Question

Staples, a chain of large office supply stores, sells a line of desktop and laptop computers. Company executives want to know whether the demands for these two types of computers are dependent on one another. Each day's demand for each type of computers is categorized as Low, Medium-Low, Medium-High, or High. The data shown in the table below is based on 205 days of operation. Based on these data, can ? Test at the 5% level of significance.
low med-low med-high high
low 4 15 14 3 36
laptops med-low 6 18 18 23 65
med-high 13 17 10 17 57
high 7 15 15 10 47
30 65 57 53 205

Required:
a. What is the test value for this hypothesis test?
b. What is the conclusion for this hypothesis test?

Answer 1

Answer:

As the calculated  χ² = 1.23  is less than the critical region  χ²≥ χ²(0.05,9)=  16.92 ,Staples can conclude that demands for these two types of computers are independent  at the 0.05 level of significance.

Step-by-step explanation:

As we have to test the independence of the two categories the chi square of independence test is applied.

Let the null and alternate hypotheses be

H0 :The demand for each type of computers is independent

against the claim

Ha: The  demand for each type of computers is not independent i.e they are associated.

The significance level alpha is chosen to be 0.05

The critical region for alpha =0.05 and (4-1)(4-1) = 3*3 =9 d.f is

χ²≥ χ²(0.05,9)=  16.92

Actual Data

                  low          med-low       med-high          high       Total

low             4                15                    14                     3         36

med-low  6                 18                    18                    23        65

med-high 13                 17                   10                     17         57

high          7                  15                  15                      10          47      

Total       30                65                 57                     53         205        

Expected Frequencies

low- low = 30*36/205= 5.27

low- med.low = 65*36/205 = 11.41

low- medium high = 57*36/205= 10.009

low- high = 53*36/205 =9.31

medium low- low = 30*65/205 = 9.51

medium low- med.low = 65*65/205= 20.61

medium low- medium high = 57*65/205= 18.073

medium low- high = 53*65/205= 16.80

medium high- low = 30*57/205= 8.34

medium high- med.low = 65*57/205= 18.07

medium high- medium high = 57*57/205= 15.85

medium high- high = 53*57/205= 14.74

high- low = 30*53/205= 7.76

high- med.low = 65*53/205= 16.80

high- medium high = 57*53/205= 14.74

high- high = 53*53/205= 13.70

Summarizing in a table

Expected Frequencies

                  low          med-low       med-high          high           Total        

low             5.27         11.41                    10.009           9.31        35.999

med-low  9.51            20.61               18.073             16.8        64.993

med-high 8.34             18.07                 15.85            14.74       57  

high          7.76              16.8                  14.74             13.7         53              

Total       30.88           66.89               58.672          54.55         210.992  

Using Excel

The Chi Square Values are

0.062823

0.270857

0.159572

0.735995791

1.229247532

As the calculated  χ² = 1.23  is less than the critical region  χ²≥ χ²(0.05,9)=  16.92 ,Staples can conclude that demands for these two types of computers are independent  at the 0.05 level of significance,