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Chapter 13

MGMT 1050 Chapter Notes - Chapter 13: Variance, F-Distribution, Test Statistic


Department
Management
Course Code
MGMT 1050
Professor
Olga Kraminer
Chapter
13

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Chapter 13 REMEMBER WHEN YOU TEST IF TWO VARIANCES ARE EQUAL THE
SIGNIFICANCE LEVEL ARE ALWAYS 0.05 AND THE DF FORMULA IS ALWAYS ROUNDED
DOWN
FOR F-DISTRIBUTION, THE CONFIDENCE LEVEL ALWAYS REMAINS THE SAME
- The F distribution ranges from 0 to infinity (like the chi-squared) and v1 and v2 are
the parameters of the distributions called degrees of freedom (like from chapter 12,
except now there are two of them); v1 is the numerator degrees of freedom and v2
is the denominator degrees of freedom
- Note that the mean only depends on the denominator degrees of freedom and that
for a large number of degrees of freedom, the mean of the F-distribution is
approximately 1
- The F-distribution, like the chi-squared distribution, also looks like a positively
skewed normal curve
- Again, its exact shape depends on the two values for degrees of freedom
- We define FA, v1, v2 as the value of F with v1 and v2 degrees of freedom such that
the area to its right under the curve is A this means:
P(F>FA, v1, v2) = A
- Since the F-random variable can only equal positive values, we define F1-A, v1, v2 as
the value such that the area to the left is A
- In your table, however, you will not see values that correspond to F-statistics on the
left side of the distribution
- Why? It’s because statisticians have shown that
F1-A, v1, v2 (right) = 1/FA, v2, v1 (left)
- Note that v1 and v2 switch!
Determining if the Two Population Variance Differ
- We can perform a statistical test to determine whether there is evidence to infer
that two population variance differ
- The hypothesis should always be stated as a ratio between the two variances, even
if the question is about standard deviations
- Notice that we are not looking at the actual differences between them; we use
o^2B/o^2A>1 instead of o^2B o^2A>0
- The test statistic is the ratio of the sample variances ratio and it is F-distributed
(required condition is again: both populations are normal or approximately normal)
- The degrees of freedom of F-distribution are identical to the degrees of freedom for
the two chi-squared distributions
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Difference of Means
- For instance, you may want to determine through hypothesis testing if the average
Canadian weighs more than the average American
- In essence, you wish to find if µCanadian > µAmerican, which is equivalent to
µCanadian µAmerican > 0 or µAmerican µCanadian < 0
- There are four types of difference of means questions: known variance, unknown
but likely equal variances, unknown but likely unequal variances, and paired samples
Difference of Means (Known Population Variances)
- When you know the variances of your two population, you should use the formulas
in row 1 for hypothesis testing and estimation (very simple)
- The distribution is normal, so you would use z-scores
- When you do not know the variances of your two populations, you must do an F-
test to determine if the variances are likely equal or not
- Therefore for difference of means questions when population variances are
unknown, there are two hypothesis tests per question (F-test to test if 2 variances
are equal and testing if means are different)
- Every difference of two sample means hypothesis testing question actually involves
two hypothesis tests: one to determine if the population variances differ and
another to determine if the means differ (and by how much)
Difference of Means (Likely Equal Population Variances)
- When the variances are equal there is a quantity labelled s^2p which is called the
pooled variance estimator
- It is the weighted average of the two sample variances with the number of degrees
of freedom in each sample used as weights
- The test statistic is student-t distributed with n1 + n2 2 degrees of freedom
provided that the two populations are normal
- The test statistics is referred to as the equal-variances test statistic
Difference of Means (Likely Unequal Population Variances)
- When the population variances are unequal, we cannot use the pooled variance
estimate
- Instead we estimate each population variance with its sample variance
- We refer to this formula as the unequal-variances test statistic
If the normality requirement is unsatisfied (and it is not a large sample, you cannot solve)
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