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8/11/2019 Anova Soalan

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4.5 Calculating ANOVA by hand

Example: A researcher was interested in studying the effects of three different

text book on mathematics achievement. To investigate the effects, the three

different books were used in three different school which had equal

demographical characteristics. The three school employed the same teaching

methods. At the end of the program, a mathematics test was administered to the

students. Five scores from each school were randomly selected and the scores

are as follows.

Saya berminat untuk mengkaji kesan tiga buah buku teks yang berbeza terhadap

pencapaian matematik. Untuk menyiasat kesan-kesan, ketiga-tiga buku yang

berbeza telah digunakan dalam tiga sekolah yang berbeza yang mempunyai

ciri-ciri demografi yang sama. Tiga sekolah yang bekerja kaedah pengajaran

sama. Pada akhir program ini, satu ujian matematik telah diberikan kepada

pelajar. Lima markah dari setiap sekolah telah dipilih secara rawak dan skor

adalah seperti berikut.

Text Book A Text Book B Text Book C

54 53 49

49 56 53

52 57 4755 51 50

48 59 54

With = .05, test if the means of the three populations are equal.

1. State the independent variable and the dependent variable in this study

2. State the assumptions for using a one-way ANOVA

3. State the null hypothesis and the alternative hypothesis

4. Compute SSB, SSw and SST

5. Compute the between and within samples variances

6. Indicate the value of Fcritical.

7. Compute the F value

8. Create and ANOVA table and fill in the above information

9. Describe the conclusion


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Solution:

Text Book A Text Book B Text Book C

54 53 49

49 56 53

52 57 47

55 51 5048 59 54

T1= 258 T2= 276 T3= 253

X21= 13350 X

22= 15276 X

23= 12835

n1 = 5 n2 = 5 n3 = 5

1= 51.6 2= 55.2 3= 50.6

1) Independent variable : Text book with three different text books

Dependent variable : scores of mathematics achievement

2) The assumption using one-way ANOVA:1. The distribution of the populations are normal,

2. The variances of the populations are equal

3. Scores are independent

4. Samples are independent

5. Samples are random

3) Null Hypothesis, H0= (the three group mean are equal)

Alternative Hyphotesis, Ha : ( at least one of the means are

unequal)

4) a) Sum of Squares Between Group (SSB)

SSB=

()

()

SSB =()

()

()

()

= 58.5333

b) Sum of Square Within Groups (SSw)

SSw= -

=

= 41,461 -()

()

()

= 111.2


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c) Sum of Squares Total (SST)

SST= SSB+ SSw = 58.5333 + 111.2 = 169.7333

5) Between Group Variance

MSB=

Within Group Variance

MSw=

=

= 9.2667

6) The value of Fcritical

Fcritical= F (0.05,2,12)= 3.89

Decision Rules: Reject Hoif F> 3.89

7) The value of FF =

=

8) One-Way ANOVA Table

Sources of Variation Sum ofSquares(SS)

DegreesofFreedom(df)

MeanSquare(MS)

TestStatisticValue (F)

F critical

Between 2 58.5333 29.2667

3.16 3.89

Within 12 112.2000 9.2667

Total 14 169.7333

9) Conclusions

F = 3.16, Fcritical= 3.89. Therefore we fail to reject the Ho. The data indicate

that the means

of populations are equal ( F(2,12)= 3.16, = 0.05). The differences

of the three sample means are simply due to sampling errors.

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