Question

If a dice is rolled one time, classical probability would indicate that the probability of a "two" should be 1/6. If the dice is rolled 60 times and comes up "two" only 9 times does that suguest that the dice is loaded? why or why not

Answer 1

A hypothesis related to the population proportion will be tested. In this case,
Theoretical proportion, π0 = 1/6,
number of observations, n = 60
Observed proportion, p = 9/n = 9/60

In the case where n>30, nπ0>5 and n(1-π0)>5, the normal distribution could be used as an approximation for the test of the null hypothesis, H0, where
H0 : the sample proportion, π = 1/6.

The sample standard error, σp
=sqrt(π0(1-π0)/n)
=sqrt((1/6)(5/6)/60)
=0.048
z=(p-π0)/σp
=(9/60-1/6)/0.048
=-0.347
critical z (α=0.05)=-1.645 (from the Normal distribution table).

Since -0.347 is greater than -1.645, the null hypothesis is not rejected at the 5% level of significance, i.e. there is no indication that the die is loaded.

Answer 2

Find the chance that if you toss a pair of dice, you get 6 for the sum.