Is this Roulette Wheel Fair? Statistical Analysis and Casino Manager's Claim

[Ackbach]

My apologies for not posting at all on time! I completely spaced it. I can claim a lot of things going on at home (just sold house). Anyway, here you go. This is an easier one. You have until Tuesday to do it.

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An American roulette wheel has 18 red slots among its 38 slots. To test if a particular roulette wheel is fair, you spin the wheel 50 times and the ball lands in a red slot 31 times. The resulting P-value is 0.0384. Are the results statistically significant at the $\alpha=0.05$ level? Explain. What conclusion would you make? The casino manager uses your data to produce a $99\%$ confidence interval for $p$ and gets $(0.44,0.80)$. He says that this interval provides convincing evidence that the wheel is fair. How do you respond?

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[Ackbach]

No one answered this POT"W". You can see my solution below.

Spoiler

By definition, the P-value is the probability, given that the null hypothesis ($p=9/19=p_0=0.4737$) is true, of getting a $\hat{p}$ value as extreme as what we actually got ($\hat{p}=31/50=0.62>p_0$). Since the P-value is below the significance level $\alpha$, we would conclude that this sample data provides convincing evidence to reject the null hypothesis, and is in favor of the alternative hypothesis $p\not= 9/19$.

The confidence interval given is suspect because, although $p_0$ is in the interval, it is on the low end of the interval, and the casino owner chose a somewhat unusually high confidence level (thus requiring a larger interval). In fact, let us compute a more standard $95\%$ C.I. as follows:

\begin{align*}
\hat{p}&=0.62 \\
\hat{q}&=0.38 \\
\sigma_{\hat{p}}&=\sqrt{\frac{\hat{p} \hat{q}}{n}}=\sqrt{\frac{0.62\times 0.38}{50}}=0.0686 \\
\text{C.I.}&=(\hat{p}-2\sigma_{\hat{p}},\hat{p}+2\sigma_{\hat{p}})
=(0.4828,0.7572).
\end{align*}
We used the critical multiplier of $2$ from the Empirical Rule.
And here, we see that $p_0\not\in\text{C.I.}$, thus refuting the casino owner's claim.