STRAIGHT FROM THE BOOK

Roulette In the casino game of roulette there is a wheelwith 19 black slots, 19 red slots, and 2 green slots. In the game,a ball is rolled around a spinning wheel and it lands in one of theslots. It is assumed that each slot has the same probability ofgetting the ball. This results in the table of probabilitiesbelow.

Fair Table Probabilities

	  black Â	  red Â	  green Â
Probability Â	19/40	19/40	2/40

You watch the game at a particular table for 130 rounds and countthe number of black, red, and green results. Your observations aresummarized in the table below.

Outcomes (n = 130)

	  black Â	  red Â	  green Â
Counts Â	  48 Â	  73 Â	  9 Â

The Test: Test the claim that this roulette tableis not fair. That is, test the claim that the distribution ofcolors for all spins of this wheel does not fit the expecteddistribution from a fair table. Test this claim at the 0.01significance level.

(a) What is the null hypothesis for this test?

H₀: The probabilities are not all equal to1/3.H₀: p₁ = 19/40, p₂ =19/40, and p₃ =2/40.    H₀: p₁= p₂ = p₃ = 1/3H₀: Theprobabilities associated with this table do not fit thoseassociated with a fair table.

(b) The table below is used to calculate the test statistic.Complete the missing cells.
Round your answers to the same number of decimal places asother entries for that column.

		Observed	Assumed Â	Expected
i	Color	Frequency(Oi)	Probability(pi)	FrequencyEi	(Oi −Ei)2 Ei
1	black		0.475	61.75	3.062
2	red	73
3	green	9	0.050	6.50	0.962
Σ		n = 130			χ2 =

(c) What is the value for the degrees of freedom?
(d) What is the critical value of χ²?Use the answer found in theχ²-table or round to 3 decimalplaces.
t_α =

(e) What is the conclusion regarding the null hypothesis?

reject H₀

fail to rejectH₀    

(f) Choose the appropriate concluding statement.

We have proven that this table is fair.

The results of this sample suggest the table is notfair.    

There is not enough data to conclude that this table is notfair.