Question

A dice is tossed 120 times with the following

results.

Number turned up 1 2 3 4 5 6 Total

Frequency 30 25 18 10 22 15 120

Test the hypothesis that the dice is un biased.

Answer 1

The term unbiased dice means that there is an equal chance of observing all the dice faces. If we have a total of 120 tosses the probability of observing each face is $\frac{1}{6}$ .Thus we expect to observe each face (120 $\cdot$  $\frac{1}{6}$ ) times .The dice is unbiased if face 1,2,3,4,5,6 are observed 20 times .

Null Hypothesis :The dice is unbiased

Alternative hypothesis :The dice is biased

X_0.05,5=11.07

Apply the chi square test

Expected for each dice face =$\frac{30+25+18+10+22+15}{6}$ =20

0 E (0-E)² (O-E)²/E

30 20 100 5

25 20 25 1.25

18 2O 4 0.2

10 20 100 5.0

22 20 4 0.2

15 20 25 1.25

$\sum\frac{((0-E)^2}{E}$ =12.9

Chi critical value =11.07

Since the test statistics(12.9) is greater than critical value (11.07) we reject the null hypothesis. There is sufficient evidence to conclude that the dice is biased.