91Ó°ÊÓ

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 subassemblies. 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. $$ \begin{array}{|cc|} \hline \begin{array}{c} \text { Number of } \\ \text { Assemblers } \end{array} & \begin{array}{c} \text { One-Hour } \\ \text { Production } \\ \text { (units) } \end{array} \\ \hline 2 & 15 \\ 4 & 25 \\ 1 & 10 \\ 5 & 40 \\ 3 & 30 \\ \hline \end{array} $$ The dependent variable is production; that is, it is assumed that different levels of production result from a different number of employees. a. Draw a scatter diagram. b. Based on the scatter diagram, does there appear to be any relationship between the number of assemblers and production? Explain. c. Compute the correlation coefficient.

Step by step solution

01

Plotting the Scatter Diagram

To begin, we will create a scatter plot using the given data points, where the x-axis represents the number of assemblers and the y-axis represents the one-hour production units. Plot each pair from the table: (2, 15), (4, 25), (1, 10), (5, 40), and (3, 30). Check the positioning of points to establish any visible pattern or trend.

02

Analyzing the Scatter Diagram

Analyze the plotted points to determine if there is a discernible pattern or trend. A positive correlation will be indicated by an upward trend (i.e., production seems to increase with more assemblers), whereas a negative correlation will show a downward trend. If the points are randomly scattered, this suggests no correlation.

03

Calculation - Mean Values

Calculate the mean of the number of assemblers and the mean of production. For assemblers: \( \text{Mean} = \frac{1+2+3+4+5}{5} = 3 \). For production: \( \text{Mean} = \frac{10+15+30+25+40}{5} = 24 \).

04

Calculation - Deviations

Compute the deviation from the mean for each paired observation. This is done by subtracting the mean from each observed value, separately for both the x-values (assemblers) and y-values (production).

05

Calculation - Covariance and Variance

Calculate the product of deviations for each pair and then determine covariance. Next, compute the variance for the x-values (assemblers). These calculations are necessary to eventually find the correlation coefficient.

06

Compute the Correlation Coefficient

The correlation coefficient is given by \( r = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}} \). Using the computed deviations, calculate covariance as the average of the product of deviations, and the same for variance. Substitute these into the formula to find \( r \). After calculations, \( r \) is found to be approximately 0.93, indicating a strong positive correlation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatter Diagram

A scatter diagram, often known as a scatter plot, is a useful graphical tool for visualizing the relationship between two numerical variables. Here, each point represents a single observation in a dataset, plotted using two variables. In our exercise, the scatter plot helps to visualize the relationship between the number of employees and the production output. The x-axis represents the number of assemblers, while the y-axis shows the production levels (units produced in one hour).

By looking at the scatter plot, one can assess patterns or trends in the data. A trend line suggesting an upward or downward movement can often be seen, indicating a potential correlation between variables. For the given dataset, plotting points such as (2, 15) or (5, 40) will help highlight any relationships. If the points roughly form a line pointing towards a direction, it indicates a correlation; if they seem scattered with no clear direction, it implies no correlation.

Correlation Coefficient

The correlation coefficient, often represented by the symbol \( r \), is a statistical measure that calculates the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1.

• A value of 1 indicates a perfect positive linear relationship.
• -1 indicates a perfect negative linear relationship.
• 0 suggests no linear relationship.

To compute the correlation coefficient \( r \), we need the covariance of the two variables and their respective variances. In our exercise, we calculated \( r \) to be around 0.93, indicating a strong positive correlation. This means as the number of assemblers increases, so does production, and vice versa. It's important to note that correlation does not imply causation; it only tells us the strength and direction of a relationship.

Covariance

Covariance provides a measure of how much two random variables change together. Specifically, it indicates the direction of a linear relationship between variables. Unlike the correlation coefficient, it does not have a standardized scale and thus can take on any real number value.

The calculation involves finding the deviation of each data point from its mean for both variables, multiplying these deviations, and then averaging the products. If the covariance is positive, it implies that both variables increase together; if negative, when one variable increases, the other decreases.

In the exercise, we calculated covariance between the number of assemblers and the units produced. A positive covariance was expected due to simultaneous increases in both variables, leading to the observed positive correlation.

Variance

Variance is a measure of how much the values of a variable deviate from the mean of that variable. It helps in quantifying the spread or dispersion of a set of data points. Variance is computed by taking the mean of the squared deviations from the mean of the data set.

For the exercise, we focused on the variance of the number of assemblers. Calculating variance is crucial in determining the correlation as it forms part of the equation for the correlation coefficient. By understanding the spread of data, one can better interpret statistical results such as correlation.

Variance is useful for understanding data variability, but it is important to remember that it measures the spread of a single variable, unlike covariance which measures the relationship between two different variables.