A gambler wants to test whether the die she has is fair so she rolls it 300 time

Question

A gambler wants to test whether the die she has is fair so she rolls it 300 times and records the outcomes as meaning she rolled a 1 on the die 54 times, etc. Use alpha = 0.05 (a) A fair die has probability 1/6 for each of the six outcomes. Use a chi-squared test to test whether or not the probabilities are all equal. (b) An oblong die will have P(1) = P(6), P(2) = P(5), and P(3) = P(4), but all of the probabilities will not necessarily be the same. Use a chi-squared test to test whether the data is consistent with an oblong die.

Explanation / Answer

Q.3 (a) Null Hypothesis : H0: Fair die has the probability of 1/6 for each of the six outcomes.

Alternative Hypothesis : Ha : Fair die has not the probability of 1/6 for each of the six outcomes.

Observed Table and Expected Table

Now we should calculate value of chi- square

X2= (54-50)2 /50 + (71-50)2 /50 + (42-50)2 /50 + (45-50)2 /50 + (51-50)2 /50 + (37-50)2 /50

X2= 14.32

so for alpha = 0.05 and dF = 5

X2critical = 11.070

so here X2 > X2critical so we can reject the null hypothesis and claim that fair die has the probability of 1/6 for each of the outcome.

(b)

Null Hypothesis : H0: The given data probability distribution is consistent with oblong die.

Alternative Hypothesis : Ha : The given data probability distribution is not consistent with oblong die.

Observed Table and Expected Table

Here Expected value of Roll 1 and 6 will be the same and it will be the average of both data values.

so Here X2= 6.558

so for alpha = 0.05 and dF = (6-1) =5

so X2critical = 11.070

so here  X2 < X2critical so we cannot reject the null hypothesis and can say that the given data probability distribution is consistent with oblong die.

Roll Count (oi) Expected(Ei) 1 54 50 2 71 50 3 42 50 4 45 50 5 51 50 6 37 50

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