Because -1.818 does not fall in the rejection region, H
0
is
NOT
REJECTED
at the .01 significance level. We have
NOT
demonstrated
that the cost-cutting measures reduced the mean cost per claim to less
than $60. The difference of $3.58 ($56.42 - $60) between the sample
mean and the population mean could be due to sampling error.
Step 4: Formulate the decision rule.
Reject H
0
if
t
< -t
D
,n-1
343
NOTE:
Calculation
discrepancy in
z-value

The current rate for producing 5 amp fuses at Neary Electric
Co. is
250
per hour.
A new machine has been purchased
and installed that, according to the supplier, will increase
the production rate.
A sample of
10
randomly selected
hours from last month revealed the mean hourly production
on the new machine was
256
units, with a sample standard
deviation of
6
per hour.
At the
.05
significance level can Neary conclude that the new
machine is faster?
Testing for a Population Mean with an Unknown
Population Standard Deviation- Example
Step 1:
State the null and the alternate hypothesis.
H
0
: μ
£
250
H
1
: μ > 250
Step 2:
Select the level of significance.
It is .05.
Step 3:
Find a test statistic.
Use the
t
distribution because the population
standard deviation is not
known and the sample size is less than 30.
Testing for a Population Mean with an Unknown
Population Standard Deviation- Example

Step 4:
State the decision rule.
There are 10 – 1 = 9 degrees of freedom.
The null
hypothesis is rejected if
t
> 1.833.
Step 5:
Make a decision and interpret the results.
The null hypothesis is rejected.
The mean number produced is
more than 250 per hour.
162
.
3
10
6
250
256
!
0
!
0
!
n
s
X
t
P
Testing for a Population Mean with an Unknown
Population Standard Deviation- Example
Tests Concerning Proportion
z
A
Proportion
is the fraction or percentage that indicates the part of
the population or sample having a particular trait of interest.
z
The sample proportion is denoted by
p
and is found by
x/n
z
The test statistic is computed as follows:
350

Assumptions in Testing a Population Proportion using
the z-Distribution
z
A random sample is chosen from the population.
z
It is assumed that the binomial assumptions discussed in Chapter 6 are
met:
(1) the sample data collected are the result of counts;
(2) the outcome of an experiment is classified into one of two mutually
exclusive
categories—a “success” or a “failure”;
(3) the probability of a success is the same for each trial;
(4) the trials are independent
z
The test we will conduct shortly is appropriate when both
n
S
and
n
(1-
S
)
are at least 5.
z
When the above conditions are met, the normal distribution can be used
as an approximation to the binomial distribution
349
Assumptions when using a Normal
approximation for hypothesis testing
with a proportion (
short version
)
1.
It meets the conditions of a
BINOMIAL
distribution
2.

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- Spring '11
- Leany
- Statistics, Normal Distribution, Null hypothesis, Statistical hypothesis testing, Hypothesis and Testing