Question

a six - sided die is rolled 120 times. fill in the expected frequency column. then, conduct a hypothesis test to determine if the die is fair. the data in table 11.34 are the result of the 120 rolls.

state chi - square test statistic.

a. 13.6
b. 14.1
c. 15
d. 12

Explanation:

Step1: Calculate expected frequency

For a fair die, probability of each face \( p=\frac{1}{6} \). Total rolls \( n = 120 \). Expected frequency \( E=np \). So \( E=\frac{120}{6}=20 \) for each face.

Step2: Calculate chi - square statistic

The formula for chi - square statistic is \( \chi^{2}=\sum\frac{(O - E)^{2}}{E} \), where \( O \) is the observed frequency.
For face 1: \( \frac{(15 - 20)^{2}}{20}=\frac{25}{20}=1.25 \)
For face 2: \( \frac{(29 - 20)^{2}}{20}=\frac{81}{20}=4.05 \)
For face 3: \( \frac{(16 - 20)^{2}}{20}=\frac{16}{20}=0.8 \)
For face 4: \( \frac{(15 - 20)^{2}}{20}=1.25 \)
For face 5: \( \frac{(30 - 20)^{2}}{20}=\frac{100}{20}=5 \)
For face 6: \( \frac{(15 - 20)^{2}}{20}=1.25 \)
Sum these values: \( 1.25+4.05 + 0.8+1.25+5+1.25=13.6 \)

Answer:

a. 13.6