MKTG 368 All Statistics PowerPoints

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MKTG 368All Statistics PowerPoints

• Setting Up Null and Alternative Hypotheses
• One-tailed vs. Two-Tailed Hypotheses
• Single Sample T-Test
• Paired Samples T-Test
• Independent Samples T-Test
• ANOVA
• Correlation and Regression
• One-Way and Two-Way Chi-Square

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Translating a Problem StatementInto the Null and
Alternative Hypotheses
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Initial Problem Statement

• Example
• Lets say we are interested in whether a flyer
  increases contributions to National Public Radio.
  We know that last year the average contribution
  was 52. This year, we sent out a flyer to 30
  people explaining the benefits of NPR and asked
  for donations. This years average contribution
  with the flyer ended up being 55, with a
  standard deviation of 12.
• How do we translate this into the null and
  alternative hypotheses (in terms of both a
  sentence and a formula)?

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Gleaning Information from the Statement

• Example
• Lets say we are interested in whether a flyer
  increases contributions to National Public Radio.
  We know that last year the average contribution
  was 52. This year, we sent out a flyer to 30
  people explaining the benefits of NPR and asked
  for donations. This years average contribution
  with the flyer ended up being 55, with a
  standard deviation of 12.

Direction of Alternative Hypothesis
Population Information
Sample Information
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Translating Information into Null and Alternative
Hypotheses
Set up Alternative Hypothesis First Null is exact
opposite of Alternative Null Alternative must
include all possibilities Hence, we say less
than or equal to rather than just less than
Ho (Null Hypothesis) A flyer does not
increase contribution to NPR Flyer Last
Year H1 (Alternative Hypothesis) A flyer
increases contributions to NPR Flyer gt Last
Year
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On One-Tailed (Directional) vs. Two-Tailed
(Non-Directional) Hypotheses
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Basics on the Normal Distribution
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One-Tailed Hypothesis (H1 Condition 1 gt
Condition 2)
Ho (Null Hypothesis) A flyer does not
increase contribution to NPR Flyer Last
Year H1 (Alternative Hypothesis) A flyer
increases contributions to NPR Flyer gt Last
Year
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One-Tailed Hypothesis (H1 Condition 1 lt
Condition 2)
Ho (Null Hypothesis) A poster does not
decrease lbs of litter in park Posterlbs Last
Yearlbs H1 (Alternative Hypothesis) A poster
decreases lbs of litter in park Posterlbs lt Last
Yearlbs
Alpha Region a .01, 1-tailed
(negative) t-critical
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Two-Tailed Hypothesis (H1 Condition 1 ?
Condition 2)
Ho (Null Hypothesis) People are willing to
pay the same for Nike vs. Adidas NikeWTP
AdidasWTP H1 (Alternative Hypothesis) People
not willing to pay same for Nike vs. Adidas
NikeWTP ? AdidasWTP
Alpha Region a .025
Alpha Region a .025
(positive) t-critical
(negative) t-critical
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T-TestsSingle SamplePaired Samples (Correlated
Groups)Independent Samples
Single Sample
Independent Samples
Paired Samples
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Single Sample T-Test(Example 1)

• Comparing a sample mean to an existing population
  mean

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Gleaning Information from the Statement

• Example
• Lets say we are interested in whether a flyer
  increases contributions to National Public Radio.
  We know that last year the average contribution
  was 52. This year, we sent out a flyer to 30
  people explaining the benefits of NPR and asked
  for donations. This years average contribution
  with the flyer ended up being 55, with a
  standard deviation of 12.

