Question

In each of Exercises 12.24-12.33, apply the chi-square goodness-of-fit test, using either the critical-value approach or the$P-$value approach, to perform the required hypothesis test.
An American roulette wheel contains $18$red numbers, $18$black numbers, and $2$green numbers. The following table shows the frequency with which the ball landed on each color in $200$trials.

At the $5\%$significance level, do the data suggest that the wheel is out of balance?

Short answer

The data does not provide enough evidence to indicate that the wheel is out of balance.

Step-by-step solution

Step 1. Given Information

A table with the distribution of trials:

Step 2. Hypotheses and calculating goodness of fit

The null and alternate hypotheses are:
$H_0:$The data does not provide enough evidence to indicate that the wheel is out of balance.
$H_a:$The data provides enough evidence to indicate that the wheel is out of balance.
Calculating the goodness of fit

${\chi }^2=\sum \frac{(O-E{)}^2}{E}$
$=1.0622$
The degrees of freedom of the given data is
$df=k-1$
$$\begin{gathered} =3-1 \\ =2 \end{gathered}$$
The critical value is ${\chi }_{0.05}^{2}2df$is $5.991$
The test statistic is ${\chi }^2=1.0622$
${\chi }^2=1.0622<{\chi }_{0.05}^2=5.991$as it does not fall in the rejection region.