Business Research Methods William G. Zikmund

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Business Research MethodsWilliam G. Zikmund

• Chapter 21
• Univariate Statistics

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Univariate Statistics

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

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Hypothesis

• An unproven proposition or supposition that
  tentatively explains certain facts or phenomena
• Null hypothesis
• Alternative hypothesis

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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
• Confidence Level
• Alpha
• Probability Level selected is typically .05 or .01

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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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Red Lion Restaurant Example
The null hypothesis that the mean is equal to 3.0
The alternative hypothesis that the mean does not
equal to 3.0
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A Sampling Distribution
LOWER LIMIT
UPPER LIMIT
a.025
a.025
m3.0
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Critical values of m
Critical value - upper limit
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Critical values of m
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Critical values of m
Critical value - lower limit
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Critical values of m
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Hypothesis Test m 3.0
LOWER LIMIT
UPPER LIMIT

2.804
3.78
3.196
m3.0
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Alternate Way of Testing the Hypothesis
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Alternate Way of Testing the Hypothesis
Thus, we reject the null.
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Choosing the Appropriate Statistical Technique

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

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NONPARAMETRIC STATISTICS
PARAMETRIC STATISTICS
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t-Distribution

• Symmetrical, bell-shaped distribution
• Mean of zero and a unit standard deviation
• Shape influenced by degrees of freedom

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Degrees of Freedom

• Abbreviated d.f.
• Number of observations
• Number of constraints

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Univariate Hypothesis Test Utilizing the
t-Distribution
Suppose that a production manager believes the
average number of defective assemblies each day
to be 20. The factory records the number of
defective assemblies for each of the 25 days it
was opened in a given month. The mean was
calculated to be 22, and the standard deviation,
,to be 5.
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Univariate Hypothesis Test Utilizing the
t-Distribution
The researcher desired a 95 percent confidence,
and the significance level becomes .05.The
researcher must then find the upper and lower
limits of the confidence interval to determine
the region of rejection. Thus, the value of t is
needed. For 24 degrees of freedom (n-1, 25-1),
the t-value is 2.064.
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Univariate Hypothesis Test t-Test
Since the observed t-value of 2 is less than the
critical t-value of 2.064, null hypothesis cannot
be rejected. Managers assumption is correct.
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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
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 Estimation for Expected Number
for Each Cell
Ri total observed frequency in the ith row Cj
total observed frequency in the jth column n
sample size
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Univariate Hypothesis Test Chi-square Example
d.f. k 1 Where, k the number of cells
associated with column or row data
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Univariate Hypothesis Test Chi-square Example
Since the calculated chi-square value of 4 is
larger than the tabular Chi-square of 3.84, the
null hypothesis is rejected.
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Hypothesis Test of a Proportion

• p is the population proportion
• p is the sample proportion
• p is estimated with p

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Hypothesis Test of a Proportion

p
5
.


H
0
¹
p
5
.


H
1
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The Z obs value of 2.04 is less than the critical
value of 2.57, so the Null hypothesis cannot be
rejected.
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Hypothesis Test of a Proportion Another Example
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Hypothesis Test of a Proportion Another Example
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Hypothesis Test of a Proportion Another Example
p
-
p

Z
S
p
-
15
.
20
.

Z
0115
.
05
.

Z
0115
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348
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4
Z
Indeed