One-Sample Tests of Hypothesis
Chapter
Ten
McGraw-Hill/Irwin
© 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
What is a Hypothesis?
A statement about the value of a population parameter.
Example:
The mean monthly income for a systems analyst is $5K.
Hypothesis testing
•Process used to determine if the hypothesis is a reasonable statement
•The hypothesis is either accepted or rejected
•Decision is based on sample data & probability theory
$5K
3K (say)
7K (say)
Process involves taking a sample & calculating the sample mean.
Then we look at how far the sample mean is from the hypothesized mean.
If it is too far, we reject it; else, we accept it.
Eg. If the mean salary of a sample of systems analysts is between 3-7K, accept hyp.
Hypothesis Testing – Formal Steps
Step 1: State null and alternate hypotheses
Step 2: Select a level of significance
Step 3: Identify the test statistic
Null: Innocent
Alternate: Guilty
Error level you are willing to
tolerate; eg. 5%
Identify method to
weigh the evidence
Step 4: Formulate a decision rule
If evidence is stronger
than error level (eg. 95%)
reject null hypothesis.
Step 5: Take a sample, arrive at a decision
Listen to lawyers on both
sides & decide based on
above criteria
Do not reject null
Reject null and accept alternate
Practice time!
Do Self-Review 10-1
Page 287-8
(2 tails)
-2.56
Step One :
State the null and alternate hypotheses
Generally H0 represents what is currently believed.
H1 represents a researcher’s claim.
H1 is accepted if H0 is shown to be false
H0: m = 0
H1: m = 0
Three possibilities
H0:The mean income of women financial
planners is $65,000, ie. μ = $65,000 .
H1: The mean income of .. is not equal to…
ie. μ ≠ $65000.
H0: m < 0
H1: m > 0
H0:The mean income of women financial
planners is ≤ $65,000
H1: The mean income of .. is > $65000.
H0: m > 0
H1: m < 0
H0:The mean income of women financial
planners is ≥ $65,000
H1: The mean income of .. is < $65000.
The null hypothesis always contains equality.
Step Two: Select a Level of Significance.
Level of Significance is the probability ( α ) of rejecting the null hypothesis
when it is actually true.
α = P (Reject H0 | H0 is true) = Type I error
We are deciding upfront how much type 1 error we are willing to tolerate.
H0: The suspect is innocent.
H1: The suspect is guilty.
>.01
p
If we set the Level of Significance (α) at 0.05 (5%), it implies that
we are willing to convict with only 95% of evidence pointing to guilt
(ie. Even though the suspect is innocent).
If we set the Level of Significance for the testing at 0.01 (1%), it
implies that we could mistakenly convict an innocent person only 1%
of the time.
.0
Two types of errors in hypothesis testing
α = P (Reject H0 | H0 is true) = Type I error
β = P (Accept H0 | H0 is false) = Type II error
Null
Hypothesis
Ho is true
Ho is false
Researcher
Accepts
Rejects
Ho
Ho
Correct
decision
Type I error
(a)
Type II
Error
(b)
Correct
decision
Step 3: Identify test statistic
Decide if you want to use z or t as the statistic.
(No need to calculate anything yet!)
X m
z
/ n
X m
z
s/ n
z
p 
 (1   )
n
t
X m
s/
n
Step Four: Formulate the decision rule.
Find the Critical Value(s) corresponding to α from the z or t table.
Mark the rejection/ acceptance regions.
Region of
rejection
Do not
rejection
reject
[Probability=.025]
[Probability =.95]
reject
[Probability=.025]
[Probability =.95]
-1.96
Critical value
H0: m = 0
H1 : m = 0
0
Region of
Region of
Do not
1.96
0
[Probability=.05]
1.65
Critical value
Critical value
2 tails testing
rejection
H0: m < 0
H1 : m > 0
1 tail testing
Step Five: Make a decision.
•Now, you compute the z or t statistic.
•Check if it falls inside the Rejection or Acceptance region
•If it falls inside the Rejection region, reject H0.
•If it falls inside the Acceptance region, do not reject H0.
p-Value
The probability of observing a sample value as extreme as, or
more extreme than the calculated test statistic value.
p-value
Cut-off Calculated
Z
Z
Decision Rule: If the p-Value is smaller than the significance
level, a, H0 is rejected.
Practice
Self-Review 10-2, Page 291
(1 tail)
(1 tail on the right)
(from table for α=0.01)
0.01
Z=1.81
Z=2.33