Staples: A Chain Of Large Office Supply Stores
Staples A Chain Of Large Office Supply Stores Sells A Line Of Deskto
Staples, a chain of large office supply stores, sells a line of desktop and laptop computers. Company executives want to determine whether the demands for these two types of computers are dependent on one another. The demand levels are categorized as Low, Medium-Low, Medium-High, or High. Data based on 200 days of operation is provided to analyze this relationship. The question asks whether executives can conclude that demand for desktops and laptops are independent or dependent at the 5% significance level, including calculation of the test statistic, critical value, and conclusion.
Paper For Above instruction
The analysis of dependency between desktop and laptop computer demand types hinges on hypothesis testing, specifically using the Chi-Square Test of Independence. This statistical test evaluates whether two categorical variables, in this case demand levels for desktops and laptops, are independent or associated. Given the data collected over 200 days, we can construct a contingency table to perform this assessment.
Constructing the Contingency Table
Assuming that the provided data is formatted into a matrix with demand levels for desktops as rows and demand levels for laptops as columns, the table might look like this:
| | Laptop Low | Med-Low | Med-High | High | Total |
|--------------|--------------|---------|----------|-------|--------|
| Desktop Low | a | b | c | d | R1 |
| Desktop Med-Low | e | f | g | h | R2 |
| Desktop Med-High | i | j | k | l | R3 |
| Desktop High | m | n | o | p | R4 |
| Total | C1 | C2 | C3 | C4 | N=200 |
The actual frequencies for each cell (a through p) are derived from the collected data.
Formulating Hypotheses
- Null hypothesis \(H_0\): Demand for desktops and laptops are independent.
- Alternative hypothesis \(H_1\): Demand for desktops and laptops are dependent.
Calculating the Test Statistic
The Chi-Square statistic is computed as:
\[
\chi^2 = \sum \frac{(O - E)^2}{E}
\]
where O represents observed frequencies from the data, and E represents expected frequencies assuming independence, calculated as:
\[
E_{ij} = \frac{(Row\,Total_i) \times (Column\,Total_j)}{Grand\,Total}
\]
Once the expected frequencies are obtained, the summation over all cells produces the χ² value.
Determining the Critical Value
The degrees of freedom for this test are:
\[
df = (r - 1) \times (c - 1)
\]
where r is the number of rows (demand levels for desktops), and c is the number of columns (demand levels for laptops). For four demand levels each, \(df = (4-1) \times (4-1) = 9\).
The critical value at the 0.05 significance level for \(df=9\) can be obtained from the Chi-Square distribution table, approximately 16.919.
Conclusion
If the computed χ² exceeds 16.919, we reject \(H_0\), concluding that demand levels are dependent. Conversely, if χ² is less than this critical value, we fail to reject \(H_0\), supporting the independence hypothesis.
Application to the Present Data
Suppose the calculated χ² value from the data is, say, 23.5, which exceeds 16.919. This indicates significant dependence between desktop and laptop demand levels at the 5% level of significance. Conversely, if the χ² value is, for example, 14.2, which is less than 16.919, then we have insufficient evidence to suggest dependence, and the demands are likely independent.
Final Answer:
Assuming the analysis yields a test statistic greater than the critical value, Staples can conclude at the 0.05 significance level that the demands for desktops and laptops are dependent. Conversely, if the test statistic is lower, the conclusion would be that demands are independent.
References
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