Question

The chief of security for the Mall of the Dakotas directed a study of theft. He selected a sample of 140 boxes that had been tampered with and ascertained that, for 60 of the boxes, the missing pants, shoes, and so on were attributed to shoplifting. For 25 boxes, employees had stolen the goods, and for the remaining 55 boxes, he blamed poor inventory control. In his report to the mall management, can he say that shoplifting is twice as likely to be the cause of the loss as compared with either employee theft or poor inventory control and that employee theft and poor inventory control are equally likely? Use the 0.02 significance level.

H₀: The proportions are as stated.      H₁: The proportions are not as stated.

State the decision rule using 0.02 significance level. (Round your answer to 3 decimal places.)

Compute the value of chi-square. (Round your answer to 3 decimal places.)

What is your decision regarding H₀?

Answer 1

Answer: The chief of security for the Mall of the Dakotas directed a study of theft. He selected a sample of 140 boxes that had been tampered with and ascertained that, for 60 of the boxes, the missing pants, shoes, and so on were attributed to shoplifting. For 25 boxes, employees had stolen the goods, and for the remaining 55 boxes, he blamed poor inventory control.

Solution:

In his report to the mall management, can he say that shoplifting is twice as likely to be the cause of the loss as compared with either employee theft or poor inventory control and that employee theft and poor inventory control are equally likely?

The hypothesis test:

H0: The proportions are as stated.

p1 = 0.50, p2 = 0.25, P3 = 0.25

H1: The proportions are not as stated.

Atleast one proportion are not as stated.

Test statistic χ2 = (observed value - expected value)^2 / expected value

	Observed (O)	Expected (E)	(O-E)^2/E
	60	70	1.4286
	25	35	2.8571
	55	35	0.0000
Total	140	140	4.2857

Test statistic χ2 = 4.286

Critical value of χ2:

df = (k-1) = 3-1 = 2

Critical value of χ2 = 9.210

Since, test statistic χ2 (4.286) < χ2 critical (9.210)

Do not reject the null hypothesis Ho.

Decision:

Do not reject Ho. There is sufficient evidence to support the claim that shoplifting is twice as likely to be the cause of the loss as compared with either employee theft or poor inventory control and that employee theft and poor inventory control are equally likely at 0.02 significance level.