The Oxnard Retailers Anti-Theft Alliance (ORATA) published a study that claimed the causes of disappearance of inventory in retail stores were 30 percent shoplifting, 50 percent employee theft, and 20 percent faulty paperwork. The manager of the Melodic Kortholt Outlet performed an audit of the disappearance of 80 items and found the frequencies shown below. She would like to know if her store's experience follows the same pattern as other retailers. $$\begin{array} { | l | c | c | c | }
\hline \text { Reason } & \text { Shoplifting } & \text { Employee Theft } & \text { Poor Paperwork } \
\hline \text { Frequency } & 32 & 38 & 10 \
\hline
\end{array}$$ Using ? = .05, what can you conclude from your analysis?
A)The store's pattern is clearly significantly different from the published data.
B)The store's pattern is almost, but not quite, significantly different from the published data.
C)The store's pattern is very close to the published data.
D)We can form no conclusion because Cochran's Rule is violated.

Question

The Oxnard Retailers Anti-Theft Alliance (ORATA) published a study that claimed the causes of disappearance of inventory in retail stores were 30 percent shoplifting, 50 percent employee theft, and 20 percent faulty paperwork. The manager of the Melodic Kortholt Outlet performed an audit of the disappearance of 80 items and found the frequencies shown below. She would like to know if her store's experience follows the same pattern as other retailers. $$\begin{array} { | l | c | c | c | } 
\hline \text { Reason } & \text { Shoplifting } & \text { Employee Theft } & \text { Poor Paperwork } \
\hline \text { Frequency } & 32 & 38 & 10 \
\hline
\end{array}$$ Using ? = .05, what can you conclude from your analysis?
A)The store's pattern is clearly significantly different from the published data.
B)The store's pattern is almost, but not quite, significantly different from the published data.
C)The store's pattern is very close to the published data.
D)We can form no conclusion because Cochran's Rule is violated.

Quizplus · Accepted Answer

To test whether the store's pattern matches the published data, we will use a goodness-of-fit chi-square test. 
The expected frequencies for each category can be calculated based on the percentages given in the study: Shoplifting = 0.3*80 = 24, Employee Theft = 0.5*80 = 40, Poor Paperwork = 0.2*80 = 16
Using these expected frequencies and the observed frequencies given in the table, we can calculate the chi-square test statistic:

$\chi^2 = \frac{(32-24)^2}{24} + \frac{(38-40)^2}{40} + \frac{(10-16)^2}{16} = 2.17$

Using a chi-square distribution table with 2 degrees of freedom (3 categories - 1), and a significance level of 0.05, we find the critical value to be 5.99. Since our calculated chi-square value (2.17) is less than the critical value (5.99), we fail to reject the null hypothesis. Therefore, we cannot say that the store's pattern is significantly different from the published data, but we can say that the store's pattern is almost, but not quite, significantly different from the published data.

The Oxnard Retailers Anti-Theft Alliance (ORATA) Published a Study That