Question

Choosing Lottery Numbers: In the Super-Mega lottery there are 50 numbers (1 to 50), a player chooses ten different numbers and hopes that these get drawn. If the player's numbers get drawn, he/she wins an obscene amount of money. The table below displays the frequency with which classes of numbers are chosen (not drawn). These numbers came from a sample of 180 chosen numbers.

Chosen Numbers (n = 180)

	1 to 10	11 to 20	21 to 30	31 to 40	41 to 50
Count	42	54	27	34	23

The Test: Test the claim that all chosen numbers are not evenly distributed across the five classes. Test this claim at the 0.01 significance level.

(a) The table below is used to calculate the test statistic. Complete the missing cells.
Round your answers to the same number of decimal places as other entries for that column.

	Chosen	Observed	Assumed	Expected
i	Numbers	Frequency (Oi)	Probability (pi)	Frequency Ei	(Oi − Ei)2 Ei
1	1 to 10	1	0.2	36.0	1.000
2	11 to 20	54	2	36.0	9.000
3	21 to 30	27	0.2	3	2.250
4	31 to 40	34	0.2	36.0	4
5	41 to 50	23	0.2	36.0	4.694
Σ		n = 180			χ2 = 5

(b) What is the value for the degrees of freedom? 6
(c) What is the critical value of

χ²

? Use the answer found in the

χ²

-table or round to 3 decimal places.
t_α = 7

(d) What is the conclusion regarding the null hypothesis?

reject H₀ fail to reject H₀    

(e) Choose the appropriate concluding statement.

We have proven that all chosen numbers are evenly distributed across the five classes. The data supports the claim that all chosen numbers are not evenly distributed across the five classes.      There is not enough data to support the claim that all chosen numbers are not evenly distributed across the five classes.

Answer 1

a). the completed table:-

Chosen	Observed	Assumed	Expected
i	Numbers	Frequency (Oi)	Probability (pi)	Frequency Ei	(Oi − Ei)2 Ei
1	1 to 10	42	0.2	36.0	1.000
2	11 to 20	54	1/5 = 0.2	36.0	9.000
3	21 to 30	27	0.2	(180*0.2) = 36	2.250
4	31 to 40	34	0.2	36.0
5	41 to 50	23	0.2	36.0	4.694
Σ		n = 180			χ2 = 17.055

b). degrees of freedom:-

=(5-1) = 4

c). critical value of chi square = 13.277

( from chi square distribution table, for df= 4 , alpha=0.01)

d).the conclusion regarding the null hypothesis:- reject H₀

[because here, chi square calculated = 17.055 > chi square critical = 13.277, so we reject the null hypothesis.]

e).the appropriate concluding statement be:-

The data supports the claim that all chosen numbers are not evenly distributed across the five classes.   

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