CHAPTER 6
Fundamentals of Hypothesis Testing: One-Sample Tests
OBJECTIVES
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To understand the basic hypothesis testing procedure
To appreciate the risk involved in making decisions about population parameters based only
on sample information
To be able to perform and apply the various tests of hypotheses in dealing with one sample
To understand the difference between the traditional critical value approach and the p-value
approach to hypothesis testing
OVERVIEW AND KEY CONCEPTS
Some Basic Concepts in Hypothesis Testing
 Null hypothesis  H 0  : The hypothesis that is always tested.
 The null hypothesis always refers to a specified value of the population parameter,
not a sample statistic.
The statement of the null hypothesis always contains an equals sign regarding the
specified value of the population parameter.
Alternative hypothesis: The opposite of the null hypothesis and represents the conclusion
supported if the null hypothesis is rejected.
 The statement of the alternative hypothesis never contains an equals sign regarding
the specified value of the population parameter.
Critical value: A value or values that separate the rejection region or regions from the
remaining values.
Type I error: A Type I error occurs if the null hypothesis is rejected when in fact it is true
and should not be rejected.
Type II error: A Type II error occurs if the null hypothesis is not rejected when in fact it is
false and should be rejected.
Level of significance   : The probability of committing a Type I error.
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The  risk (the consumer’s risk level): The probability of committing a Type II error.
Factors that affect the  risk: Holding everything else constant,
  increases when the difference between the hypothesized parameter and its true
value decreases.
  increases when  decreases.
  increases when  increases.
  increases when the sample size n decreases.
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The confidence coefficient 1    : The probability that the null hypothesis is not rejected
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when in fact it is true and should not be rejected.
The confidence level: 100 1    %
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The power of a test 1    : The probability of rejecting the null hypothesis when in fact it
is false and should be rejected.
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Risk in decision making: There is a delicate balance between the probability of committing
a Type I error and the probability of a Type II error.
H0: Innocent
E.g. Jury Trial
Hypothesis Test
The Truth
The Truth
Verdict Innocent Guilty Decision H0 True H0 False
Innocent Correct
Guilty
Error
Error
Do Not
Reject
H0
Correct Reject
H0
1-
Type II
Error ( )
Type I
Error
( )
Power
(1 -  )
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Reducing the probability of Type I error will inevitably increase the probability of
committing a Type II error holding everything else constant.
 One should choose a smaller Type I error when the cost of rejecting the maintained
hypothesis is high.
 One should choose a larger Type I error when there is an interest in changing the
status quo.
p-value (the observed level of significance): The probability of obtaining a test statistic
equal to or more extreme than the result obtained from the sample data, given the null
hypothesis is true.
 It is also the largest level of significance at which the null hypothesis will not be
rejected.
 Roughly speaking, it measures the amount of evidence against the null hypothesis.
The smaller the p-value, the stronger is the evidence against the null hypothesis.
 The statistical decision rule is to reject the null hypothesis for any level of
significance   greater than the p-value, and do not reject otherwise.
General Steps in the Traditional Critical Value Approach to Hypothesis Testing
1. State the null hypothesis.
2. State the alternative hypothesis.
3. Choose the level of significance   .
4.
5.
6.
7.
8.
9.
Choose the sample size, n.
Choose an appropriate test.
Collect data and compute the sample value of the appropriate test statistic.
Obtain the critical value(s) based on the level of significance.
Compare the computed test statistic to the critical value(s).
Make a statistical decision: Reject H 0 when the computed test statistic falls in a rejection
region; do not reject H 0 otherwise.
10. Draw the conclusion.
General Steps in the p Value Approach to Hypothesis Testing
1. State the null hypothesis.
2. State the alternative hypothesis.
3. Choose the level of significance   .
4.
5.
6.
7.
8.
9.
10.
Choose the sample size, n.
Choose an appropriate test.
Collect data and compute the sample value of the appropriate test statistic.
Obtain the p-value based on the computed test statistic.
Compare the p-value to 
Make a statistical decision: Reject H 0 when the p-value <  ; do not reject H 0 otherwise.
Draw the conclusion.
Z Test for the Population Mean    when  is Known
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Assumptions:
 Population is normally distributed or large sample size.
  is known.
Test statistic:
X  X
Z
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The alternative hypothesis can be one-tail with a right-tail rejection region, one-tail
with a left-tail rejection region or two-tail with both right-tail and left-tail rejection
regions.
X

X 
/ n
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t Test for the Population Mean    when  Is Unknown
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Assumptions:
 Population is normally distributed or large sample size.
  is unknown.
Test statistic:
X 
with (n – 1) degrees of freedom.
S/ n
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t
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The alternative hypothesis can be one-tail with a right-tail rejection region, one-tail
with a left-tail rejection region or two-tail with both right-tail and left-tail rejection
regions.
Z Test for the Population Proportion (p)
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Assumptions:
 Population involves 2 categorical values.
 Both np and n(1-p) are at least 5.
Test statistic:
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Z
pS   pS
p
S
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
pS  p
p 1  p 
n
The alternative hypothesis can be one-tail with a right-tail rejection region, one-tail
with a left-tail rejection region or two-tail with both right-tail and left-tail rejection
regions.