Question

In a four-digit lottery, each of the four digits is supposed to have the same probability of occurrence. The table shows the frequency of occurrence of each digit for 83 consecutive daily four-digit drawings. Digit Frequency 0 26 1 32 2 27 3 40 4 31 5 31 6 33 7 44 8 33 9 35 Total 332 Click here for the Excel Data File

(a) The hypothesis for the given issue is H0: The digits come from a uniform population. No Yes

(b) Calculate the chi-square test statistic, degrees of freedom, and the p-value. (Round your test statistic value to 2 decimal places and the p-value to 4 decimal places.) Test statistic d.f. p-value

(c) Find the critical value of the chi-square for α = .01. (Round your answer to 2 decimal places.) Critical value

(d) At α = .01, can you reject the hypothesis that the digits are from a uniform population? Yes No

Answer 1

Hypothesis :

H0:The digits come from a uniform population.

Vs

H1:The digits does not come from a uniform population.

Expected Feq=332/10=33.2 times

Test Statistics:

=8.0602

Calculation :

Digit	Original Frq(O)	Expected Frq (E)	O-E	(O-E)2	(O-E)2/E
1	26	33.2	-7.2	51.84	1.5614
2	32	33.2	-1.2	1.44	0.043373
3	27	33.2	-6.2	38.44	1.157
4	40	33.2	6.8	46.24	1.39277
5	31	33.2	-2.2	4.84	0.1457
6	31	33.2	-2.2	4.84	0.1457
7	33	33.2	-0.2	0.04	0.001
8	44	33.2	10.8	116.64	3.5132
9	33	33.2	-0.2	0.04	0.001
10	35	33.2	1.8	3.24	0.09759
				Total	8.0602

P value : Is the vaue of the lowets level of singnificance at which we could reject the Null hypothesis H0

P value =CHISQ.TEST(observed array,expected array) Excel commond

P value =0.5081 >0.01

Critical value

is the value appering in the table for the specifies statistical test with what computed null hypothesis can be rejected

Critical value with right tailed with 1% LOS is given by  21.666 (from chi squre table)

Conclusion and decision:

Test statistics is less than Critical value Fail to reject H0 that digit are from the uniform population

(a) The hypothesis for the given issue is H0: The digits come from a uniform population.=Yes

b) Calculate the chi-square test statistic =8.0602, degrees of freedom=9, and the p-value =0.5081

(c) Find the critical value of the chi-square for α = .01 =21.666

(d) At α = .01, can you reject the hypothesis that the digits are from a uniform population = No