Chapter 12, Problem 1RE

To determine

To test: The claim that is the wheel out of balance.

Answer to Problem 1RE

Solution: There is an inadequate indication to discard the null hypothesis.

Explanation of Solution

Given: The sample size is $n=500$.

Explanation: To test the hypothesis, the null and alternate hypotheses are defined in the following manner:

$\begin{array}{c}{H}_0:\text{The roulette wheel is balanced}\text{.}\\ H_1:\text{The wheel is not in balance}\text{.}\end{array}$

The observed frequencies $O_i$ are shown in the table below:

Outcome	Observed frequency $\left(O_i\right)$
Black	233
Red	237
Green	30
Total	500

The expected frequencies $E_i$ are obtained using the formula

$E_i=n{p}_i$

where n is the number of trials and $p_i$ is the probability of success in the $i^{th}$ trial.

In the given problem, $n=500$.

The expected frequencies $E_i$ corresponding to each outcome are shown below:

Outcome	Number of slots	$n{p}_i$	Expected frequency $\left(E_i\right)$
Black	18	$500\times \frac{18}{38}$	$236.84$
Red	18	$500\times \frac{18}{38}$	$236.84$
Green	2	$500\times \frac{2}{38}$	$26.32$
Total	38		500

The test statistic is calculated as follows:

$\begin{array}{c}{\chi }_0^2={\displaystyle \sum \frac{{\left(O_i-E_i\right)}^2}{E_I}}\\ =\frac{{\left(233-236.84\right)}^2}{236.84}+\frac{{\left(237-236.84\right)}^2}{236.84}+\frac{{\left(30-26.32\right)}^2}{26.32}\\ =0.577\end{array}$

where ${\chi }_0^2$ follows the chi-square distribution with $\left(k-1\right)$. The critical values are obtained using the Table VIII provided in the book.

$\begin{array}{c}k-1=3-1\\ =2\end{array}$

Therefore, the critical values are observed as follows:

$\begin{array}{c}{\chi }_{\alpha }^2={\chi }_{0.05}^2\\ ={\chi }_{0.05}^2\\ =5.991\end{array}$

Therefore, there is an inadequate indication to discard the null hypothesis because the value of test statistic $\left({\chi }_0^2\right)$ is less than the critical value $\left({\chi }_{0.05}^2\right)$.