Chapter 9, Problem 9.64RE

a)

To determine

To find the null and alternative hypothesis.

Answer to Problem 9.64RE

  $H_0:p=\frac{18}{38}$

  $H_a:p\ne \frac{18}{38}$

Explanation of Solution

Given:

Number of possible outcomes = 38

Number of favorable outcomes = 18

Claim: The proportion if spins that result in red.

As per given information, the proportion p is true proportion of spins that result red.

Therefore, using formula of probability,

  $\begin{array}{l}Probability=\frac{Numberofpossibleoutcomes}{Numberoffavorableoutcomes}\\ Probability=\frac{18}{38}=0.4737\end{array}$

Therefore, null and alternative hypotheses:

  $H_0:p=\frac{18}{38}$

  $H_a:p\ne \frac{18}{38}$

b)

To determine

To perform hypothesis test.

Answer to Problem 9.64RE

There is sufficient evidence to reject the claim that the roulette wheel is fair.

Explanation of Solution

Given:

n = 50

  $x$ = 31

Formula:

Sample proportion:

  $\widehat{p}=\frac{x}{n}$

Test statistic

  $z=\frac{\widehat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}$

Calculation:

  $\widehat{p}=\frac{31}{50}=0.62$

First need to find test statistic:

  $z=\frac{0.62-0.4737}{\sqrt{\frac{0.4737(1-0.4737)}{50}}}=2.07$

Therefore,

  $2\times P(Z<2.07)=2\times 0.0192=0.0384$

Using excel formula, =NORMSDIST(2.07)

Therefore, p-value = 0.0384

Decision: The p-value <0.05, reject H0.

Conclusion: There is sufficient evidence to reject the claim that the roulette wheel is fair.

c)

To determine

To explain result using confidence interval

Answer to Problem 9.64RE

The casino manager’s claim is false.

Explanation of Solution

Given:

The 99% confidence interval is (0.44,0.80)

The claim should be like, there is no convincing evidence that the wheel is not fair. That is, casino manager’s claim is false. We always have evidence in favor of alternative hypothesis.