Transcript document

Chapter 9
Continued...
III. One-Tailed Tests (large sample)
• Hilltop Coffee states that each can has at least 3
pounds of coffee.
• The Fed. Trade Commission randomly tests
corporate claims.
• If Hilltop’s claim is correct,   3.
Ho:   3
Ha:  < 3
If we reject Ho, Hilltop is violating their claim.
A. Sampling Distribution
x 
=3

n
x
If we take a random sample of n=36, we use the
C.L.T. to assume a normal sampling distribution.
B. How low is too low?
• Suppose we measured out each coffee can and
calculated a sample mean weight of 2.99 pounds.
• Do you think this is enough evidence to reject Ho
and conclude that Hilltop is underfilling their
cans? Probably not.
• If we did reject Ho, we might make a type I error.
• What if x-bar was 1.99 pounds? Maybe this is too
low and we should reject Ho.
C. The role of z-scores
• Remember a z-score tells us how many standard
deviations a sample mean falls from the expected
value, or population mean.
• So would a sample mean that was 1 standard
deviation below =3 be far enough below to reject
Ho?
• We need to consider the probability involved in
calculating such a sample mean.
D. The Rejection Range
 = .05
Z=-1.645
=3
x
If we get a sample and calculate Z=1.645 below
the mean, only a probability of .05 remains in
the lower tail of the sampling distribution.
Maybe this is low enough?
• In other words, whenever the value of Z is less
than -1.645, the probability of making a type I
error would be .05.
• Thus, we would reject Ho if Z<-1.645, if we
believed that .05 was an acceptable degree of risk.
• If we wanted to lower that to .01, our rejection
range would lie below Z=-2.33.
E. Methodology
1. Specify a maximum allowable probability of a
type I error (). This is the probability of
rejecting Ho when it is true.
2. Find Z that corresponds to . This is the critical
Z score. If =.01, then Z corresponds to the area
under the curve of .4900. Z=.01=2.33
Thus, reject Ho if Z<-2.33.
Methodology continued
3. Take a sample, calculate the mean and standard
error.
4. Calculate Z and compare to the critical Z.
5. If your Z is greater (in absolute value) than the
critical Z, reject Ho.
Example
Ho:   3 lbs.
Ha:  < 3 lbs.
A sample of 36 cans is taken and the sample mean is
2.92 lbs. Previous studies have found that
historically cans are filled with a standard
deviation of .18 lbs.
The standard error of the sampling distribution is:
 .18
x 

.03
6
n
Calculate the Z-score, Z  x    2.92  3  2.67
x
.03
Since this Z-score is greater (in absolute value)
than the critical value of 2.33, we reject Ho
and conclude that they are underfilling their
coffee cans.