Problem 53 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that every customer order taken by the XYZ Company requires exacty 5 hours of labor for handling the paperwork; this length of time is fixed and does not vary from lot to lot. The total number of hours \(y\) required to manufacture and sell a lot of size \(x\) would then be \(y=(\) number of hours to produce a lot of size \(x)+5\) Some data on XYZ's bookcases are given in the following table. $$ \begin{array}{ccc} \hline \text { Order } & \text { Lot Size } x & \begin{array}{c} \text { Total Labor } \\ \text { Hours } y \end{array} \\ \hline 1 & 11 & 38 \\ 2 & 16 & 52 \\ 3 & 8 & 29 \\ 4 & 7 & 25 \\ 5 & 10 & 38 \\ \hline \end{array} $$ (a) From the description of the problem, the least-squares line should have 5 as its \(y\)-intercept. Find a formula for the value of the slope \(b\) that minimizes the sum of squares $$ S=\sum_{i=1}^{n}\left[y_{i}-\left(5+b x_{i}\right)\right]^{2} $$ (b) Use this formula to estimate the slope \(b\). (c) Use your least-squares line to predict the total number of labor hours to produce a lot consisting of 15 bookcases.

Step-by-Step Solution

Answer

Slope \(b \approx 0.668\) and predicted labor hours for 15 bookcases is approximately 15.02 hours.

Step 1: Understand the Given Problem

The problem asks to find the least-squares line's slope that minimizes the sum of squares, using given data. The line represents the relationship between the lot size \(x\) and the total labor hours \(y\). The line's intercept should be 5, given each order requires a fixed 5 hours for paperwork. The relationship generally is \( y = 5 + bx \). You are asked to find the value of \(b\), predict further, and solve the sum of squares.

Step 2: Represent the Model Equation

The model equation provided is \( y = 5 + bx \), where \(y\) is the total labor hours, and \(x\) is the lot size. We need to find the value of \(b\) which minimizes the sum \( S = \sum_{i=1}^{n}\left(y_i - (5 + bx_i)\right)^2 \).

Step 3: Apply Least Squares Method

In order to minimize \(S\), find the derivative of \(S\) with respect to \(b\), and equate it to zero. Begin with writing \(S\): \( S = \sum_{i=1}^n (y_i - 5 - bx_i)^2 \). Simplify and differentiate:\[ \frac{dS}{db} = \sum_{i=1}^n -2x_i (y_i - 5 - bx_i) = 0 \]. Solving for \(b\):\[ \sum_{i=1}^n x_i (y_i - 5 - bx_i) = 0 \]. This further expands and simplifies to:\[ \sum_{i=1}^n x_iy_i - 5\sum_{i=1}^n x_i - b\sum_{i=1}^n x_i^2 = 0 \].

Step 4: Solve for the Slope b

Rearrange the equation from Step 3 to find \(b\):\[ b = \frac{\sum_{i=1}^n x_i y_i - 5\sum_{i=1}^n x_i}{\sum_{i=1}^n x_i^2} \]. Substitute values from the table:\( \sum_{i=1}^n x_i = 52, \sum_{i=1}^n y_i = 182, \sum_{i=1}^n x_i y_i = 625, \sum_{i=1}^n x_i^2 = 546 \).

Step 5: Calculate b Using Data

Substitute the sums into the formula to find the slope:\[ b = \frac{625 - 5(52)}{546} = \frac{625 - 260}{546} = \frac{365}{546} \approx 0.668 \].Thus, the slope \(b\) is approximately 0.668.

Step 6: Use the Least-Squares Line to Predict

To predict the total labor hours for a lot size of 15, substitute \(x = 15\) into the formula \( y = 5 + bx \): \( y = 5 + 0.668 \times 15 \). Calculate:\( y = 5 + 10.02 = 15.02 \). Thus, the predicted total labor hours are approximately 15.02.

Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. In the context of this exercise, linear regression helps establish a relationship between the lot size of bookcases, denoted by \(x\), and the total labor hours \(y\). The linear equation provided is \[y = 5 + bx\], where 5 is the fixed labor hours required for paperwork for each order.
What makes this method powerful is its simplicity combined with its predictive value. It allows us to make estimations about the value of the dependent variable based on the values of the independent variables. In our case, it helps predict future labor costs based on lot sizes by calculating the most accurate slope \(b\), through the least squares method. This slope reflects how a change in the bookcase lot size impacts the total labor hours.

Sum of Squares

The sum of squares is a measure that captures the variance in data points from a defined model or line of best fit, which in this case, is our linear regression line. The sum of squares tells us how far off each data point is from the predicted value on our line \(y = 5 + bx\).
For linear regression, we aim to minimize this sum of squares to find the best fit line. This means we adjust the slope \(b\) so that \(S = \sum_{i=1}^n(y_i - (5 + bx_i))^2\) is as small as possible. By minimizing this difference, we ensure that our line is the best representation of the data, leading to more accurate predictions and insights.
Understanding the sum of squares enables us to comprehend the accuracy of our model; a smaller sum suggests our linear model is closely aligned with actual data, thereby increasing its reliability.

Slope Calculation

The slope of a line in linear regression indicates the rate at which the dependent variable \(y\) changes with every one-unit change in the independent variable \(x\). In this problem, the slope \(b\) tells us how much the total labor hours increase per additional bookcase in the lot.
To find the slope, we need a method to minimize the sum of squares. Through calculus, we derive the formula \(b = \frac{\sum_{i=1}^n x_iy_i - 5\sum_{i=1}^n x_i}{\sum_{i=1}^n x_i^2}\). This equation helps us accurately calculate \(b\) using the dataset provided; the numbers from our table were substituted into the equation resulting in \(b \approx 0.668\).
By calculating this slope, we effectively determine the labor cost incurred with each additional bookcase in the order, which is essential for planning and budget forecasting.

Labor Cost Analysis

Labor cost analysis evaluates the relationship between product quantities and the corresponding labor hours required. This understanding is crucial for any business to manage and predict labor expenses accurately.
In the exercise, we find that each bookcase requires an additional 0.668 hours of labor, aside from the fixed 5 hours for paperwork. By constructing our linear regression model, we can extend this analysis to predict labor costs. For instance, the total labor for producing a lot of 15 bookcases is calculated as \(y = 5 + 0.668 \times 15\), which yields approximately 15.02 hours.
Such analysis helps businesses like XYZ Company in financial planning, ensuring they allocate adequate human resources, control costs, and maximize efficiency for different lot sizes.