Suppose that every customer order taken by the XYZ Company...

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. $$ \begin{array}{ccc} \hline & & \text { Total Labor } \\ \text { Order } & \text { Lot Size } x & \text { Hours } y \\ \hline 1 & 11 & 38 \\ 2 & 16 & 52 \\ 3 & 8 & 29 \\ 4 & 7 & 25 \\ 5 & 10 & 38 \\ \hline \end{array} $$ (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.

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.
$$
\begin{array}{ccc}
\hline & & \text { Total Labor } \\
\text { Order } & \text { Lot Size } x & \text { Hours } y \\
\hline 1 & 11 & 38 \\
2 & 16 & 52 \\
3 & 8 & 29 \\
4 & 7 & 25 \\
5 & 10 & 38 \\
\hline
\end{array}
$$
(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.

Key Concepts

Regression Prediction

Once the regression equation is obtained, it can be used to predict the dependent variable for a given value of the independent variable. This involves substituting the new value into the regression formula, thereby estimating the corresponding output. The prediction takes into account both the fixed intercept and the estimated slope, providing an estimate based on the relationship identified in the data.

Least-Squares Regression

Least-squares regression is a method used to find the line or curve that best fits a set of data points by minimizing the sum of the squares of the vertical deviations (residuals) between the data points and the line. This technique helps in deriving a functional relationship between an independent variable and a dependent variable while accounting for error or variability in measurements.

Fixed Intercept in Regression

In some regression problems the y-intercept is predetermined or known rather than being estimated from the data. This is known as a fixed intercept situation. In such cases, the estimation process focuses solely on determining the best-fit slope that minimizes the squared errors, given that the intercept is already set at a specific value.

Minimization of Sum of Squared Errors

The minimization of sum of squared errors is the process used in regression analysis where the sum of the squared differences between the observed values and the values predicted by the model is minimized. This approach ensures that the overall error across all data points is as small as possible, resulting in an optimal curve or line that best represents the data.

Slope Estimation Formula

When the intercept is fixed, the slope is estimated by differentiating the sum of squared errors with respect to the slope and setting the derivative to zero. This leads to the formula b = [? x_i (y_i - fixed intercept)] / [? x_i²]. This formula isolates the contribution of the variable part of the model to the total error and provides the slope value that minimizes the error given the fixed intercept.

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