Direction of Alternative Hypothesis
Single Sample T-test df N-1 30-1 29
Population Information
Sample Information
Use alpha .05
How do we get a t-critical value? ?
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Critical T-Table
For single sample t-test, df N-1
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One-Tailed Hypothesis (H1 Condition 1 gt
Condition 2)
Ho (Null Hypothesis) A flyer does not
increase contribution to NPR Flyer Last
Year H1 (Alternative Hypothesis) A flyer
increases contributions to NPR Flyer gt Last
Year
If t-obtained gt t-critical, reject Ho (i.e., if
t-obtained falls in the critical region, reject
Ho).
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Computation of Single Sample T-test
Decision? Because t-obtained (1.37) lt t-critical
(1.699), retain Ho. Conclusion? The flyer did
not increase contributions to NPR.
t-obtained 1.37
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Single Sample T-Test(Example 2)

• Comparing a sample mean to an existing population
  mean

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Gleaning Information from the Statement

• Example
• Lets say we are interested in whether a poster
  decreases amount of litter in city parks. We know
  that last year the average amount of litter in
  city parks was 115 lbs. This year, we placed
  flyers in 25 parks that said Did you know that
  95 of people dont litter? Join the crowd.
  Later, when we weighed the litter, the average
  amount of litter was 100 lbs, with a standard
  deviation of 10 lbs.

Direction of Alternative Hypothesis
Single Sample T-test df N-1 25-1 24
Population Information
Sample Information
Use alpha .01
How do we get a t-critical value? ?
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Critical T-Table
For single sample t-test, df N-1
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One-Tailed Hypothesis (H1 Condition 1 lt
Condition 2)
Ho (Null Hypothesis) A poster does not
decrease lbs of litter in park Posterlbs Last
Yearlbs H1 (Alternative Hypothesis) A poster
decreases lbs of litter in park Posterlbs lt Last
Yearlbs
Alpha Region a .01, 1-tailed
t-critical -2.492
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Computation of Single Sample T-test
Decision? Because t-obtained (-7.50) lt
t-critical (-2.492), reject Ho. Conclusion?
The signs did decrease lbs of trash in the park.
Alpha Region a .01, 1-tailed
t-critical -2.492
t-obtained -7.50
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Paired Samples T-Test

• Comparing two scores from the same
• Individual (or unit of analysis)

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Gleaning Information from the Statement

• Example
• Lets say we are interested in whether a brand
  name (Nike vs. Adidas) affects willingness to pay
  for a sweatshirt. To explore this question, we
  take 9 people and have them indicate their WTP
  for a Nike sweatshirt and for an Adidas
  sweatshirt. The only difference between the
  sweatshirts is the brand name.

Non-Directional (Two-Tailed) Alternative
Hypothesis doesnt say is higher or is
lower just says affects
Paired Samples T-test df N-1 9-1 8
Paired Scores From Same Person
Use alpha .05
How do we get a t-critical value? ?
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Two-Tailed Hypothesis (H1 Condition 1 ?
Condition 2)
Ho (Null Hypothesis) People are willing to
pay the same for Nike vs. Adidas NikeWTP
AdidasWTP H1 (Alternative Hypothesis) People
not willing to pay same for Nike vs. Adidas
NikeWTP ? AdidasWTP
Alpha Region a .025
Alpha Region a .025
(negative) t-critical
(positive) t-critical
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Critical T-Table
For paired samples t-test, df N-1
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Two-Tailed Hypothesis (H1 Condition 1 ?
Condition 2)
Ho (Null Hypothesis) People are willing to
pay the same for Nike vs. Adidas NikeWTP
AdidasWTP H1 (Alternative Hypothesis) People
not willing to pay same for Nike vs. Adidas
NikeWTP ? AdidasWTP
Alpha Region a .025
Alpha Region a .025
t-critical 2.306
t-critical -2.306
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Defining Symbols in Paired T-test
_ D average difference score. D difference
score (eg., time 1 vs. time 2 midterm vs. final
husband vs. wife) N Paired Scores (not the
of numbers in front of you). ? average
difference score in the Null Hypothesis
Population (most often 0) SSD Sum of
Squared Deviations for the Difference Scores
?D2 (?D)2/N tobt the t statistic which is
compared to tcrit with N-1 df
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Paired Samples T-testNike vs. Adidas Sweatshirt
Example
First, Compute SSD
Then, Compute t
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Decision and Conclusion?
t-obtained 3.07
Decision? Because t-obtained (3.07) lt t-critical
(2.306), reject Ho. Conclusion? People willing
to pay more for Nike than for Adidas.
We know this, because the average difference
score was positive. (Nike Adidas)
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Independent Samples T-Test

