Q. 2 - The Practice of Statistics for AP

Question

Roulette Casinos are required to verify that their games operate as advertised. American roulette wheels have $38$ slots$18$ red, $18$ black, and $2$ green In one casino, managers record data from a random sample of  spins of one of their American roulette wheels. The one-way table below displays the results.

(a)  State appropriate hypotheses for testing whether these data give convincing evidence that the distribution of outcomes on this wheel is not what it should be.

(b) Calculate the expected counts for each color. Show your work.  

Step-by-Step Solution

Answer

(a) $H_0$There is no significant difference between expected and observed count of red, black and green slots in 200 trails.

$H_1$: There is significant difference between expected and observed count of red, black and green slots in 200 trails.

(b) The expected counts are listed below.

Part (a) Step 1: Given Information

Given in the question that, There are $38$ slots on an American roulette wheel. $2$ green, $18$ red, $18$ black Managers at one casino keep track of statistics from a random sample of American roulette wheel spins. The findings are shown in the one-way table below.

Part (a) Step 2: Explanation

Alternative Hypothesis: $H_1$: The hypothesis we're attempting to prove.

$H_0$: The null hypothesis is the inverse of the alternative hypothesis.

The hypotheses here would be

$H_0$: In $200$ rails, there is no statistically significant difference between the expected and observed count of red, black, and green slots.

Versus

$H_1$ : In $200$ trails, the expected and observed count of red, black, and green slots varies significantly.

Part (b) Step 1: Given Information

Expected count for each colour slot.

Part (b) Step 2: Explanation

When the roulette wheel is fair, we have an equal probability of landing on any of the 38 slots.

As a result, the odds of earning a red slot in 200 trials are $18/38\times 200=94.74.$

In the same way, the odds of getting black slot in 200 trials are$18/38\times 200=94.74$

The green slot costs$2/38\times 200=10.52.$

Expected frequencies or counts are what they're called.

These can be found in the table below.