CHAPTER 10:
HYPOTHESIS TESTS REGARDING A
PARAMETER.
LANGUAGE OF HYPOTHESIS TESTING (10.1)
Hypothesis Testing is a procedure, based on
sampling data and probability, used to test
statements regarding a characteristic of one or
more populations.
 General Procedure involves:

 Making
a statement regarding populations
 Collecting Sample Data
 Analyzing the data to access the probability of the
statement’s validity
LANGUAGE OF HYPOTHESIS TESTING
 HYPOTHESIS
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TESTING IS A STEP-BY-STEP PROCEDURE:
1. From a statement of the problem (the claim), determine the
null and alternative hypothesis and the type (direction) of test.
2. Determine the significance at which the test should be run.
3. Test the null hypothesis from the experimental data using
either the Classical Approach (Critical Value Method) or the PValue Method.
4. Determine the conclusion concerning the null hypothesis.
5. Determine the conclusion concerning the original claim
LANGUAGE OF HYPOTHESIS TESTING
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The Null Hypothesis ( Ho ) is a statement that nothing has
changed, that populations are the same regarding some
characteristic.
The Alternative Hypothesis ( H1 ) is a that which we are trying to
find evidence to support, that the characteristic of a population
has changed.
There are four possible claims and the resulting Null and
Alternative Hypothesis to go with them.
And from the Alternative Hypothesis, the type of test (direction
of test) can be determined.
LANGUAGE OF HYPOTHESIS TESTING
CLAIM
NULL H 0
ALTERNATIVE H1 DIRECTION OF TESTS
A=B
A=B
A≠B
TWO SIDED
A≠B
A=B
A≠B
TWO TAILED
A<B
A=B
A<B
LEFT TAILED
A>B
A=B
A>B
RIGHT TAILED
EXAMPLE:
The claim is made that the mean of a population
has changed; in other words the current mean is different
than the former mean.
Null Hypothesis:   0
Alternative Hypothesis:   0
Type of Test: Two Tailed
LANGUAGE OF HYPOTHESIS TESTING
 There
are four possible outcomes to an
Hypothesis Test:
 1. The claim was true and the test found
it to be true.
 2. The claim was false and the test found
it was false.
 3. The claim was true but the test found
it false.
 4. The claim was false but the test found
it to be true.
LANGUAGE OF HYPOTHESIS TESTING

Types of Errors:
REALITY ABOUT CLAIM
CONCLUSION
ABOUT CLAIM



TRUE
FALSE
TRUE
CORRECT
CONCLUSION
TYPE II ERROR
FALSE
TYPE I ERROR
CORRECT
CONCLUSION
Type I Error = P(rejecting the Claim when it is true) = 
Type II Error = P(not rejecting the Claim when it is False) =
Which is most important not to make?

LANGUAGE OF HYPOTHESIS TESTING
is  , the Type I error. It is the Probability
of being wrong about the Claim when it is true and is
typically low: 0.20, 0.10, 0.05, 0.02, 0.01.
 As The significance (type I error) decreases, the type II
error increases.
 Significance is the area of the tail(s) of a normal
distribution. If the direction of the test is right sided,
the area in the right tail is the significance. If the
direction of the test is left sided, the area in the left
tail is the significance. If the direction of the test is
two sided, the significance is divided evenly between
the two tails.
 Significance
LANGUAGE OF HYPOTHESIS TESTING
 Types
of Tests:
 Classical Approach (Critical Value Method) – From
the experimental data, find a test statistic. From
the significance, find the Critical Value. If the Test
Statistic is in the area of the tail of the
significance, then the Null Hypothesis is
REJECTED.
 P-Value Method – From the experimental data
find the test statistic and from that the p-value
(area to the right or left of the test statistic in the
tail) and if the p-value is less than the
significance, then the Null Hypothesis is
REJECTED.
LANGUAGE OF HYPOTHESIS TESTING
Conclusions: If the Null Hypothesis is
REJECTED, the Alternative Hypothesis is
ACCEPTED.
 THE NULL HYPOTHESIS CAN NEVER BE
ACCEPTED SO THE ALTERNATIVE
HYPOTHESIS CAN NEVER BE REJECTED.
 The only thing we can do is FAIL TO
REJECT the Null and FAIL TO SUPPORT
the Alternative.
 Why?

LANGUAGE OF HYPOTHESIS TESTING

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Conclusion about the claim:
If the Null is Rejected and the Null is the Claim,
the claim is Rejected.
If the Null is Rejected (then the Alternative is
Accepted) and the Alternative is the Claim, the
Claim is accepted.
If the Null is Not Rejected and the Null is the
Claim, the Claim is Not Rejected.
If the Null is Not Rejected (then the Alternative is
not Supported) and the Alternative is the Claim,
the Claim is Not Supported.
Examples.
LANGUAGE OF HYPOTHESIS TESTING
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Example:
The mean score on all SAT tests for Math reasoning is
516. A certain company states the mean score of
students who take their SAT prep course is higher
than 516.
Find both hypothesis.
If the conclusion about the Null is not rejected, what is
the conclusion about the claim?
If the conclusion about the Null is wrong, what type of
error has been made?
HYPOTHESIS TESTING A POPULATION PROPORTION (10.2)