• Comparing means
• of two conditions or groups

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Gleaning Information from the Statement

• Example
• Lets say we are interested in how consumers
  respond to service failures, so we decide to run
  an experiment. We ask people to read about a
  hypothetical service failure scenario (e.g.,
  delayed service at a restaurant). Then we
  randomly assign half of the subjects to the
  apology condition (well call this Group 1),
  and the other half to a control condition
  (well call this Group 2). Those in the apology
  condition read that the restaurant owner offered
  a sincere apology for having to wait so long.
  After this, we assess subjects self-reported
  anger (1 not at all angry, 11 fuming mad). We
  hypothesize that subjects will report less anger
  in the apology condition.

Directional (One-Tailed) Alternative Hypothesis
Scores come from two Independent groups
Independent Samples t-test df N-2 20-2 18
Use alpha .05
How do we get a t-critical value? ?
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One-Tailed Hypothesis (H1 Condition 1 lt
Condition 2)
Ho (Null Hypothesis) An apology does not
decrease anger ApologyAnger ControlAnger H1
(Alternative Hypothesis) Anger will be lower
in the Apology Condition ApologyAnger lt
ControlAnger
Alpha Region a .05, 1-tailed
(negative) t-critical
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Critical T-Table
For independent t-test, df N-2
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One-Tailed Hypothesis (H1 Condition 1 lt
Condition 2)
Ho (Null Hypothesis) An apology does not
decrease anger ApologyAnger ControlAnger H1
(Alternative Hypothesis) Anger will be lower
in the Apology Condition ApologyAnger lt
ControlAnger
Alpha Region a .05, 1-tailed
t-critical -1.734
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Defining Symbols in Independent T-test
_ _ X1 and X2 the means of X1 and X2 (our
two conditions), respectively SS1 and SS2 Sum
of Squared Deviations for X1 and X2 whereSS
?X2 (?X)2/N for each group n the number of
subjects in each conditions. n1 n2 N. In
other words, n ? N! tobt the t statistic
which is compared tcrit with N-2 df.
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Independent Samples T-testApology vs. No Apology
Example
First, Compute SS for Each Condition
Then, Compute t
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Decision and Conclusion?
Decision? Because t-obtained (-2.50) lt
t-critical (-1.734), reject Ho. Conclusion?
People report less anger after an apology
t-obtained -3.07
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Analysis of Variance (ANOVA)

• Comparing means
• of three or more conditions or groups

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The F-Ratio A Ratio of Variances Between and
Within Groups
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Variance Within Group 3
Variance Within Group 1
Variance Within Group 2
Mean 9
Mean 3
Mean 5
Between Groups Variance (Numerator of F-ratio)
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F-Distribution

• Probability distribution
• All values positive (variance ratio)
• Positively skewed
• Median 1
• Shape varies with degrees of freedom (within and
  between)

Alpha Region a .05
0
1
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F-critical TableIf we have 3 conditions, N
14, alpha .05 F-crit 3.98
Alpha Level
df numerator K-1
df denominator N-K
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Null and Alternative Hypotheses
Lets say a marketing researcher is interested in
the impact of music on sales at a new clothing
store targeted to tweens. She sets up a mock
store in her universitys research lab, gives
each subject 50 spending money, and then
randomly assigns subjects to one of three
conditions. One third of the subjects browse the
mock store with no music. One third browse the
store with soft music. And the final third browse
the store with loud music. The sales figures are
shown below. Assume the researcher decides to use
an alpha level of .01.
Null Hypothesis (Ho) All of the means are equal
(ucontrol usoft music uloud
music) Alternative (H1) At least two means are
different F-critical (based on alpha .01
df-numerator 2 df-denominator 9)
8.02 Decision Rule If Fobt Fcritical, then
reject Ho. Otherwise, retain Ho
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The Data Sales as a Function of Music Condition
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(No Transcript)
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Do this for each of the three conditions
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(See Statistics Notes Packet)
Summarize in a Source Table
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ANOVA - Source Table
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F-critical in our example 8.02 N 12 K
3 Alpha .01
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Decision Rule and Conclusion?
Reject Null Hypothesis At least two means are
different
Alpha Region a .01
F-critical 8.02
F-obtained 23.47
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Correlation
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Differences Between Correlation and Regression

