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Chapter Eleven Part 3 (Sections 11. 4 & 11.5) Chi-Square and F Distributions PowerPoint PPT Presentation


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Understandable Statistics S eventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College. Chapter Eleven Part 3 (Sections 11. 4 & 11.5) Chi-Square and F Distributions. Testing Two Variances.

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Chapter Eleven Part 3 (Sections 11. 4 & 11.5) Chi-Square and F Distributions

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Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Understandable StatisticsSeventh EditionBy Brase and BrasePrepared by: Lynn SmithGloucester County College

Chapter Eleven Part 3

(Sections 11. 4 & 11.5)

Chi-Square and F Distributions


Testing two variances

Testing Two Variances

Use independent samples from two populations to test the claim that the variances are equal.


Assumptions for testing two variances

Assumptions for Testing Two Variances

  • The two populations are independent

  • The two populations each have a normal probability distribution.


Notations used

Notations Used:


Define population i as the population with the larger or equal sample variance

Define population I as the population with the larger (or equal) sample variance


Set up hypotheses

Set Up Hypotheses


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Set Up Hypotheses


Equivalent hypotheses may be stated about standard deviations

Equivalent hypotheses may be stated about standard deviations.


Use the f statistic

Use the F Statistic


The f distribution

The F Distribution

  • Not symmetrical

  • Skewed right

  • Values are always greater than or equal to zero.

  • A specific F distribution is determined from two degrees of freedom.


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

An F Distribution


Degrees of freedom for test of two variances

Degrees of Freedom for Test of Two Variances

  • Degrees of freedom for the numerator =

    d.f. N = n1 - 1

  • Degrees of freedom for the denominator =

    d.f. D = n2 - 1


Values of the f distribution

Values of the F Distribution

Given in Table 8 of Appendix II


Some values of the f distribution

Some Values of the F Distribution


Find critical value of f from table 8 appendix ii

Find critical value of F from Table 8 Appendix II

d.f.N = 3

d.f.D = 5

Right tail area =  = 0.025


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

F Distributiond.f.N = 3, d.f.D = 5,  = 0.025


Testing two variances1

Testing Two Variances


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Assume we have the following data and wish to test the claim that the population variances are not equal.


Hypotheses

Hypotheses


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

The Sample Test Statistic


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Degrees of Freedom for Test of Two Variances

  • Degrees of freedom for the numerator =

    d.f. N = n1 - 1 = 9 - 1 = 8

  • Degrees of freedom for the denominator =

    d.f. D = n2 - 1 = 10 - 1 = 9


Critical values of f distribution

Critical Values of F Distribution

  • Use  = 0.05

  • For a two-tailed test , the area in the right tail of the distribution should be  /2 = 0.025.

  • With d.f. N = 8 and d.f. D = 9 the critical value of F is 4.10.


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Critical Value of F: Two-Tailed Test

Area = /2

F = 4.10


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Our Test Statistic Does not fall in the Critical Region

Area = /2

F = 1.108

F = 4.10


Conclusion

Conclusion

At 5% level of significance, we cannot reject the claim that the variances are the same.


P value approach

P Value Approach

  • Our sample test statistic was F = 1.108

  • Looking in the block of entries in table 8 where d.f. N = 8 and d.f. D = 9, we find entries ranging from 3.23 to 5.47 for  ranging from 0.050 to 0.010.

  • F = 1.108 is less than even the smallest of these results.


P value conclusion

P Value Conclusion

For a two-tailed test, double the area in the right tail.

Therefore P is greater than 0.100.


Analysis of variance

Analysis of Variance

A technique used to determine if there are differences among means for several groups.


One way analysis of variance

One Way Analysis of Variance

Groups are based on values of only one variable.


Anova

ANOVA

Analysis of Variance


Assumptions for anova

Assumptions for ANOVA

  • Each of k groups of measurements is from a normal population.

  • Each group is randomly selected and is independent of all other groups.

  • Variables from each group come from distributions with approximately the same standard deviation.


