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 (i)
9.35
8.91
91.03
494
7
To test the claim that the day of the week is independent of the number of defective parts, we can use the chi-squared test for independence.
First, we need to calculate the expected values for each category.
To calculate the expected value for each day of the week, we use the formula:
Expected value = (row total * column total) / grand total
Row total for Monday = 80 + 15 = 95
Row total for Tuesday = 100 + 5 = 105
Row total for Wednesday = 95 + 5 = 100
Row total for Thursday = 93 + 7 = 100
Row total for Friday = 82 + 12 = 94
Column total for Acceptable products = 80 + 100 + 95 + 93 + 82 = 450
Column total for Defective products = 15 + 5 + 5 + 7 + 12 = 44
Grand total = 450 + 44 = 494
Expected value for Monday, Acceptable products = (95 * 450) / 494 ≈ 86.05
Expected value for Monday, Defective products = (95 * 44) / 494 ≈ 8.49
Expected value for Tuesday, Acceptable products = (105 * 450) / 494 ≈ 95.74
Expected value for Tuesday, Defective products = (105 * 44) / 494 ≈ 9.33
Expected value for Wednesday, Acceptable products = (100 * 450) / 494 ≈ 91.03
Expected value for Wednesday, Defective products = (100 * 44) / 494 ≈ 8.91
Expected value for Thursday, Acceptable products = (100 * 450) / 494 ≈ 91.03
Expected value for Thursday, Defective products = (100 * 44) / 494 ≈ 8.91
Expected value for Friday, Acceptable products = (94 * 450) / 494 ≈ 85.25
Expected value for Friday, Defective products = (94 * 44) / 494 ≈ 8.41
The expected values for each category are as follows:
(i) The expected value for Monday, Defective products is approximately 8.49.
Therefore, the correct answer is: 8.49
To test the claim that the day of the week is independent of the number of defective parts, you need to calculate the expected value.
The expected value (E) is calculated by multiplying the row total (RT) and column total (CT), and then dividing it by the grand total (GT). In this case, the row total represents the total number of acceptable and defective products for each day of the week, and the column total represents the total number of acceptable and defective products across all days of the week. The grand total is the total number of all products.
Here's how you can calculate the expected value:
1. Calculate the total number of acceptable products for each day of the week:
Monday: 80 (a)
Tuesday: 100 (b)
Wednesday: 95 (c)
Thursday: 93 (d)
Friday: 82 (e)
2. Calculate the row totals:
Monday: 80 (a) + 15 (f) = 95
Tuesday: 100 (b) + 5 (g) = 105
Wednesday: 95 (c) + 5 (h) = 100
Thursday: 93 (d) + 7 (i) = 100
Friday: 82 (e) + 12 (j) = 94
3. Calculate the column totals:
Acceptable products: 80 (a) + 100 (b) + 95 (c) + 93 (d) + 82 (e) = 450
Defective products: 15 (f) + 5 (g) + 5 (h) + 7 (i) + 12 (j) = 44
4. Calculate the grand total:
GT = 450 (Acceptable products) + 44 (Defective products) = 494
5. Calculate the expected value (E) for each cell:
(Expected value for Monday and Acceptable products) = (95/494) * 450
(Expected value for Monday and Defective products) = (95/494) * 44
Repeat this calculation for all cells.
6. Calculate the overall expected value (i) by summing up all the expected values.
Based on the provided data, the expected value (E) for each cell will be different. The expected value (i) will depend on the calculations performed in Step 5. However, without performing the specific calculations, it is not possible to provide a precise expected value (i).
To test the claim that the day of the week is independent of the number of defective parts, you can perform a chi-square test for independence. The expected value (i) represents the expected number of defective parts for each day of the week.
To calculate the expected value, you can use the formula:
Expected value = (row total * column total) / grand total
In this case, the row totals are the total number of acceptable products and defective products, while the column totals are the total number of products for each day of the week. The grand total is the total number of products overall.
Using the given data:
Row totals (acceptable products):
80 + 100 + 95 + 93 + 82 = 450
Row totals (defective products):
15 + 5 + 5 + 7 + 12 = 44
Column totals (Monday):
80 + 15 = 95
Column totals (Tuesday):
100 + 5 = 105
Column totals (Wednesday):
95 + 5 = 100
Column totals (Thursday):
93 + 7 = 100
Column totals (Friday):
82 + 12 = 94
Grand total:
450 + 44 = 494
Now, let's calculate the expected value for each day of the week:
Expected value (Monday) = (450 * 95) / 494 ≈ 86.05
Expected value (Tuesday) = (450 * 105) / 494 ≈ 95.68
Expected value (Wednesday) = (450 * 100) / 494 ≈ 91.08
Expected value (Thursday) = (450 * 100) / 494 ≈ 91.08
Expected value (Friday) = (450 * 94) / 494 ≈ 85.60
Therefore, the expected value (i) is approximately 91.03.