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Chapter Thirteen Part II
Hypothesis Testing:
Means
Where does this come from
• Suppose Tide is contemplating a new ad campaign. They
feel that mean consumer attitude to the old campaign is 2
on a [1 (dislike very much) -5 (like very much) scale]
• Past studies have confirmed that the average variability in
the target audience is 1.30
• Tide wants to verify this feeling so they survey 500
consumers and measure consumer attitudes to the old
campaign. The mean attitudes in the sample are 3.
• Should Tide continue with the old campaign (and save
money) or make a new campaign?
Hypothesis Testing About
a Single Mean - Step-by-Step
1) Formulate Hypotheses
Is this a one-tailed or two-tailed test?
Ho:  = hypothesized value of population
Ha:   hypothesized value of population
Our problem: What is Ho and Ha?
Which test to apply?
•
Is the population standard deviation known?
•
•
If ‘yes’ – use the Z test
If ‘no’ use the t test
•
Our problem: Which test do we use?
Hypothesis Testing About
a Single Mean - Step-by-Step
1) Formulate Hypotheses
2) Select appropriate formula
T = sample mean – population mean / standard error of the
mean
Z = sample mean – population mean / standard error of the
mean
Where, standard error of the mean = population standard
deviation / sq.root of N, (z test) OR
Sample standard deviation / sq. root of N (t test)
Our problem: What is the standard error of the mean?
Hypothesis Testing About
a Single Mean - Step-by-Step
1) Formulate Hypotheses
2) Select appropriate formula
3) Select significance level
1%, 5% or 10%
a
What is the usual significance level in the social
sciences?
Hypothesis Testing About
a Single Mean - Step-by-Step
1)
2)
3)
4)
Formulate Hypotheses
Select appropriate formula
Select significance level
Calculate z or t statistic
Our problem: What is the observed test statistic?
( X  )
t
sx
where
sx =
s
n
Hypothesis Testing About
a Single Mean - Step-by-Step
1)
2)
3)
4)
5)
Formulate Hypotheses
Select appropriate formula
Select significance level
Calculate z or t statistic
Calculate degrees of freedom (for t-test)
Our problem: What are the degrees of freedom?
d.f. = n-1
Hypothesis Testing About
a Single Mean - Step-by-Step
1)
2)
3)
4)
5)
6)
Formulate Hypotheses
Select appropriate formula
Select significance level
Calculate z or t statistic
Calculate degrees of freedom (for t-test)
Obtain critical value from table
Our problem: What is the critical value from the
table?
Hypothesis Testing About
a Single Mean - Step-by-Step
One-tailed: if tts > ta then reject
1) Formulate Hypotheses
Ho
2) Select appropriate formula
Two-tailed: if |tts| > ta/2 then
3) Select significance level
reject Ho
4) Calculate z or t statistic
5) Calculate degrees of freedom (for t-test)
6) Obtain critical value from table
7) Make decision regarding the Null-hypothesis
Our problem: What would you advise Tide?
Hypothesis Testing About
two means from independent samples
• Suppose your survey of brand preferences
towards Coke revealed that the sample drawn
from Charlotte had a mean preference score of
3.5 and the sample drawn from Columbia had a
mean preference score of 3.0.
• Is this difference statistically significant?
• Do the two samples originate from two different
populations (two different cities) or are they from
a single population (the US South East)
Hypothesis Testing About
two means from independent samples
1) Formulate Hypotheses
Ho: X 1 = X 2
Ha: X 1  X 2
Standard Error of the Differences Between Means
• (to recap) Same logic as the standard error of the
mean
– Very large number of samples with replacement
possible (each sample will have a mean)
– This generates a hypothetical distribution of means
– standard deviation of the distribution of means
• We have two sets of means (since we have two
independent samples)
– Here we have two distributions of means
– Take the difference between the means and we have
the third distribution of difference between means
– Standard deviation of the differences between means
Hypothesis Testing About
two means from independent samples
1) Formulate Hypotheses
2) Select appropriate formula
t ts
 X 2 ) - 0 where S
(X
1

