Question

Below are 50 random numbers taken from a random number generator.

0.534	0.401	0.401	0.445	0.445
0.125	0.094	0.094	0.104	0.104
0.345	0.259	0.259	0.288	0.288
0.785	0.589	0.589	0.654	0.654
0.975	0.731	0.731	0.813	0.813
0.834	0.626	0.626	0.695	0.695
0.683	0.512	0.512	0.569	0.569
0.322	0.242	0.242	0.268	0.268
0.526	0.395	0.395	0.438	0.438
0.234	0.176	0.177	0.198	0.196

a) Using 5 class intervals, determine the computed chi-square value. Answer in 1 decimal place.   

b) What is the critical value of chi-square at 0.05 level of significance. Answer in 2 decimal places.

c) Are the random numbers uniform? Answer YES or NO.

Answer 1

Before we start working on the data, we convert random numbers into integers by multiplying each one

by 1000. ( For better readability of data )

We get following data;

534	401	401	445	445
125	94	94	104	104
345	259	259	288	288
785	589	589	654	654
975	731	731	813	813
834	626	626	695	695
683	512	512	569	569
322	242	242	268	268
526	395	395	438	438
234	176	177	198	196

a)

Step-1 : Construction of Frequency table :

Minimum data value = 94

Maximum data value = 975

So, we create 5 class intervals as below; Minimum value should be covered in the first class and intervals should multiples of 10.

Class

0-100

100-350

350-600

600-850

850-1100

By using Tally mark for each above class intervals , we get the following Frequency table;

Class	Frequency
0-100	2
100-350	18
350-600	16
600-850	13
850-1100	1
Total	50

Step-2 : Calculation Chi-square value:

We will use the following Chi-square formula :

where E = expected values = 50/5 = 10 for each class interval. ( ie: Expected value is the average )

Class	Observed values (O)	Expected value (E)	( O-E)^2	[ ( O-E)^2] / E
0-100	2	10	64	6.4
100-350	18	10	64	6.4
350-600	16	10	36	3.6
600-850	13	10	9	0.9
850-1100	1	10	81	8.1
Total	50	50	Chi-square==>	25.4

b)

Critical value of Chi-square for degrees of freedom = (5-1) = 4 and alpha = 0.05.

Critical value of Chi-square = 9.49 ( Refer below table )

c) Answer : No

Explanation:

Since Chi-square value=25.4 > Chi-square critical=9.49,

It implies that Reject the null hypothesis that "Random numbers are Uniform".

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