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1. Let
[tex]\[ \sum_{i=1}^{10} \left( x_i - \bar{x} \right) \left( y_i - \bar{y} \right) = 121, \][/tex]
calculate
[tex]\[ \text{Cov}(x, y) = \frac{1}{n} \sum_{i=1}^{10} x_i y_i - \bar{x} \cdot \bar{y}. \][/tex]

2. Assume that [tex]\( y \)[/tex] depends on [tex]\( x \)[/tex] and they are related as [tex]\( Y = \beta_1 + \beta_2 X + U \)[/tex].

\begin{tabular}{|l|l|l|l|l|l|}
\hline
Hours of study (x) & 3 & 2 & 5 & 4 & 1 \\
\hline
Test Result (y) & 6 & 7 & 4 & 5 & 8 \\
\hline
\end{tabular}

Calculate:
A) Estimators [tex]\(\beta_1, \beta_2\)[/tex].
B) Give the equation of the least squares regression line (LSL) or [tex]\(\hat{y}\)[/tex].
C) The probable value of [tex]\( y \)[/tex] if [tex]\( x = 10 \)[/tex].
D) Compute the coefficient of determination [tex]\( r \)[/tex] and comment on the degree of fit of the regression line to the data set.
E) Compare observed and predicted values, show error explained briefly.
F) Sketch the graph.

3. Let [tex]\( Y = \beta_1 + \beta_2 X + U \)[/tex], then calculate:
A) [tex]\( \sum_{i=1}^n U_i = \)[/tex].
B) The least squared residual sum is [tex]\(\)[/tex].

4. Prove that
[tex]\[ 4 \sum_{i=1}^n \left( x_i - \bar{x} \right)^2 = n \left[ \sum_{i=1}^n x_i^2 - n \overline{x^2} \right]. \][/tex]

5. Let
[tex]\[ n = 50, \sum_{i=1}^{50} x_i = 150, \sum_{i=1}^{50} y_i = 200, \sum_{i=1}^{50} x_i^2 = 500, \sum_{i=1}^{50} y_i^2 = 1000, \sum_{i=1}^{50} x_i y_i = 650. \][/tex]

Calculate [tex]\(\beta_1, \beta_2\)[/tex], regression equation line, and [tex]\( r \)[/tex].

6. From the data of [tex]\( n \)[/tex] pairs of observations [tex]\( x, y \)[/tex], the following results are obtained:
[tex]\[ \sum_{i=1}^n \left( x_i - \bar{x} \right)^2 = 1298, \sum_{i=1}^n \left( y_i - \bar{y} \right)^2 = 600, \sum_{i=1}^n \left( x_i - \bar{x} \right) \left( y_i - \bar{y} \right) = -262. \][/tex]
Obtain the correlation coefficient [tex]\( r \)[/tex].

7. If [tex]\( r = 0.15 \)[/tex],
[tex]\[ \sum_{i=1}^n \left( x_i - \bar{x} \right) \left( y_i - \bar{y} \right) = 12, \sum_{i=1}^n \left( y_i - \bar{y} \right)^2 = 40, \sigma_x = 4, \sigma_y = 1, \][/tex]
then calculate [tex]\( n \)[/tex].

8. Given the production function
[tex]\[ f(L, Q) = 10 L^{0.3} \times 10 Q^{0.7}, \][/tex]
calculate:
A) [tex]\( \frac{\partial f}{\partial L} \)[/tex].
B) [tex]\( \frac{\partial f}{\partial Q} \)[/tex].

9. Given the non-linear regression line
[tex]\[ y = a \cdot R^b S^c T^d, \][/tex]
rewrite it in linear form.

10. Given the data:

\begin{tabular}{|l|l|l|l|l|l|}
\hline
[tex]\( X_i \)[/tex] & 11 & 7 & 19 & 15 & 23 \\
\hline
[tex]\( Y_i \)[/tex] & 7 & 5 & 11 & 9 & 13 \\
\hline
\end{tabular}

Calculate:
A) The slope [tex]\(\beta_2\)[/tex].
B) [tex]\(\beta_1\)[/tex].
C) Regression equation.
D) Interpret.
E) Correlation coefficient.
F) TSS, ESS, RSS.

