Question

2. (25 P) A random number generator was used to generate a 100 numbers listed below. Perform x2 goodness of fit test to check whether the data distributed uniformly in the interval [0, 1] (a= 0.05, state the hypothesis first). 0,01 0,01 0,07 0,08 0,08 0,09 0,12 0,21 0,24 0,24 0,25 0,25 0,26 0,27 0,27 0,27 0,28 0,02 0,03 0,03 0,05 0,05 0,06 0,06 0,06 0,28 0,28 0,29 0,29 0,3 0,31 0,32 0,32 0,33 0,331 0,33 0,34 0,35 0,35 0,35 0,39 0,41 0,42 0,42 0,44 0,47 0,48 0,48 0,49 0,49 0,5 0,5 0,51 0,52 0,53 0,53 0,54 0,54 0,55 0,55 0,58 0,58 0,6 0,6 0,61 0,63 0,64 0,65 0,68 0,68 0,68 0,7 0,71 0,73 0,73 0,74 0,76 0,76 0,78 0,8 0,83 0,84 0,84 0,84 0,86 0,86 0,86 0,86 0,87 0,88 0,91 0,91 0,94 0,94 0,97 0,13 0,15 0,16 0,18 0,19

Answer 1

H0: The data is distributed to uniform distribution

H1: The data is not distributed to uniform distribution

Let the los be alpha = 0.05

From the given data,

Class Interval	Observed
LL	UL	freq (f)
0.01	0.1	14
0.11	0.2	6
0.21	0.3	15
0.31	0.4	11
0.41	0.5	11
0.51	0.6	12
0.61	0.7	8
0.71	0.8	7
0.81	0.9	11
0.91	1	5
	Total =	100

The expected frequencies can be obtained by using 100 / 10 = 10 for all class

	Observed	Expected
Digit	Freq (Oi)	Freq Ei	(Oi-Ei)^2 /Ei
0	14	10	1.6
1	6	10	1.6
2	15	10	2.5
3	11	10	0.1
4	11	10	0.1
5	12	10	0.4
6	8	10	0.4
7	7	10	0.9
8	11	10	0.1
9	5	10	2.5
Total:	100	100	10.2

Num Categories: 10
Degrees of freedom: 9

Test Statistic, X^2: 10.2000

Critical X^2: 16.91895

P-Value: 0.3345

since Chi-square value < Chi-square critical value and P-value > alpha 0.05 so we accept H0

Thus we conclude that there is sufficient evidence that the data is distributed to uniform distribution