Question

An Edge in Roulette? Of the $38$numbers on an American roulette wheel, $18$are red, $18$are black, and $2$are green. If the wheel is balanced, the probability of the ball landing on red is $\frac{18}{38}=0.474$. A gambler has been studying a roulette wheel. If the wheel is out of balance, he can improve his odds of winning. The gambler observes $200$spins of the wheel and finds that the ball lands on red $93$times, At the$10\%$ significance level, do the data provide sufficient evidence to conclude that the ball is not landing on red the correct percentage of the time for a balanced wheel?

Short answer

At the $5\%$level, the test results are not statistically significant.

As a result, the data is insufficient to infer that the ball is not falling on red at the correct percentage of the time for a balanced wheel.

Step-by-step solution

Given information

The significant level is $10\%$

The total number of spins is $n=200$

The probability of the ball landing on red is $p_0=0.474$
$x=93$

Explanation

Form the given values

Calculate the value of $n{p}_0$

$n{p}_0=\left(200\right)(0.474)$

$=94.8$

Calculate the value of $n\left(1-p_0\right)$

$n\left(1-p_0\right)=\left(200\right)(1-0.474)$

$=105.2$

Both the values are greater than $5$.

So one proportion $z-$test is appropriate to use.

The null hypothesis is

$H_0:p_0=0.474$

The alternate hypothesis is

$H_0:p_0\ne 0.474$

Calculate the sample proportion

$\hat{p}=\frac{x}{n}$

$\hat{p}=\frac{93}{200}$

$=0.465$

Write the expression for $z$

$z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}$

The $z$value is calculated as

$z=\frac{0.465-0.474}{\sqrt{\frac{0.474(1-0.474)}{200}}}$

$=\frac{-0.009}{0.035}$

$=-0.255$

$\alpha =0.10$

The tail is two-tailed

When $\alpha =0.05$, the critical value from the standard table is

$z_{0.05}=\pm 1.645$

The test statistic is within acceptable bounds. As a result, the hypothesis $H_0$is not ruled out.

At the$5\%$ level, the test results are not statistically significant.

As a result, the data is insufficient to infer that the ball is not falling on red at the correct percentage of the time for a balanced wheel.