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8.2 The Z-Test for the Population Mean  :
Definition
The critical value separates the critical region from the noncritical region. The
symbol for critical value is C.V.
The critical or rejection region is the range of values of the test value that
indicates that there is a significant difference and that the null hypothesis
should be rejected.
The noncritical or non-rejection region is the range of values of the test value
that indicates that the difference was probably due to chance and that the null
hypothesis should not be rejected.
A one-tailed test indicates that the null hypothesis should be rejected when the
test value is in the critical region on one side of the mean. A one-tailed test is either a righttailed test or left-tailed test, depending on the direction of the inequality of the alternative
hypothesis.
Example
Critical and noncritical regions on the Standard Normal Distribution N(0,1)
(here   0.01)
Example
Finding the Critical Value for  =0.01 (Right-Tailed Test) using the standard normal distribution
N(0,1).
Solution
The critical value is for  =0.01 is 2.33.
Let’s Do It! Critical Values.
a. Finding the Critical Value for  =0.05 (Left-Tailed Test) using The standard normal
distribution N(0,1).
b. Finding the Critical Value for  =0.05 (Two-Tailed Test) using The standard normal
distribution N(0,1).
Case 1: Testing the Mean Using a Large Sample n  30
The Decision Tool (THE TEST STATISTIC)
In order to perform the hypothesis test, we first determine the the value of the
observed statistic of our sample. Many hypotheses are tested using a statistical
test based on the following general formula:
There are two cases:
 If the sample used is large, then the distribution used for testing the
hypothesis is Z. The standardized observed test statistic is given by:
Z
x  0
s/ n
 If the sample is small (n<30) and the population is normally distributed
then the distribution used for testing is T. The standardized observed test
statistic is:
T
x  0
s/ n
(we will discuss the second case in the next section)
Example
A researcher reports that the average salary of assistant professors is more than $42,000. A
sample of 30 assistant professors has a mean salary of $43,260. Test the claim that assistant
professors earn more than $42,000 a year. The standard deviation of the sample is $5230. Use
 =0.05.
a) Write the hypothesis of the researcher.
b) What type of test? (Circle one)
One-sided to the right
One-sided to the left
Two-Sided
c) Find the critical value and the p-value for  =0.05 on the Z-distribution.
d) Is the observed average an extreme under the null hypothesis?
e) Summarize your result?
Solution
a)
H0 :   $42,000
(Or H0 :   $42,000 )
H1 :   $42,000  This is the researcher’s claim
b) This test is a one-Sided to the right (b/c the direction of the extreme under the null is to the
right. Values on far right are unlikely under the null and more likely under the alternative)
c) Critical value C.V. = InvNorm (0.95, 0, 1) = 1.65
d) To determine if the observed sample average $43,260 is extreme under the null, first
standardize under the null. The standardized value is called The Test Statistic.
Test Statistic=
=
x  0
s
n
$43,260  $42,000
5230 / 30
= 1.32
Notice how the standardized value of the observed sample average is outside the critical
region. This implies that $43,260 is NOT extreme under the null hypothesis.
Also, P-value= area to the right of th test statistic
= normcdf( 1.32, 10699,0,1)=0.093 > 
Therefore we don’t reject he null hypothesis.
e) Result: There is not enough evidence to support the researchers claim that the average
salary is more than $42,000.
Let’s Do it! Weight Loss
A diet pill promises a weight loss of 10lbs in the first week (the fine print on the bottle
indicates that the reported amount is an average weight loss). A researcher wishes to test the
effectiveness of the pill. 36 subjects we selected randomly and given the pill. The amount of
their weight lost in the first week is given below:
3
2
1
1
0
5
5
5
7
2
2
0
4
8
8
1
0
0
2
2
2
4
6
2
2
1
1
0
0
0
3
2
1
1
0
2
a) Write the hypothesis.
b) What type of test? (Circle one)
One-sided to the right
One-sided to the left
Two-Sided
c) Find the critical value for  =0.10 on the Z-distribution.
d) Find the p-value.
e) Is the observed average an extreme under the null hypothesis?
f) Summarize your result?
Let’s Do it!
Teen Spending
According to Teenage Research Unlimited, in 2007, teens spent an average of $103 per week
(over the course of the year that equals $5356). A researcher believes that in the year of 2008,
this average has changed. He collected information from 125 randomly selected teenagers. His
sample resulted in an average spending of $118 per week with a standard deviation of $122.
a) Write the hypothesis.
b) What type of test? (Circle one)
One-sided to the right
One-sided to the left
Two-Sided
c) Find the critical value for  =0.10 on the Z-distribution.
d) Find the p-value.
e) Is the observed average an extreme under the null hypothesis?
f) State your conclusion.
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