Question

A computer random number generator was used to generate 950 random digits from 0 to 9. The observed frequencies of the digits are given in the table below.

0	1	2	3	4	5	6	7	8	9
88	82	97	84	87	87	95	93	90	147

Using a 0.05significance level, test the claim that all of the digits are equally likely.


(a) Find the rejection region.
Reject H0 if χ2>

(b) Find the test statistic. (Round your final answer to 2 decimal places.)
χ2=

(c) Do these data provide significant evidence that the the digits are not equally likely? (Type: Yes or No )

Answer 1

Here we are to test that all the digits are equally likely or not.

Hence we compute the probabilities of the given digits and then see whether the distribution matches with uniform distribution or not.

Digit	0	1	2	3	4	5	6	7	8	9	Total
Frequency	88	82	97	84	87	87	95	93	90	147	950

(a)

The rejection region is given as:

Reject H0 when

The critical value of is obtained from the Biometrika table as 16.919

(b)

For the data to be equally likely, the probability of the 10 digits to occur must be 1/10=0.1

So the expected frequency for all the digits must be 0.1*950=95

Now we make a continuation of the table to compute the test statistics.

Digits	Observed Frequency(O)	Expected Frequency(E)
0	88	95	0.515789
1	82	95	1.778947
2	97	95	0.042105
3	84	95	1.273684
4	87	95	0.673684
5	87	95	0.673684
6	95	95	0
7	93	95	0.042105
8	90	95	0.263158
9	147	95	28.46316
Total	950	950	33.72632

Hence the test statistic is =33.72632

(c)

As , we reject the null hypothesis at 5% level of significance and hence conclude that the digits are not equally likely at 5% level of significance.

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