Cost Curve Dean ^25 made a statistical estimation of the...

Cost Curve Dean $^{25}$ made a statistical estimation of the cost-output relationship for a shoe chain. The data for the firm is given in the following table. $$ \begin{aligned} &\begin{array}{|c|cccccccc|} \hline x & 16 & 22 & 35 & 48 & 53 & 70 & 100 & 150 \\ \hline y & 4.8 & 5.9 & 7 & 9 & 10 & 12 & 18 & 30 \\ \hline \end{array}\\ &\text { Here } x \text { is the output in thousands of dollars, and } y \text { is } \end{aligned} $$ a. Determine the best-fitting line (using least squares) and the square of the correlation coefficient. Graph. b. Determine the best-fitting quadratic (using least squares) and the square of the correlation coefficient. Graph. c. Which curve is better? Why?

Cost Curve Dean $^{25}$ made a statistical estimation of the cost-output relationship for a shoe chain. The data for the firm is given in the following table. $$ \begin{aligned} &\begin{array}{|c|cccccccc|} \hline x & 16 & 22 & 35 & 48 & 53 & 70 & 100 & 150 \\ \hline y & 4.8 & 5.9 & 7 & 9 & 10 & 12 & 18 & 30 \\ \hline \end{array}\\ &\text { Here } x \text { is the output in thousands of dollars, and } y \text { is } \end{aligned} $$
a. Determine the best-fitting line (using least squares) and the square of the correlation coefficient. Graph.
b. Determine the best-fitting quadratic (using least squares) and the square of the correlation coefficient. Graph.
c. Which curve is better? Why?

Key Concepts

Model Comparison

An evaluation technique used to determine which among different statistical models best fits a given set of data. This involves comparing criteria such as the sum of squared residuals, R² values, and simplicity of the model, to choose the model that provides a more accurate and interpretable representation of the data.

Polynomial Regression

An extension of linear regression used to model nonlinear relationships by fitting a polynomial equation to the data. Quadratic regression is a common case that involves a second-degree polynomial, which can capture curvature in the relationship between variables.

Least Squares Regression

A standard approach in regression analysis that minimizes the sum of the squared differences between observed values and the values predicted by a model. This method is used to find the best-fitting line or curve through a set of data points, providing estimates for the parameters of the model.

Coefficient of Determination (R²)

A statistical measure that represents the proportion of variance in the dependent variable that can be explained by the independent variable(s) in the model. It is used to assess the goodness-of-fit of a statistical model, with higher R² values indicating a better explanatory power.

Transcript

00:01 Here guys, let's two problem in this problem the cost curve is given and we need to find for what values of b are cost, average cost, average variable cost, these are positive.

00:17 The cost function is given as c equal to f plus 10 q minus b k squared plus k to the where b is a positive number the cost average cost the average variable cost these are positive when the post function is greater than zero when cost function is greater than zero then these values are going to be positive therefore when this function is greater than zero here we can write f plus 10 q plus quib and take the constant b on the right hand side here and we can now divide everything the left and right hand side by q square when we divide it by q square what we get f divided by q is square plus 10 divided by q these cues will be gone then plus q greater than b the whole thing is greater than b therefore, the value of cost, the value of cost, average cost and average variable, these are going to be positive when the derivative, when this derived condition, this is satisfied for b.

01:57 Then the next problem says, then the next one says, what is the shape of ac curve, at what output level is the ac minimized? let's do that.

02:18 Now the ac curve will be u -shaped and the average cost ac is minimized when the shape of ac curve it is equal to 0.

02:28 And the shape of ac curve is equal to 0 when the slope is equal to 0.

02:33 When we take the derivative of ac, then we get when we take the derivative of this average cost equation and we set it to 0.

02:47 Then we get the minimum value and we know average cost is equal to cost divided by q the quantity then c is the value the equation for the cost is given as f plus 10 q.

03:06 Minus bq squared plus kube and then we get f divided by q plus 10 minus bq plus q plus when we take the derivative, what we get? when we take the derivative of 1 divided by q, we get negative 1 by q squared.

03:24 Therefore we get f divided by q squared.

03:28 This is the negative and f is a constant.

03:31 When we get the derivative of 10, we get 0.

03:35 And then the derivative of bq.

03:38 When we take the derivative of q, we get 1.

03:42 Therefore we are left with and the b and then when we take the derivative of q square we get twice q and here we get this twice q equal to 0 therefore when this condition is made we get the average cost at the minimum value and the ac curve is having this q shape then the last problem asks us at what output level does mc curve crosses the ac curve and the abc curve.

04:15 The average cost and average variable cost when these two are crossing the marginal cost.

04:25 Now marginal cost mc it is defined as the first derivative of cost.

04:32 When we take the first derivative of cost we get 10 minus 2bq plus 3 kb square.

04:40 The mc curve will cross both the average and average variable cost curves at their minimum...

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