Hypothesis testing Univariate

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• Hypothesis testing Univariate Bivariate
• Statistical Analysis

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Univariate Statistics

• Test of statistical significance
• Hypothesis testing one variable at a time

Hypothesis

• Unproven proposition
• Supposition that tentatively explains certain
  facts or phenomena
• Assumption about nature of the world

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Types of hypotheses

• Null hypothesis
• Statement about the status quo
• No difference
• Alternative hypothesis
• Statement that indicates the opposite of the null
  hypothesis

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Significance Level

• Critical probability in choosing between the null
  hypothesis and the alternative hypothesis

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Linear Transformation of Any Normal Variable
Into a Standardized Normal Variable
Sometimes the scale is shrunk
Sometimes the scale is stretched
s
s
m
X
m
-2 -1 0 1 2
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Standardized Normal Distribution
34.13
34.13
13.59
13.59
2.14
2.14
Z
0
2
3
-2
-1
1
-3
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Region of Rejection
UPPER LIMIT
LOWER LIMIT
95
m3.0

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Hypothesis Test an example
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Hypothesis Test an example
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Hypothesis Test an example
m3.0
X 3.78
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Type I and Type II Errors
Accept null
Reject null
Null is true
Correct- no error
Type I error
Null is false
Type II error
Correct- no error
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Choosing the Appropriate Statistical Technique

• Type of question to be answered
• Number of variables
• Univariate
• Bivariate
• Multivariate
• Scale of measurement

NONPARAMETRIC STATISTICS
PARAMETRIC STATISTICS
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Questionnaire An Example
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Questionnaire An Example
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Hypothesis Test Using the t-Distribution
3. How would you rate the quality of the
following attributes of Starbucks?
HYPOTHESIS Consumers perceive the quality of
Starbucks as high
H0 µ 3.5 Ha µ ? 3.5
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T-Test
From the table, t-values of atmosphere, services,
place, sales promotion, advertisement, taste, and
quality of products are exceeding 1.96, with
p-values of less than 0.05, therefore, those null
hypotheses could be rejected. Thus, consumers
perceive Starbucks atmosphere, services, place,
sales promotion, advertisement, taste, and
quality of products as significantly high while
consumers perception on reputation, price, and
variety of product are not high.
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Differences Between Groups when Comparing Means

• Ratio/Interval scaled dependent variables
• t-test for comparing means
• When groups are small
• When population standard deviation is unknown
• Null Hypothesis About Mean Differences Between
  Groups

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Hypothesis Frequent and non frequent customers
perceive the quality of Starbucks differently.
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Hypothesis Ho µ1 µ2 Ha µ1 ?
µ2 Conclusion..
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Differences among groups when comparing means

• Ratio/interval scaled dependent variables
• Analysis of Variance
• Hypothesis when comparing three groups
• H0 m1 m2 m3
• H1 At least one group is different from others.

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ANOVA
Discount coupon
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Homogeneous Subsets
Ho µ1 µ2 µ3 µ4 Ha At least one µ is not
equal Conclusions .
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Measures of Association

• A general term that refers to a number of
  bivariate statistical techniques used to measure
  the strength of a relationship between two
  variables.
• Relationships Among Variables
• Correlation analysis
• Bivariate regression analysis

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Correlation Coefficient

• is a statistical measure of the covariation or
  association between two variables.
• The Correlation coefficient for two variables, X
  and Y is r

Regression analysis

• is a measure of linear association that
  investigates a straight line relationship
• Useful in forecasting

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Bivariate Linear Regression

• A measure of linear association that investigates
  a straight-line relationship
• Y a bX or Y ß0 ßX e
• where
• Y is the dependent variable
• X is the independent variable
• a and b (ß0 ß) are two constants to be
  estimated (unstandardized ß)
• a Y intercept
• which is an intercepted segment of a line
• The point at which a regression line intercepts
  the Y-axis
• b Slope
• The inclination of a regression line as compared
  to a base line
• Rise over run
• Can be positive or negative direction

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Simple Regression Analysis
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Multiple Regression Analysis
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Non-parametric Test

• Testing a Hypothesis about a Distribution
• Chi-Square test
• Test for significance in the analysis of
  frequency distributions
• Compare observed frequencies with expected
  frequencies
• Goodness of Fit

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Chi-Square Test an example
Starbucks Survey 2.How many times did you go to
Starbucks last month ? ? Less than 1 time.
? 1-2 times. ? 3-4 times. ? More
than 4 times

• HYPOTHESIS
• The numbers of customers who went to Starbucks in
  the last month with different frequency are
  different.
• Ho The numbers of customers who went to
  Starbucks in the last month with different
  frequency are equal.
• Ha The numbers of customers who went to
  Starbucks in the last month with different
  frequency are different.

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Chi-square test

• Less than 1 time.

Expected counts
Observed counts
Expected counts
Observed counts
1-2 times.
Expected counts
Observed counts
3-4 times.
Expected counts
Observed counts
More than 4 times
x² chi-square statistics Oi observed
frequency in the ith cell Ei expected frequency
on the ith cell
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Chi-Square Test an example
From the table, chi-square value is 20.9 with a
p-value of less than 0.05, therefore, the null
hypothesis could be rejected and alternative
hypothesis is accepted. Thus, the numbers of
customers who went to Starbucks in the last month
with different frequency are significantly
different?
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Differences Between Groups

• Contingency Tables
• Cross-Tabulation
• Chi-Square allows testing for significant
  differences between groups
• Goodness of Fit

x² chi-square statistics Oi observed
frequency in the ith cell Ei expected frequency
on the ith cell
d.f. (R-1)(C-1)
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Hypothesis The frequency to visit Starbucks is
different among various income groups.
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Error? collapse scale
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To transform data by collapsing the scale, recode
command would be used
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Hypothesis The frequency to visit Starbucks is
different among the group of customers with
different income levels.
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Chi-square of 6.992 with 2 degree of freedom
yields a p-value of .030 which is less than 0.05.
Thus Null hypothesis can be rejected, the
frequency to visit Starbucks is different among
the group of customers with different income
levels.
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Do you have to perform any other data analyses?
RankingMultiple response analysis (for the
checklist)Etc.