• Correlation (r)
• assessing direction ( or -) and degree (strong,
  medium, weak) of relationship between two
  variables
• Linear Regression (slope, y-intercept)
• assessing nature of relationship between an
  outcome variable and one or more predictors
• making predictions for Y (cfc) based on X (impuss)

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Reading Scatterplots
Negative Correlation
Zero Correlation
Positive Correlation
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(No Transcript)
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Two Interpretations of Correlation Coefficient

• Direction Degree of Relationship Between Two
  Variables
• Range from 1 to 1
• Stronger correlations at the extremes
• r -1 (perfect negative relationship)
• r 0 (no relationship)
• r 1 (perfect positive relationship)
• Variance Explained
• r2, Ranges from 0 to 1.0
• What percent of the variance in Y is explained by
  X?
• Model Comparison Approach

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Problem Statement - A

• Lets say we survey 5 shoppers about their level
  of satisfaction with the service they received
  from a furniture store (X satisfaction
  w/service) and their intention to return to the
  store in the future (Y future intentions).
  Presumably, there should be a positive
  correlation between these variables.
• Null Hypothesis (Ho) Satisfaction with service
  and future shopping intentions are not positively
  correlated
• Alternative (H1) Satisfaction with service and
  future shopping intentions are positively
  correlated (this is a directional hypothesis)
• r-critical (based on alpha .05(one-tailed),df
  N-2 3) r-critical .8054
• Decision Rule (in this example, because r is
  predicted to be positive)If robt rcritical,
  then reject Ho. Otherwise, retain Ho

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r-critical TableIf alpha .05 (1-tailed), N
5, df 3, r-critical .8054
Decision Rule (when r is predicted to be
positive) If robt rcritical, then reject Ho.
Otherwise, retain Ho Decision Rule (when r is
predicted to be negative) If robt rcritical,
then reject Ho. Otherwise, retain Ho Decision
Rule (when H1 is non-directional) If robt
rcritical, then reject Ho. Otherwise, retain Ho
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Data and Scatterplot
Data
Scatterplot
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Raw Score Formula for Pearsons r Correlation
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Computing Pearsons r (and variance explained)

• Compute SSx and SSy
• Then compute r

r2 (.313.313) .098 So, satisfaction explains
9.8 of variance in future intentions
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Regression
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Problem Statement - B

• Lets use the data we just worked with for
  correlation. Five shoppers were asked their
  satisfaction with the service they received and
  their intention to shop at the store in the
  future.
• Regression would be used to make predictions for
  future shopping intentions (Y) based on peoples
  satisfaction with service (X).
• For example, what would we predict if a shopper
  rated their satisfaction with service at a 3?
• First need to compute regression equation, then
  use it to make a prediction

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Slope and Y-Intercept
Y-intercept (bo) (value of Y, when X 0)
Slope (b1) (change in Y for 1 unit change in X)
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Raw Score Formula for Slope and Y-Intercept
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Computing Regression Equation
First compute slope Then compute y-intercept
So, the regression equation is
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Data and Scatterplot
Data
Scatterplot
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Using the Regression Equationto Make a Prediction

• Lets say a customer rates their satisfaction as
  a 3 on our 7-point scale.
• What is their predicted future intention of
  shopping at the store in the future?

