Download Point Estimate - Milan C-2

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
is used as a point estimate for  .
•
x
•
s is used as a point estimate for .
Error of Estimate
the magnitude of the difference
between the point estimate and
the true parameter value
The error of estimate using
as a point estimate for 
x
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
This area = c.
– zc
0
zc
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
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 
x  E  30)
  xE
x  Sample Mean
s
Ez
n
s  sample standard deviation
where
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)
z  critical value for confidence level c
xE
where
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.