S X1 X 2
Where
X
1
 X
2

sp
1 + 1
n1
n2
(n1 – 1) s12 + (n2 – 1) s22
sp 2 
n1 + n2 -2
Note: This assumes that variance are equal across the two
independent samples
Hypothesis Testing About
two means from independent samples
1) Formulate Hypotheses
2) Select appropriate formula
3) Select significance level
a
Hypothesis Testing About
two means from independent samples
1)
2)
3)
4)
t ts
Formulate Hypotheses
Select appropriate formula
Select significance level
Calculate t statistic
 X 2 ) - 0 where S
(X
1

S X1 X 2
Where
X
1
 X
2

sp
1 + 1
n1
n2
(n1 – 1) s12 + (n2 – 1) s22
sp 2 
n1 + n2 -2
Hypothesis Testing About
two means from independent samples
1)
2)
3)
4)
5)
Formulate Hypotheses
Select appropriate formula
Select significance level
Calculate z or t statistic
Calculate degrees of freedom (for t-test)
d.f. = n1 + n2 -2
Hypothesis Testing About
two means from independent samples
1)
2)
3)
4)
5)
6)
Formulate Hypotheses
Select appropriate formula
Select significance level
Calculate z or t statistic
Calculate degrees of freedom (for t-test)
Obtain critical value from table
Hypothesis Testing About
two means from independent samples
1)
2)
3)
4)
5)
6)
7)
Formulate Hypotheses
Select appropriate formula
Select significance level
Calculate z or t statistic
Calculate degrees of freedom (for t-test)
Obtain critical value from table
Make decision regarding the Null-hypothesis
One-tailed: if tts > ta then reject Ho
Two-tailed: if |tts| > ta/2 then reject Ho
Hypothesis Testing About
Differences in Means (dependent samples)
1) Formulate Hypotheses
Ho: X 1 - X 2 = 0
Ha: X 1- X 2  0
Hypothesis Testing About
Differences in Means (dependent samples)
1) Formulate Hypotheses
2) Select appropriate formula
t ts
 d)
 (D
s D/
where
D
n
Where
1
= n
s D2 =
1
n-1
n
n
S Di
i=1
(S Di2 – nD2)
i=1
Hypothesis Testing About
Differences in Means (dependent samples)
1) Formulate Hypotheses
2) Select appropriate formula
3) Select significance level
a
Hypothesis Testing About
Differences in Means (dependent samples)
1)
2)
3)
4)
t ts
Formulate Hypotheses
Select appropriate formula
Select significance level
Calculate t statistic
 d)
(D

s D/
where
D
n
Where
1
= n
s D2 =
1
n-1
n
n
S Di
i=1
(S Di2 – nD2)
i=1
Hypothesis Testing About
Differences in Means (dependent samples)
1)
2)
3)
4)
5)
Formulate Hypotheses
Select appropriate formula
Select significance level
Calculate z or t statistic
Calculate degrees of freedom (for t-test)
d.f. = n - 1
Hypothesis Testing About
Differences in Means (dependent samples)
1)
2)
3)
4)
5)
6)
Formulate Hypotheses
Select appropriate formula
Select significance level
Calculate z or t statistic
Calculate degrees of freedom (for t-test)
Obtain critical value from table
Hypothesis Testing About
Differences in Means (dependent samples)
1)
2)
3)
4)
5)
6)
7)
Formulate Hypotheses
Select appropriate formula
Select significance level
Calculate z or t statistic
Calculate degrees of freedom (for t-test)
Obtain critical value from table
Make decision regarding the Null-hypothesis
One-tailed: if tts > ta then reject Ho
Two-tailed: if |tts| > ta/2 then reject Ho
Example: Test of Difference in Means
Before: 5
2
3
3
4
2
5
4
After:
6
4
3
6
3
4
5
6
D:
1
2
0
3
-1
2
0
2
2
1
4
0
9
1
4
0
4
D:
Rules to remember
1. State Ho (Ho is the hypothesis of no difference / no
relationship between the sample and population or two
samples)
2. State Ha (Ha is the mirror image of Ho)
3. Identify critical region (this the region beyond the critical
value on either side. Ho is rejected if observed value falls
in the critical region)
4. Decide distribution (if ơ is known use Z distribution, if
not, use t distribution)
5. Identify the test (1 sample = one sample z or t; two
independent samples = z or t; two dependent samples
= z or t on differences.
6. Decide one / two tailed (choose tail according to
direction of Ha)