So, a person who gives a 3 on the satisfaction
scale has a predicted future intention score of
3.544
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Y-Predicted Residuals
If Satisfaction (X) 3 Predicted Intention (Y)
3.54
Y predicted 3.54
Residual (Y-Y predicted) When r is
strong, residuals are small
X 3
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Chi-Square
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One-Way vs. Two-Way Chi-Square

• Chi-square is appropriate when our data are
  frequency (count) data
• In one-way chi-square, we have one categorical
  variable (type of shoe) with several levels
  (Adidas, Asics, Nikes, Pumas) and we want to know
  whether the frequency of observations differs
  between the groups (or conditions, or levels)
• In two-way chi-square, we have two categorical
  variables (Gender x Support for New Stadium) and
  we want to know if these two variables are
  related

1
2
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One-way Chi-Square
Where
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Problem Statement

• Lets say we ask 100 people to pick their
  favorite brand of shoes among four types. The
  data are shown below. Clearly, the frequencies
  are not equal (25 in each). Here, 15 pick Adidas,
  30 pick Asics, 45 pick Nike, and 10 pick Puma.
  The question is whether these frequencies are
  significantly different.
• Null Hypothesis (Ho) Frequencies of people
  choosing different brands is equal
• Alternative (H1) Not all the frequencies are
  equal (doesnt mean theyre all different)
• X2-critical (based on alpha .05 df K-1
  4-13) X2-critical 7.815 (X2 critical always
  positive) In df, K stands for the number of
  groups. (see critical table next page)
• Decision Rule is always as follows (b/c
  chi-square is always positive) If X2obt
  X2critical, then reject Ho. Otherwise, retain Ho

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Chi-Square Critical Table(for one-way chi-square)
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Formula and Frequency Expected
Frequency Observed (Actual Frequencies)
Frequency Expected (Typically Total N/K)
To compute chi-square, we need to know fe
expected frequency. Typically, well just assume
this represents an equal distribution across the
conditions (Total N/K). So, we have a total of
100 people and 4 conditions (brands of shoe).
Based on chance alone, an equal distribution
across the conditions would mean 25 people would
select each type of shoe. So, here well assume
fe 25.
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Computation
Decision Reject Ho, because X2obt (30)
X2critical (7.815). Conclusion People do not
show an equal preference among the four brands of
shoes.
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Two-Way Chi-Square
Where
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Problem Statement

• Lets say were interested in whether males and
  females differ in their support for building a
  new football stadium. We survey 40 people (10
  men, and 30 women) and we ask them a simple
  (categorical) yes/no question Do you support
  building a new football stadium? Now we want to
  know if there is a relationship between gender
  (male/female) and support for the stadium
  (yes/no).
• Null Hypothesis (Ho) There is no relationship
  between gender and support for football stadium
• Alternative (H1) There is a relationship between
  gender and support for football stadium
• X2-critical (based on alpha .05 df
  (Rows-1)(Columns-1)(2-1)(2-1)1 X2-critical
  3.841 (X2 critical always positive) (see critical
  table next page)
• Decision Rule is always as follows (b/c
  chi-square is always positive) If X2obt
  X2critical, then reject Ho. Otherwise, retain Ho

Of the 10 men surveyed, 8 supported it, and 2
didnt Of the 30 women surveyed, 5 supported it
and 25 didnt
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Data
Of the 10 men surveyed, 8 supported it, and 2
didnt Of the 30 women surveyed, 5 supported it
and 25 didnt
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Chi-Square Critical Table(works for two-way
chi-square)
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Formula and Frequency Expected
Frequency Observed (Actual Frequencies)
Frequency Expected (Row NColumn N)/Total N
To compute chi-square, we need to know fe
expected frequency. Typically, well just assume
this represents an equal distribution across the
conditions (Row NColumn N)/Total N. The next
slide illustrates the computation of frequency
expected.
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Computing Frequency Expected and Chi-Square
Decision Reject Ho, because X2obt (13.7)
X2critical (3.841). Conclusion Gender is related
to support for football stadium (men gt women)