Question

In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and , or critical value, and state the conclusion.

Testing a Slot Machine The author purchased a slot machine (Bally Model 809) and tested it by playing it 1197 times. There are 10 different categories of outcomes, including no win, win jackpot, win with three bells, and so on. When testing the claim that the observed outcomes agree with the expected frequencies, the author obtained a test statistic of\({\chi ^2} = 8.185\). Use a 0.05 significance level to test the claim that the actual outcomes agree with the expected frequencies. Does the slot machine appear to be functioning as expected?

Short answer

There is not enough evidence to support the statement that the observed frequencies differ from the expected frequency.

Thus, the slot machine is operating as expected.

Step-by-step solution

Given information

A slot machine is tested to examine whether the observed outcomes agree with the expected frequencies. The machine is run 1197 times to test the 10 different categories. The test statistic \({\chi ^2} = 8.185\).

State the hypotheses

Assume that the samples are randomly selected, where the data includes a frequency counts. Also, it is assumed that the expected frequency is lesser than 5.

The hypotheses for conducting the given test is as follows:

\({H_0}:{p_0} = {p_1} = ... = {p_{10}}\)

\({H_a}:\)at least one of the proportions measure is different from others.

Where,\({p_0},{p_1},...,{p_{10}}\)are the proportions corresponding to different categories.

The test is right-tailed.

Determine the test statistic

The value of the test statistic is given to be equal to 8.185.

Let k be the number of categories.

The degrees of freedom for computing the critical value are:

\(\begin{aligned}{c}df = k - 1\\ = 10 - 1\\ = 9\end{aligned}\)

Thus, the critical value of\({\chi ^2}\)corresponding to\(\alpha = 0.05\)and degrees of freedom equal to 9 is equal to 16.919.

The p-value is,

\(\begin{aligned}{c}p - value = P\left( {{\chi ^2} > 8.815} \right)\\ = 0.516\end{aligned}\)

Since the test statistic value is less than the critical value along with the p-value which is greater than 0.05, then the null hypothesis is failed to be rejected.

State the conclusion

There is not enough evidence to support the statement that the observed frequencies differ from the expected frequency.

Thus, the slot machine appears to be working as expected.