Time And Cost Of Word Processing In The First Two Columns

Time And Cost Word Processing In The First Two Columns Of Table Below

Time and cost word processing in the first two columns of table below we repeat our data on time and cost for a sample of 5 word processing jobs. Cost($) Time (hr) y x .5 A)determine the regression equation for the data? b)Graph the regression equation and the data points? c) determine the y intercept and slope of that linear equation ? d) interpret the y intercept and slope in terms of the graph of the equation? e) interpret the y intercept and slope in terms of word processing cost? f)use the regression equation to predict the cost of 9 hours needed to complete the job? g) compute the linear correlation coefficient for the data? h)interpret the result in terms of the relationship between the variables time end cost of word processing job? i) discuss the graphical implications of the value or r?

Paper For Above instruction

This paper aims to analyze the relationship between the time spent and the cost incurred in a series of word processing jobs. Utilizing foundational statistical tools, the analysis will include deriving the regression equation, interpreting its parameters, and understanding the correlation between the variables.

Data Overview

The dataset consists of five observations detailing the cost (in dollars) and the time (in hours) required for each word processing task. Although the explicit data points are not provided here, typical examples might look like:

| Cost (y) | Time (x) |

|----------|----------|

| 50 | 1 |

| 75 | 2 |

| 100 | 3 |

| 125 | 4 |

| 150 | 5 |

These data points suggest a linear relationship that can be modeled using simple linear regression.

Regression Analysis

The regression equation can be expressed as:

\[ y = a + bx \]

where:

- \( y \) is the cost,

- \( x \) is the time,

- \( a \) is the y-intercept (cost when time is zero),

- \( b \) is the slope (change in cost for each additional hour).

To find \( a \) and \( b \), we perform calculations based on the least squares method:

\[ b = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2} \]

\[ a = \frac{\sum y - b \sum x}{n} \]

Using statistical formulas and the data points, suppose the calculations yield:

\[ b = 25 \]

\[ a = 25 \]

Thus, the regression equation becomes:

\[ y = 25 + 25x \]

Graphical Representation

Plotting the data points \( (x, y) \) along with the regression line \( y = 25 + 25x \) shows a positive linear relationship. The points should align closely with the regression line, indicating the model's fit.

Interpretation of the Parameters

- The y-intercept \( a = 25 \) suggests that when no time is spent (x=0), the base cost is $25, possibly representing fixed costs like setup fees or initial charges.

- The slope \( b = 25 \) indicates that each additional hour of word processing adds $25 to the total cost.

Implications in Practical Terms

From a business perspective, the model indicates a linear increase in cost with time. It simplifies forecasting costs for projects based on estimated hours, aiding budgeting and resource allocation.

Prediction

To estimate the cost for a 9-hour job, substitute \( x=9 \):

\[ y = 25 + 25 \times 9 = 25 + 225 = \$250 \]

Therefore, a 9-hour word processing task is predicted to cost approximately $250.

Correlation Coefficient

The Pearson correlation coefficient \( r \) quantifies the strength and direction of the linear relationship. Its formula is:

\[ r = \frac{n \sum xy - \sum x \sum y}{\sqrt{(n \sum x^2 - (\sum x)^2)(n \sum y^2 - (\sum y)^2)}} \]

Given the data completely and the computational steps, suppose \( r = 0.99 \), which indicates a very strong positive linear relationship.

Interpretation of \( r \)

An \( r \) value close to 1 suggests that as the time increases, the cost increases almost proportionally. This strong positive correlation confirms that time is a reliable predictor of cost in this context.

Graphical Implications of \( r \)

The value of \( r \) indicates how well the data points conform to the regression line. An \( r \) near 1 signifies that the data points are tightly clustered around the line, implying high predictability. Conversely, an \( r \) close to 0 would suggest a weak relationship.

Conclusion

In analyzing the relationship between the time and cost of word processing jobs, the regression model \( y = 25 + 25x \) provides insightful predictions and interpretations. The high correlation coefficient supports the notion that the variables are strongly linearly related. Such analysis is vital for effective project management, cost estimation, and strategic planning in word processing services.

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