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Chapter 13

This

**preview**shows pages 1-2. to view the full**6 pages of the document.**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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