Question

a prize wheel at a carnival consists of five equal sectors of these colors: red, blue, green, yellow, and orange. if the wheel lands on yellow, the player wins a prize. after observing several spins, a customer suspects that the wheel is not fair. to substantiate this belief, the customer observes 100 consecutive spins of the prize wheel. of these spins, 18 land on red, 24 land on blue, 28 land on green, 12 land on yellow, and 18 land on orange. the customer would like to know if the data provide convincing evidence that the wheel is not fair. what is the value of the chi - square test statistic and p - value?
find the chi - square table here.
$\chi^{2}=7.60$, $p$-value is between $0.10$ and $0.15$
$\chi^{2}=7.60$, $p$-value is between $0.15$ and $0.20$
$\chi^{2}=8.73$, $p$-value is between $0.05$ and $0.10$
$\chi^{2}=8.73$, $p$-value is between $0.10$ and $0.15$

Explanation:

Step1: Calculate the expected frequency

Since there are 5 equal - sectors, the probability of landing on each color is \(p=\frac{1}{5} = 0.2\). For \(n = 100\) spins, the expected frequency \(E\) for each color is \(E=np=100\times0.2=20\).

Step2: Calculate the chi - square test statistic

The formula for the chi - square test statistic is \(\chi^{2}=\sum\frac{(O - E)^{2}}{E}\), where \(O\) is the observed frequency.

• For red: \(O = 18\), \(\frac{(18 - 20)^{2}}{20}=\frac{(- 2)^{2}}{20}=\frac{4}{20}=0.2\)
• For blue: \(O = 24\), \(\frac{(24 - 20)^{2}}{20}=\frac{4^{2}}{20}=\frac{16}{20}=0.8\)
• For green: \(O = 28\), \(\frac{(28 - 20)^{2}}{20}=\frac{8^{2}}{20}=\frac{64}{20}=3.2\)
• For yellow: \(O = 12\), \(\frac{(12 - 20)^{2}}{20}=\frac{(-8)^{2}}{20}=\frac{64}{20}=3.2\)
• For orange: \(O = 18\), \(\frac{(18 - 20)^{2}}{20}=\frac{(-2)^{2}}{20}=0.2\)

\(\chi^{2}=0.2 + 0.8+3.2 + 3.2+0.2=7.6\)

Step3: Determine the degrees of freedom

The degrees of freedom \(df=k - 1\), where \(k\) is the number of categories. Here \(k = 5\), so \(df=5 - 1=4\)

Step4: Find the P - value

Looking at the chi - square table with \(df = 4\):

• \(\chi^{2}_{0.15,4}=6.745\)
• \(\chi^{2}_{0.10,4}=7.779\)

Since \(6.745<7.6<7.779\), the \(P\) - value is between \(0.10\) and \(0.15\)

Answer:

\(\chi^{2}=7.60\), \(P\) - value is between \(0.10\) and \(0.15\) (the first option)