
$1232.4.
Section 8.1, Problem 3
Diagnostic Tests: Plasma Volume Total plasma volume is important
in determining the required plasma the overall health and physical
activity of an individual. (Reference: See Problem 2.) Suppose that a
random sample of 45 male firefighters are tested and that they have
a plasma volume sample mean of x-bar = 37.5 mL/kg (milliliters
plasma per kilogram body weight). Assume that σ = 7.50 mL/kg for
the distribution of blood plasma.
(a) Find a 99% confidence interval for the population mean blood
plasma volume in male firefighters. What is the margin of error?
(b) What conditions are necessary for your calculations?
(c) Give a brief interpretation of your results in the context of this
problem.
Solution
Section 8.1, Problem 7
Salaries: College Administrators How much do college
administrators (not teachers or service personnel) make
each year? Suppose you read the local newspaper and
find that the average annual salary of administrators in
the local college is x-bar = $58,940. Assume that σ is
known to be $18,490 for college administrator salaries
(Reference: The Chronicle of Higher Education).
(a) Suppose that x-bar = $58,940 is based on a random
sample of n = 36 administrators. Find a 90% confidence
interval for the population mean annual salary of local
college administrators. What is the margin of error?
Section 8.1, Problem 7
(b) Suppose that x-bar = $58,940 is based on a random sample of n =
64 administrators. Find a 90% confidence interval for the population
mean annual salary of local college administrators. What is the
margin of error?
(c) Suppose that x-bar = $58,940 is based on a random sample of n =
121 administrators. Find a 90% confidence interval for the
population mean annual salary of local college administrators. What
is the margin of error?
(d) Compare the margins of error for parts (a) through (c). As the
sample size increases, does the margin of error decrease?
(e) Compare the lengths of the confidence intervals for parts (a)
through (c). As the sample size increased, does the length of a 90%
confidence interval decrease?
Solution