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 χ 2 (chi-squared).

(c)If the critical value is χ 2 (0.05) = 14.067 should you accept or reject the null hypothesis? Why or why not?

Answer 1

Solution:

Here, we have to use chi square test for goodness of fit.

Null hypothesis: H0: Data follows uniform distribution.

Alternative hypothesis: Ha: Data do not follow uniform distribution.

We assume/given level of significance = α = 0.05

Part a

We are given

N = 8

Degrees of freedom = df = N - 1 = 7

Part b

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

Calculation tables for test statistic are given as below:

No.	O	E	(O - E)^2/E
1	5	6.25	0.25
2	3	6.25	1.69
3	10	6.25	2.25
4	13	6.25	7.29
5	7	6.25	0.09
6	2	6.25	2.89
7	4	6.25	0.81
8	6	6.25	0.01
Total	50	50	15.28

Test Statistic = Chi square = ∑[(O – E)^2/E] = 15.28

χ2 statistic = 15.28

Part c

α = 0.05

Critical value = 14.06714043

(by using Chi square table or excel)

Test statistic value > Critical value

So, we reject the null hypothesis

There is not sufficient evidence to conclude that Data follows uniform distribution.

There is not sufficient evidence to conclude that humans can choose a number randomly.