Question

Assume that you suspect that a dice has been tampered with. You throw the dice 36 times and record the number of times each value on the dice occurs in the table below. Conduct a hypothesis test to see if this dice is considered to be a fair dice (good dice and, therefore, has not been tampered with) at an α=0.01

Table 1.

Observed frequency (O) of the number of dots of die

Number of dots	Observed frequency
1	9
2	5
3	7
4	4
5	3
6	8

2.   A large big-box retailer is accused of sex discrimination in promotion decisions. The personnel files show there were 3300 male employees among them 960 were promoted in the past 6 months. During the same time, there were 2700 female employees, and among them, 700 were promoted. Do the data substantiate the claim that this large big-box retailer discriminated against female workers in promotion decisions? Conduct a hypothesis test to evaluate if there is a relationship between promotion decisions and employee’s sex at an α=0.05.

Table 2.

Observed frequency (O) between promotion decision and employee’s sex

	Promoted	Not promoted	Row total
Male	960	2340	3300
Female	700	2000	2700
Column total	1660	4340	6000

Answer 1

(1)

H0: Null Hypothesis: This dice is considered to be a fair dice (good dice and, therefore, has not been tampered with) (Claim)

HA: Alternative Hypothesis: This dice is considered to be not a fair dice (bad dice and, therefore, has been tampered with)

Test Statistic ($\chi ^{2}$) is got asfollows:

Observed (O)	Expected (E)	(O - E)2/E
9	6	1.500
5	6	0.167
7	6	0.167
4	6	0.667
3	6	1.500
8	6	0.667
Total = $\chi ^{2}$ =		4.668

Degrees of Fredom =6 - 1 = 5

From Table, critical value of $\chi ^{2}$ = 15.086

Since calculated value of $\chi ^{2}$ = 4.668 is less than critical value of $\chi ^{2}$ = 15.086, the difference is not significant. Fail to reject the null hypothesis.

Conclusion:
The data support the claim that this dice is considered to be a fair dice (good dice and, therefore, has not been tampered with).

(2)
H0: Null Hypothesis: There is no relationship between promotion decisions and employee’s sex

HA: Alternative Hypothesis: There is a relationship between promotion decisions and employee’s sex (Claim)

Expected Frequencies are got as follows:

	Promoted	Not promoted	Row total
Male	1660 X3300/6000 = 913.00	4340 X3300/6000 = 2387.00	3300
Female	1660 X2700/6000 = 747.00	4340 X2700/6000 = 1953.00	2700
Column total	1660	4340	6000

Test Statistc ($\chi ^{2}$) is got as follows:

Observed (O)	Expected (E)	(O - E)2/E
960	913	2.42
230	2387	0.93
700	747	2.96
2000	1953	1.13
Total = $\chi ^{2}$ =		7.43

df = (r - 1) X(c - 1) = (2 - 1) X(2 - 1) = 1

By Technology, p - value = 0.0064

Since p - value = 0.0064 is less than α=0.05, the differenc is significant. Reject null hypothesis.

Conclusion:
The data support the claim that there is a relationship between promotion decisions and employee’s sex