The drug Lipator is given to reduce
Chloresterol. In trials 19 out of 863
patients taking Lipator complained of flu
like systems. It is known that 1.9% of
patients taking a competing drug had flu
like symptoms. Is there reason to
believe that Lipator users experienced
flu like symptoms more than 1.9% of the
time to a significance level of 0.01.
HYPOTHESIS TESTING A POPULATION PROPORTION


Find the two hypothesis, direction of test,
significance and experimental data.
First use the Classical (Critical Value) Method
to test the Null:
 Find the Critical Value – the value of Z for
the area of significance (i.e. Use
InvNorm(α) for left tailed, InvNorm(1-α) for
right tailed or InvNorm(α/2) for 2 tailed).
p  p0
 Find the Test Statistic p0 q0
n
HYPOTHESIS TESTING A POPULATION PROPORTION

Second use the P-value method to test the Null:
 Find
 Use
the Test Statistic
p  p0
p0 q0
n
the Test Statistic to find the P-value (i.e. Use
Normalcdf(-10^99, Test Statistic) for right tailed or
Normalcdf(Test Statistic, 10^99) for left tailed or
2*Normalcdf(Test Statistic) for two tailed).
Use 1-PropZTest.
HYPOTHESIS TESTING A POPULATION PROPORTION
From the Classical (Critical Value) Method, if
the Test statistic is in the Critical Region, then
REJECT the Null, otherwise FAIL TO REJECT
the Null.
 If the P-value is less than the significance,
then REJECT the Null, otherwise FAIL TO
REJECT the Null.
 Determine the conclusion about the Claim.

HYPOTHESIS TESTING A POPULATION PROPORTION

A poll of 676 adults aged 18 and older
found that 352 believed they would not
have enough money to live comfortably
in retirement. Does this suggest that
half of the population of adults aged 18
and older believe they will not live
comfortably in retirement? Use a
significance of 0.05.
HYPOTHESIS TESTING A POPULATION PROPORTION

58% of females 15 years old or more lived
alone in 2000. Recently a survey of 500
females aged 15 and older found that 285
lived alone. Has the proportion of females
living alone changed to a level of
significance of 0.1.

Do other examples just giving values.
HYPOTHESIS TESTING A POPULATION MEAN
WITH UNKNOWN POPULATIONS STD. DEV (10.3)
The process is the same but will use the tdistribution.
x 
t 
s
 Test Statistic:
or TTest
n
 Critical Values: InvT or t-table as was done in
Chapter 9.
 P-Value: t-table or TTest

0
0
HYPOTHESIS TESTING A POPULATION MEAN
WITH UNKNOWN POPULATIONS STD. DEV.

Do students who learned English and another
language score lower on SAT Critical Reading
exam. A sample of 100 such persons were
given the test and scored a mean of 485 with a
std. dev. of 116. Test the hypothesis that such
students do score lower than the mean of all
test takers (501) to a significance of 0.10.
HYPOTHESIS TESTING A POPULATION MEAN
WITH UNKNOWN POPULATIONS STD. DEV.
The mean household energy expenditure was
$1493in 2001. The administrating
organization believes this has changed. A
random sample of 35 households found a
mean (adjusted to 2001 dollars) household
energy expenditure of $1621 and a std. dev. Of
$321. Test their belief to a significance of
0.05.
 Do problems just from numbers

HYPOTHESIS TESTING A POPULATION
STANDARD DEVIATION (10.4)
To Test a Hypothesis about a Standard
Deviation, there is not a calculator function and
there is no p-value test.
 Test Statistic:   (n  1) s
.

2
2
0

2
Critical Value: Use Chi-Squared Table
HYPOTHESIS TESTING A POPULATION
STANDARD DEVIATION

A machine fills bottle with 64 oz. of liquid. The
quality control manager has found the volumes in
the bottle to be normally distributed with a std. dev
of 0.42. The process engineer makes some
changes to the machine and believes the standard
deviation will be reduced. The manager picks a
sample of 19 bottles and has a sample deviation
of 0.38. Test his hypothesis to a 0.01 significance.
HYPOTHESIS TESTING A POPULATION
STANDARD DEVIATION


The NCAA as requirements for the circumference of a
softball for competition. One is that a manufacturer
must have a standard deviation of the circumference
less than 0.05 inches. A representative of the NCAA
believes a manufacture does not meet the
requirement. A sample of 20 softballs are measured
and the sample standard deviation is 0.09. Is there
sufficient evidence to support the representative’s
claim to a significance of 0.05?
Do problems with numbers only.
HYPOTHESIS TESTING SUMMARY
Proportion
Must be given:
x, n,( p), p0 , 
Distribution
Test Statistic
x, n, s, 0 , 
Z
Z0 
Std. Dev.
n, s,  0 , 
2
t
pˆ  p0
t0 
p0 1  p0 
n
Critical Value
Mean (w/s)
x  0
s
n
 
2
0
(n  1) s 2
2
 2  table
InvNorm()
InvT() or t-table
p-value
Normalcdf()
Tcdf()
N/A
Calculator Function
1-PropZTest
Ttest
N/A
Reject Null if p-value < significance or if test statistic is in the Critical Region (Tails)