Question

Can humans choose a number randomly? Fifty adults were asked to chose a random number from one to eight. Use α = 0.05 as the level of significance. The table below summarizes the results.
Number 1 2 3 4 5 6 7 8 Frequency 5 3 10 13 7 2 4 6

(a) Calculate the degrees of freedom

(b)Calculate the test statistic x^2

(c)If the critical value is χ2 0.05 =14.067 should you accept or reject the null hypothesis?

Answer 1

(a) Calculate the degrees of freedom

As n = 8

the df = n - 1 = 8 - 1 = 7

df = 7

(b)Calculate the test statistic x^2

Event	Observed Frequency(O)	Expected Probability	Expected Frequency ( E)	( O-E)^2	( O-E)^2/E
1	5	0.125	6.25	1.5625	0.25
2	3	0.125	6.25	10.5625	1.69
3	10	0.125	6.25	14.0625	2.25
4	13	0.125	6.25	45.5625	7.29
5	7	0.125	6.25	0.5625	0.09
6	2	0.125	6.25	18.0625	2.89
7	4	0.125	6.25	5.0625	0.81
8	6	0.125	6.25	0.0625	0.01
Sum	50	1	50	-	15.28

$\chi^2$ = 15.28

(c)If the critical value is χ2 0.05 =14.067 should you accept or reject the null hypothesis?

Null hypothesis: The probability are equal

Alternate hypothesis: At least one probability is different

We reject the null hypothesis

If $\chi^2$ > χ2 0.05 ( 15.28 > 14.067 )

hence the null hypothesis is rejected