The production department of Celltronics International wants to explore the relationship between the number of employees who assemble a subassembly and the number produced. As an experiment, two employees were assigned to assemble the parts. They produced 15 during a one-hour period. Then, four employees assembled them. They produced 25 during a one-hour period. The complete set of paired observations follows:
 
Number of
Assemblers

One-Hour
Production (units)

2

15

4

25

1

10

5

40

3

30
 
The dependent variable is production; that is, it is assumed that the level of production depends on the number of employees.
 
**a.** Compute the correlation coefficient. **(Round the final answer to 3 decimal places.)**
 
*r* =            
 
**b.** Evaluate the strength of the relationship by computing the coefficient of determination. **(Round the intermediate calculations to 3 decimal places. Round the final answer to 4 decimal places and the percentage to nearest whole percent.)**
 
The *r*2 is  , so about  percent of the variation in production is explained by the variation in the number of assemblers.




To solve this problem, we will calculate the correlation coefficient \( r \) and the coefficient of determination \( r^2 \) using the provided dataTo solve this problem, we will calculate the correlation coefficient \( r \) and the coefficient of determination \( r^2 \) using the provided 

### Data:
\[
\begin{array}{|c|c|}
\hline
\text{Number of Assemblers (X)} & \text{Production (Y)} \\
\hline
2 & 15 \\
4 & 25 \\
1 & 10 \\
5 & 40 \\
3 & 30 \\
\hline
\end{array}
\]

### a. Compute the correlation coefficient \( r \)

The correlation coefficient \( r \) can be calculated using the formula:
\[
r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{\left[n\sum X^2 - (\sum X)^2\right]\left[n\sum Y^2 - (\sum Y)^2\right]}}
\]

Where:
- \( n \) is the number of observations.
- \( \sum X \) is the sum of the \( X \) values.
- \( \sum Y \) is the sum of the \( Y \) values.
- \( \sum XY \) is the sum of the products of corresponding \( X \) and \( Y \) values.
- \( \sum X^2 \) is the sum of the squares of the \( X \) values.
- \( \sum Y^2 \) is the sum of the squares of the \( Y \) values.

#### Step 1: Calculate the necessary sums

\[
\sum X = 2 + 4 + 1 + 5 + 3 = 15
\]
\[
\sum Y = 15 + 25 + 10 + 40 + 30 = 120
\]
\[
\sum XY = (2 \times 15) + (4 \times 25) + (1 \times 10) + (5 \times 40) + (3 \times 30) = 30 + 100 + 10 + 200 + 90 = 430
\]
\[
\sum X^2 = 2^2 + 4^2 + 1^2 + 5^2 + 3^2 = 4 + 16 + 1 + 25 + 9 = 55
\]
\[
\sum Y^2 = 15^2 + 25^2 + 10^2 + 40^2 + 30^2 = 225 + 625 + 100 + 1600 + 900 = 3450
\]

#### Step 2: Calculate the correlation coefficient \( r \)

Substitute the values into the formula:
\[
r = \frac{5(430) - (15)(120)}{\sqrt{\left[5(55) - 15^2\right]\left[5(3450) - 120^2\right]}}
\]
\[
r = \frac{2150 - 1800}{\sqrt{(275 - 225)(17250 - 14400)}} = \frac{350}{\sqrt{50 \times 2850}} = \frac{350}{\sqrt{142500}} = \frac{350}{377.60} \approx 0.927
\]

### b. Compute the coefficient of determination \( r^2 \)

The coefficient of determination \( r^2 \) is the square of the correlation coefficient \( r \).
\[
r^2 = (0.927)^2 \approx 0.859
\]
So, about 86% of the variation in production is explained by the variation in the number of assemblers.

### Summary:
- **a.** The correlation coefficient \( r = 0.927 \) (rounded to 3 decimal places).
- **b.** The coefficient of determination \( r^2 \approx 0.859 \), so about 86% of the variation in production is explained by the variation in the number of assemblers.

Do you need further details, or do you have any questions? Here are some related questions you might find useful:

1. How do we interpret the correlation coefficient value?
2. What is the significance of the coefficient of determination?
3. Can the correlation coefficient ever exceed 1 or be less than -1?
4. How would the correlation coefficient change if the number of observations increased?
5. What are the assumptions behind the linear correlation coefficient?
6. How does outlier data affect the correlation coefficient?
7. How can we calculate the regression line based on this data?
8. What other methods can be used to assess the relationship between two variables?

**Tip:** Correlation does not imply causation; a high correlation coefficient does not mean that one variable causes the other to change.

Learn how to calculate the correlation coefficient and coefficient of determination using a provided dataset. Understand the relationship between variables and how much variation in production is explained by the number of assemblers. Suitable for advanced high school and college students.

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