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 $200$ spins of one of their American roulette wheels. The table 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 count for each color.

c. Calculate the value of the chi-square test statistic.

Short answer

Part (a)$$\begin{gathered} H_0:p_{red}=\frac{18}{38}=0.47,P_{black}=\frac{18}{38}=0.47,P_{green}=\frac{2}{38}=0.05 \\ {H}_a:Atleastoneofthe{p}_i’sisincorrect. \end{gathered}$$

Part (c) The chi-square test statistic $=4.04$

Part (b)

Step-by-step solution

Part (a) Step 1: Given information

Part (a) Step 2: Explanation

The null and alternative hypotheses:

$$\begin{gathered} H_0:p_{red}=\frac{18}{38}=0.47,P_{black}=\frac{18}{38}=0.47,P_{green}=\frac{2}{38}=0.05 \\ {H}_a:Atleastoneofthe{p}_i’sisincorrect. \end{gathered}$$

Part (b) Step 1: Explanation

$Expectedvalue=n\times frequency$

Where, $n=200$

Expected count will be:

Part (c) Step 1: Explanation

Using excel,