To develop the estimated regression equation, we will use the least squares regression method to find the relationship between the production volume ($x$) and the total cost ($y$). The regression equation is of the form:

$y = b_0 + b_1x$

Where:

• $b_1$ (slope) represents the change in total cost for each unit increase in production.
• $b_0$ (intercept) represents the total cost when production volume is zero.

Step 1: Calculate $b_1$ (Slope)

The slope $b_1$ is calculated as: $b_1 = \frac{n\sum(xy) - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$

Step 2: Calculate $b_0$ (Intercept)

The intercept $b_0$ is calculated as: $b_0 = \bar{y} - b_1\bar{x}$

Where:

• $n$ = number of data points
• $\bar{x}$ and $\bar{y}$ are the means of $x$ and $y$
• $\sum x$, $\sum y$, $\sum x^2$, and $\sum xy$ are sums derived from the data.

Data Summary Table

$x$ (Production Volume)	$y$ (Total Cost)	$x^2$	$xy$
400	4000	$160,000$	$1,600,000$
450	5100	$202,500$	$2,295,000$
550	5500	$302,500$	$3,025,000$
600	5800	$360,000$	$3,480,000$
700	6500	$490,000$	$4,550,000$
750	6900	$562,500$	$5,175,000$

Step 3: Calculate Required Sums

$\sum x = 400 + 450 + 550 + 600 + 700 + 750 = 3450$ $\sum y = 4000 + 5100 + 5500 + 5800 + 6500 + 6900 = 33,300$ $\sum x^2 = 160,000 + 202,500 + 302,500 + 360,000 + 490,000 + 562,500 = 2,077,500$ $\sum xy = 1,600,000 + 2,295,000 + 3,025,000 + 3,480,000 + 4,550,000 + 5,175,000 = 20,125,000$ $n = 6$

Step 4: Compute $b_1$ and $b_0$

$b_1 = \frac{6(20,125,000) - (3450)(33,300)}{6(2,077,500) - (3450)^2}$ $b_1 = \frac{120,750,000 - 114,885,000}{12,465,000 - 11,902,500} = \frac{5,865,000}{562,500} = 10.42$

$b_0 = \frac{33,300}{6} - 10.42 \cdot \frac{3450}{6}$ $b_0 = 5550 - 10.42 \cdot 575 = 5550 - 5986.5 = -436.5$

Step 5: Regression Equation

$\text{Estimated regression equation: } y = -436.5 + 10.42x$

Final Answer:

$y = -436.5 + 10.42x$

Would you like a detailed explanation of any part of this process?

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5 Related Questions:

1. How do we interpret the slope ($b_1$) in this regression equation?
2. What are the assumptions of linear regression?
3. How can we compute the goodness-of-fit (R-squared) for this model?
4. How would predictions be made for a production volume of 800 units?
5. What are potential limitations of this regression model?

Tip: Always visualize your data with a scatter plot before applying regression to verify the relationship is linear.