Topic 14 Notes and Examples - The Test of Independence (with solutions) PDF

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Notes, examples, and solutions to problems working with chi tests of indepence...

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Topic 14. The Test of Independence: See the relevant formulas in Chapter 12 on page F-7 of your formula booklet. Example 14.1: Students from a secondary school in Great Britain were classified by sex and whether they were overachievers or underachievers. Test at the .05 level to determine whether sex and achievement level are related.

Underachievers Boys Girls Total

26 8 34

Overachiever s 13 22 35

Total

39 30 69

H0: Sex and achievement are independent. HA: Sex and achievement are dependent. Category

Observed ( f i j)

Underachieving Boys Overachieving Boys Underachieving Girls Overachieving Girls Total

(39 )( 34 ) =19.2174 69 (39)(3 5) =19.7826 69 (30)(34) =14.7826 69 (30 )(35 ) =15.2174 69 69

26 13 8 22 69 2

Expected ( e i j)

2

χ =∑ ∑ 2

( f ij−e ij )

2 ( f i j−ei j ) /ei j

19.2174 =¿ 2.3939 2 ( 26−19.2174 ) /¿

( 13−19.7826 )2 /19.7826 =2.3255 ( 8−14.7826)2 /14.7826 =3.1120 ( 22−15.2174 )2 /15.2174 =3.0231 10.8545

2

=10.8545 e ij Rejection Region: Based on right tail of χ 2 distribution with df =( r−1 ) ( c−1) =( 2−1 ) (2−1 )=1 and α=.05 , using Table F-18, the rejection region consists of values to the right of 3.841. Test Statistic:

i=1 j=1

Conclusion: Since χ 2=10.8545 is within the rejection region, we would reject H0. The data indicate that sex and achievement are dependent. Additional Comment: Looking across the row for df =1 in Table F-18, since χ 2=10.8545 is to the right of 7.879, the area to the right of 10.8545 would be less than .005. Therefore, the p-value for this test would be less than .005. So, based on the p-value, we would also reject H0. Example 14.2: In a study to determine whether professional advancement among bank employees depends on their typical standard of dress, a random sample of 180 bank employees yielded the results shown in the table. What conclusion should we reach at the .01 significance level?

Professional Slow Average Fast Total

28 69 22 119

Casua l 15 33 13 61

Total 43 102 35 180

H0: Sex and achievement are independent. HA: Sex and achievement are dependent. Category

Observed ( f ij )

Expected ( e ij)

2 ( f ij−eij ) /e ij

28.4278=¿ 0.0064 (43 )(119) =28.4278 2 ( 28− 28.4278 ) /¿ 180 (43 )(61) =14.5722 15 ( 15 −14.5722)2 /14.5722=0.0126 180 (102)( 119 ) =67.4333 69 ( 69− 67.4333)2 /67.4333 =0.0364 180 (102)( 61) =34.5667 33 ( 33−34.5667 )2 /34.5667 =0.0710 180 (35 )(119) ( 22−23.1389 )2 /23.1389= 0.0561 =23.1389 22 180 (35 )(61 ) ( 13−11.8611)2 /11.8611=0.1094 =11.8611 13 180 180 180 0.2919 2 2 2 ( f −e ) Test Statistic: χ 2=∑ ∑ ij ij =0.2919 e ij i=1 j=1 Rejection Region: Based on right tail of χ 2 distribution with df =( r−1 ) ( c−1) =( 3−1 ) (2−1 )=2 and α=.0 1 , using Table F-18, the rejection region consists of values to the right of 9.210. Slow & Professional Slow & Casual Average & Professional Average & Casual Fast & Professional Fast & Casual Total

28

Conclusion: Since χ 2=0.2919 is outside the rejection region, we would not reject H0. The data do not indicate that standard of dress and advancement are dependent. Example 14.3: Does a taste for spicy food depend on a person’s sex? Food science researchers collected data from random samples of 40 men and 60 women, classifying them according to whether they liked or disliked a spicy food dish that was served to them. The data is summarized in the table at the right. Using the .05 significance level, conduct a hypothesis test to decide whether sex and liking spicy food are dependent or independent. H0: Sex and liking spicy food are independent.

Men Women Total Liked spicy food Disliked spicy food Total

32

28

60

8

32

40

40

60

100

HA: Sex and liking spicy food are dependent. Observed ( f ij )

Category Men who liked spicy food Women who liked spicy food Men who disliked spicy food Women who disliked spicy food Total

32 28 8 32 100 2

2

χ 2 =∑ ∑

Expected ( e ij) (60)(40) =2 4 100 (60)(60) =3 6 100 (40 )( 40 ) =1 6 100 (40)(60) =2 4 100 100

( f ij−e ij )

2 ( f ij−eij ) /e ij

24 =¿ 2.6667 2 ( 32−24 ) /¿

( 28−36) 2 /36=1.7778 ( 8−16)2 /16= 4.0000 ( 32−24) 2 /24= 2.6667 11.1112

2

=11.1112 e ij Rejection Region: Based on the right tail of a χ 2 distribution with df =( r−1 ) ( c−1) =( 2−1 ) (2−1 )=1 and α=.05 , using Table F-18, the rejection region consists of values to the right of 3.841. Test Statistic:

i=1 j=1

Conclusion: Since χ 2=11.1112 is within the rejection region, we would reject H0. The data indicate that sex and liking spicy food are dependent. Additional Comment: Looking across the row for df =1 in Table F-18, since χ 2=11.1112 is to the right of 7.879, the area to the right of 11.1112 would be less than .005. Therefore, the p-value for this test would be less than .005. So, based on the p-value, we would also reject H0....