Purpose of anova

Purpose of ANOVA

To determine the existence (or nonexistence) of a statistically significant difference among the group means


Null hypothesis

Null Hypothesis

All the group populations are the same.

All sample groups come from the same population.


Alternate hypothesis

Alternate Hypothesis

Not all the group populations are equal.


Hypotheses1

Hypotheses

  • H0:1 = 2 = . . . = k

  • H1:At least two of the means 1, 2, . . . , k are not equal.


Steps in anova

Steps in ANOVA

  • Determine null and alternate hypotheses

  • Find SS TOT= the sum of the squares of entire collection of data

  • Find SS BET which measures variability between groups

  • Find SS W which measures variability within groups


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Steps in ANOVA

  • Find the variance estimates within groups: MS W

  • Find the variance estimates between groups: MS BET

  • Find the F ratio and complete the ANOVA test


Data for three groups

Data for three groups:


Sample sizes

Sample sizes

  • The sample sizes for the groups may be the same or different from one another.

  • In our example, each sample has four items.


Population means

Population Means

Let 1, 2, and 3 represent the population means of groups 1, 2, and 3.


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Hypotheses and Level of Significance

  • H0:1 = 2 = 3

  • H1:At least two of the means 1, 2, and 3 are not equal.

    Use  = 0.05.


Find ss tot the sum of the squares of entire collection of data

Find SS TOT= the sum of the squares of entire collection of data


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Find SS TOT= the sum of the squares of entire collection of data


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Find SS TOT= the sum of the squares of entire collection of data


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Find SS TOT= the sum of the squares of entire collection of data


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Squares and Sums


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Finding SS TOT


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Find SS TOT


Find ss bet

Find SS BET


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Finding SS BET


Variability within ith group

Variability Within ith Group


Variation within first group

Variation Within First Group


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Variation Within Second Group


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Variation Within Third Group


Variability within all groups

Variability Within All Groups

SS W = SS 1 + SS 2 + SS 3


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Finding SS W

SS W = SS 1 + SS 2 + SS 3=

4.75 + 2.00 + 2.75 =

9.50


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Note:

SS TOT = SS BET + SS W


We can check

We can check:

SS TOT = SS BET + SS W =

94 = 84.5 + 9.5


Degrees of freedom

Degrees of Freedom

  • Degrees of freedom between groups =

    d.f. BET = k - 1, where k is the number of groups.

  • Degrees of freedom within groups =

    d.f. W = N - k, where N is the total sample size.

  • d.f. BET + d.f. W = N - 1.


Variance estimate between groups

Variance Estimate Between Groups

Mean Square Between Groups =


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Variance Estimate Within Groups

Mean Square Within Groups =


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Mean Square Between Groups


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Mean Square Within Groups


Find the f ratio

Find the F Ratio


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Finding the F Ratio


Null hypothesis all the groups are samples from the same distribution

Null Hypothesis: All the groups are samples from the same distribution

  • If H0 is true MS BET and MS W would estimate the same quantity.

  • The F ratio should be approximately equal to one.


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Using Table 8, Appendix II

  • d.f. BET = number of groups - 1 = d.f. for numerator.

  • d.f. W = total sample size - number of groups = d.f. for denominator.

  • Rejection region is the right tail of the distribution.


Chapter eleven part 3 sections 11 4 11 5 chi square and f distributions

Using Table 8, Appendix II

  • d.f. BET = number of groups - 1 = d.f. for numerator = 2.

  • d.f. W = total sample size - number of groups = d.f. for denominator = 9.

  •  = 0.05.

  • Critical F value = 4.26


Conclusion1

Conclusion

  • Since our observed value of F (40.009) is greater than the critical F value (4.26) we reject the null hypothesis.

  • We conclude that not all of the means are equal.


P value approach1

P Value Approach

  • For d.f. BET = d.f. for numerator = 2 and d.f. W = d.f. for denominator = 9, our observed F value (42.009) exceeds even the largest critical F value.

  • Thus P < 0.01.

  • We conclude that we would reject the null hypothesis for any   0.01.


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