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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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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

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


Assumptions for Testing Two Variances

  • The two populations are independent

  • The two populations each have a normal probability distribution.


Notations Used:


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


Set Up Hypotheses


Set Up Hypotheses


Equivalent hypotheses may be stated about standard deviations.


Use the F Statistic


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.


An F Distribution


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

Given in Table 8 of Appendix II


Some Values of the F Distribution


Find critical value of F from Table 8 Appendix II

d.f.N = 3

d.f.D = 5

Right tail area =  = 0.025


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


Testing Two Variances


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


Hypotheses


The Sample Test Statistic


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

  • 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.


Critical Value of F: Two-Tailed Test

Area = /2

F = 4.10


Our Test Statistic Does not fall in the Critical Region

Area = /2

F = 1.108

F = 4.10


Conclusion

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


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

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

Therefore P is greater than 0.100.


Analysis of Variance

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


One Way Analysis of Variance

Groups are based on values of only one variable.


ANOVA

Analysis of Variance


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

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


Null Hypothesis

All the group populations are the same.

All sample groups come from the same population.


Alternate Hypothesis

Not all the group populations are equal.


Hypotheses

  • H0:1 = 2 = . . . = k

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


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


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:


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

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


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


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


Squares and Sums


Finding SS TOT


Find SS TOT


Find SS BET


Finding SS BET


Variability Within ith Group


Variation Within First Group


Variation Within Second Group


Variation Within Third Group


Variability Within All Groups

SS W = SS 1 + SS 2 + SS 3


Finding SS W

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

4.75 + 2.00 + 2.75 =

9.50


Note:

SS TOT = SS BET + SS W


We can check:

SS TOT = SS BET + SS W =

94 = 84.5 + 9.5


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

Mean Square Between Groups =


Variance Estimate Within Groups

Mean Square Within Groups =


Mean Square Between Groups


Mean Square Within Groups


Find the F Ratio


Finding the F Ratio


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.


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.


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


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 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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