11. In a perfectly competitive market [tex]\( TC = 4Q^2 - 48Q + 100 \)[/tex], [tex]\( p = 20 \)[/tex]:

Calculate:
A) TR.
B) AC.
C) MC.
D) Profit [tex]\( (\pi) \)[/tex].
E) Profit-maximizing level output.
F) The maximum profit level.



Answer :

GinnyAnswer
To tackle the given problem systematically, we will address each of the sub-questions step-by-step. In this case, we are dealing with a perfectly competitive market described by the total cost (TC) and price (P) functions, and we need to calculate several economic metrics.

### 11. In a perfectly competitive market, given [tex]\( TC = 4Q^2 - 48Q + 100 \)[/tex] and [tex]\( P = 20 \)[/tex]:

#### A) Calculate Total Revenue (TR)
Total Revenue (TR) is calculated as the product of the price (P) and the quantity (Q) sold.
[tex]\[ TR = P \cdot Q = 20 \cdot Q = 20Q \][/tex]

#### B) Calculate Average Cost (AC)
Average Cost (AC) is calculated by dividing the total cost (TC) by the quantity (Q).
[tex]\[ AC = \frac{TC}{Q} = \frac{4Q^2 - 48Q + 100}{Q} = 4Q - 48 + \frac{100}{Q} \][/tex]

#### C) Calculate Marginal Cost (MC)
Marginal Cost (MC) is calculated by differentiating the total cost (TC) with respect to the quantity (Q).
[tex]\[ MC = \frac{d(TC)}{dQ} = \frac{d(4Q^2 - 48Q + 100)}{dQ} = 8Q - 48 \][/tex]

#### D) Calculate Profit ([tex]\(\pi\)[/tex])
Profit ([tex]\(\pi\)[/tex]) is calculated as total revenue (TR) minus total cost (TC).
[tex]\[ \pi = TR - TC = 20Q - (4Q^2 - 48Q + 100) \][/tex]
[tex]\[ \pi = 20Q - 4Q^2 + 48Q - 100 \][/tex]
[tex]\[ \pi = -4Q^2 + 68Q - 100 \][/tex]

#### E) Calculate the profit-maximizing level of output
The profit-maximizing level of output occurs where marginal cost (MC) equals the price (P).
[tex]\[ MC = P \][/tex]
[tex]\[ 8Q - 48 = 20 \][/tex]
[tex]\[ 8Q = 68 \][/tex]
[tex]\[ Q = \frac{68}{8} = 8.5 \][/tex]

So, the profit-maximizing level of output is [tex]\( Q = 8.5 \)[/tex].

#### F) Calculate the maximum profit level
We previously calculated the profit function as:
[tex]\[ \pi = -4Q^2 + 68Q - 100 \][/tex]
By substituting the profit-maximizing level of output, [tex]\( Q = 8.5 \)[/tex], we find the maximum profit:
[tex]\[ \pi = -4(8.5)^2 + 68(8.5) - 100 \][/tex]
[tex]\[ \pi = -4(72.25) + 68(8.5) - 100 \][/tex]
[tex]\[ \pi = -289 + 578 - 100 \][/tex]
[tex]\[ \pi = 189 \][/tex]

So, the maximum profit is 189.

### Summary of Results:
A) Total Revenue (TR) = [tex]\( 20Q \)[/tex]

B) Average Cost (AC) = [tex]\( 4Q - 48 + \frac{100}{Q} \)[/tex]

C) Marginal Cost (MC) = [tex]\( 8Q - 48 \)[/tex]

D) Profit ([tex]\(\pi\)[/tex]) = [tex]\( -4Q^2 + 68Q - 100 \)[/tex]

E) Profit-maximizing level of output = [tex]\( Q = 8.5 \)[/tex]

F) Maximum profit = 189

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