BUS105 Statistics
Seminar 4

Learning Objectives
1. Execute the Hypothesis Testing (two-samples) for the
population mean and proportion population
•
Select and execute an appropriate hypothesis testing for two-sample mean
or proportion.
•
Execute a hypothesis to determine whether the variances of two
populations are equal.
2. Apply an Analysis of Variance (ANOVA) procedure to compare
the means of independent random samples
•
Use hypothesis testing to compare means of independent random samples.
2

Topics
Unit 2
1 Point Estimation and Confidence Intervals
2.1 Hypothesis Tests: One-Sample
2.2 Hypothesis Tests: Two-Sample
3 Analysis of Variance (ANOVA)
3

Unit 2 - Chapter 2
Hypothesis tests:
Two-Sample Tests

Recap: Hypothesis and Hypothesis Testing
HYPOTHESIS
A statement about the value of a population parameter
developed for the purpose of testing.
HYPOTHESIS TESTING A procedure based on sample evidence and
probability theory to determine whether the hypothesis is a
reasonable statement.
5

Comparing Two Population Means
H
0
: µ
1
= µ
2
H
0
: µ
1
≥ µ
2
µ
1
≤ µ
2
H
1
: µ
1
≠ µ
2
H
0
: µ
1
< µ
2
µ
1
> µ
2
where
Reject H
0
if p-value <
α
Equal and Known
Variance
Equal but Unknown
Variance
Use
t
distribution if:
•
One or both of the samples have less than 30
observations, and
•
The population standard deviations are unknown.
What are the
assumptions that
one needs to check
before performing
the two tests?
6
Test
statistics
Hypothesis
Decision
Rule
2
2
2
1
2
1
2
1
n
n
X
X
z
2
)
1
(
)
1
(
2
1
2
2
2
2
1
1
2
n
n
s
n
s
n
s
p
2
1
2
2
1
2
1
1
2
1
n
n
s
X
X
t
p
n
n

Two-Sample Tests of Proportions
•
We investigate whether two samples came from
populations with an equal proportion of successes.
•
The formula for computing the value of
z
is:
7

Comparing Two Population Proportions
H
0
:
1
=
2
H
0
:
1
≥
2
H
0
:
1
≤
2
H
1
:
1
≠
2
H
1
:
1
<
2
H
1
:
1
>
2
8
Test
statistics
Hypothesis
Decision
Rule

Dependent vs. Independent Samples
How do we tell between dependent and independent samples?
1.
A dependent sample is characterized by a measurement
followed by an intervention of some kind and then another
measurement. This could be called a “before” and “after”
study.
2.
Dependent sample is characterized by matching or pairing
observation. Dependent samples are samples that are paired
or related in some fashion.