The manager of an assembly operation wants to determine whether the number of defective parts is dependent on the day of the week. The following sample data is obtained:
Day of the week
Monday
Tuesday
Wednesday
Thursday
Vrydag/ Friday
Acceptable products
80 (a)
100 (b)
95 (c)
93 (d)
82 (e)
Defective products
15 (f)
5 (g)
5 (h)
7 (i)
12 (j)
You have to test the claim that the day of the week is independent of the number of defective parts at the 5% significance level. Give the expected value (e)
To test the claim that the day of the week is independent of the number of defective parts, we will perform a chi-squared test of independence.
First, we need to calculate the expected values for each cell in the contingency table. The expected value for each cell is calculated by multiplying the row total by the column total and dividing by the overall total.
Here is the contingency table:
Monday Tuesday Wednesday Thursday Friday Total
Defective 15 5 5 7 12 44
Acceptable 80 100 95 93 82 450
Total 95 105 100 100 94 494
To calculate the expected values:
Expected (f) = (44/494) * 95 ≈ 8.54
Expected (g) = (44/494) * 105 ≈ 9.46
Expected (h) = (44/494) * 100 ≈ 8.95
Expected (i) = (44/494) * 100 ≈ 8.95
Expected (j) = (44/494) * 94 ≈ 8.40
Therefore, the expected values are approximately: (e) = 8.54, (f) = 9.46, (g) = 8.95, (h) = 8.95, (i) = 8.40.
To determine the expected value (e) in order to test the claim that the day of the week is independent of the number of defective parts, you need to calculate the expected number of defective products for each day of the week.
The formula to calculate the expected value is: E = (row total * column total) / grand total.
First, create a contingency table with the given data:
| | Acceptable Products | Defective Products | Row Total |
|--------------|---------------------|--------------------|-----------|
| Monday | 80 | 15 | 95 |
| Tuesday | 100 | 5 | 105 |
| Wednesday | 95 | 5 | 100 |
| Thursday | 93 | 7 | 100 |
| Vrydag/ Friday | 82 | 12 | 94 |
| Column Total | 450 | 44 | 494 |
The grand total is 494, calculated by summing all the values in the contingency table.
To calculate the expected value for each cell, use the formula: E = (row total * column total) / grand total.
For example, for the first cell (Monday, Acceptable Products), the expected value is:
E = (95 * 450) / 494
E ≈ 86.07
Repeat this calculation for all the other cells in the contingency table to obtain the expected values.
To test the claim that the day of the week is independent of the number of defective parts, we can use a chi-squared test for independence. This test will help us determine if there is a significant association between the variables.
To calculate the expected values (e), we need to assume that the null hypothesis of independence is true. Under the assumption of independence, we can calculate the expected number of defective parts for each day of the week.
The formula to calculate the expected value for a specific cell in a contingency table is:
e = (row total * column total) / total number of observations.
Let's calculate the expected values for each day of the week using the sample data given:
Day of the week Acceptable products Defective products Total
Monday 80 15 (a+f)
Tuesday 100 5 (b+g)
Wednesday 95 5 (c+h)
Thursday 93 7 (d+i)
Friday 82 12 (e+j)
Total 450 44
Now we can calculate the expected values (e) for each cell:
e(a) = (row total for Monday * column total for Acceptable products) / Total number of observations
= (95 * 450) / 494
≈ 86.44
e(b) = (row total for Tuesday * column total for Acceptable products) / Total number of observations
= (95 * 450) / 494
≈ 105.64
e(c) = (row total for Wednesday * column total for Acceptable products) / Total number of observations
= (95 * 450) / 494
≈ 100.24
e(d) = (row total for Thursday * column total for Acceptable products) / Total number of observations
= (95 * 450) / 494
≈ 98.62
e(e) = (row total for Friday * column total for Acceptable products) / Total number of observations
= (95 * 450) / 494
≈ 84.07
Therefore, the expected values (e) for each day of the week are approximately:
e(a) ≈ 86.44
e(b) ≈ 105.64
e(c) ≈ 100.24
e(d) ≈ 98.62
e(e) ≈ 84.07