Introduction to Hypothesis Testing

Chapter 10

Introduction to
Hypothesis Testing

1


Pop dist:
Normal

 known
μ?

Z-dist
n  30

Pop dist:
Non-Normal

?

n > 30 Z-dist


The purpose of hypothesis testing is to determine
whether there is enough statistical evidence supporting
a certain belief (or claim) about a parameter.
Examples
• Is there statistical evidence that support the
hypothesis that more than p% of all potential

customers will purchase a new products?
• Is the hypothesis that a certain drug is effective?

3


11.1 Concepts of Hypothesis Testing
The Reject-Region method
Step 1: Two hypotheses are defined.
H0: The null hypothesis specifies our current belief
about the parameter we test. ( = 170, p = .4, etc.)
Must be a specific value.
H1: The alternative hypothesis specifies a range of
values for the parameter tested ( > 170; p .4;
etc.) effected by the belief.
4


A person is assumed
to be innocent
before court trial
H0

Needed in the test

Not guilty
Do not reject H0

Reject H0
guilty

Claim
Claim

HH000

HHAA

=A
=A

=A
=A

≠A
≠A

–tail
22-tail
-tail
or
or 2-side
2-sidetest
test

>A
>A

=A

=A

A
A

≥A
≥A

=A
=A

A

11 –tail
–tail
or
or 1-side
1-sidetest
test
1 –tail
or 1-side test

A
A
A

=A
=A
=A

A

1 –tail
or 1-side test

≤A
≤A
≤A

=A
=A
=A

A

1 –tail
or 1-side test


10.2 Testing the Population Mean when the Population
Standard Deviation is Known
Example 1: The manager of a department determines
that new billing system will be cost-effective only if the
mean monthly account is more than $170. A random
sample of 400 monthly accounts is drawn, for which the
sample mean is $178. Assume a standard deviation of
$65. Can the manager conclude from this that the new
system will be cost-effective?
Step 1: H0: µ = 170
H1: µ > 170

7


Step 2: Choose the significance level 

reject
reject H
H000
Verdict: Guilty

Do
not H
reject
H00
accept
0
accept
Verdict:HNot
Guilty
0

H00 is really
true true
HAA is really
false false
Really innocent
Type
Type II error
error
Probability

Probability of
of committing
committing the
the
type
type II error
error
Send an innocent person
to jail
OK
OK

H00 is really
false false
HAA isisreally
true true
true
Really guilty
OK
OK

Type II error
Type
II error of committing the
Probability
Probability
type II error of committing the
type
errorperson go free
Let aIIguilty

The more severe the consequence of committing the
type-I error, the smaller/higher the value of  .
Example 1:  = 0.05


Step 3: Determine the sample size n and hence the
sampling distribution.
Example 1:n=400; N(0,1)

9


Step 4: Depending on the sampling distribution, the HA,
and the value of  , find the suitable critical value and
the reject region.
HA: μ > 170 (1-sided test)

N(0,1)

Reject region: Z > 1.645
1.645
Critical value


Right-Tail Testing


Left-Tail Testing

Two–Tail Testing


Step 5: Collect data. x 178
Calculate the standard test statistic .
Example 1:
HA: μ =170

x   178  170
z

2.46
 / n 65 / 400


Step 6: If the test statistic is in the reject region, then
reject H0, otherwise do not reject H0.
Example 1: Reject H0 at  = .05
Reject H0: There is enough statistical evidence to
conclude that the alternative hypothesis is true.
Do not reject H0: There is not enough statistical
evidence to conclude that the alternative hypothesis is
true.


A Left Hand Tail Test
H0:   0
H1:  <  0

Reject H0 if x falls here

Critical
value

16


The SSA envelop plan example: The chief financial
officer in FedEx believes that including a stamped selfaddressed (SSA) envelop in the monthly invoice sent
to customers will decrease the amount of time it take
for customers to pay their monthly bills. Currently,
customers return their payments in 22 days on the
average, with a standard deviation of 6 days.
A random sample of 220 customers was selected and
SSA envelops were included with their invoice packs.
The mean time it took customers to pay their bill was
21.63
Can the CFO conclude that the plan will be successful
at 10% significance level?
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Step 1: H0:  H1:  
Step 2:  = 0.10
Step 3: n= 220
= .10
Step 4:
Step 5:
−1.28

Z

x  21.63 22
z

 .91
 / n 6/ 220
Step 6: Do not reject the null hypothesis
18


A Two Tail Test
H 0:    
H1:  





Reject H0 if x falls here
Critical
value

Reject H0 if x falls here
Critical
value

19

Example 2: A statistician believes the monthly mean of
the long-distance bills for all AT&T residential customers
is $17.09.
A random sample of 100 customers of its leading
competitor yields x 17.55
Assuming the standard deviation of the bills of the
competitors is 3.87, can we infer that there is a
difference between AT&T’s bills and the competitor’s
bills (on the average)?

20


Step 1: H0:  H1:  
Step 2:  = 0.05
Step 3: n= 100
= .025
Step 4:
−1.96
Step 5:
Critical value

= .025
1.96
Critical value

x   17.55 17.09
z


1.19
 / n 3.87/ 100
Step 6: Do not reject the null hypothesis
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