Question 1

Given the data for Carats (-5, 6, 7, 8, 9, 1.0) and Price (2790, 3191, 3694, 4154, 5018, 5898), a linear regression is performed to estimate the relationship between carat and price. If the regression equation is $$y = mx + b$$, what is the value of $$m ($$slope) to two decimal places?

Accepted Answer

$$m = 654.80$$

Question 2

Using the regression equation from the previous question, what is the estimated price for a diamond with 1.1 carats?

Accepted Answer

$$	ext{Estimated price} = 5978$$

Question 3

Given the Price-Demand data: Demand = (14, 9, 14, 20), Price = (72, 63, 55, 33), what is the slope $$m of $$the best-fit line for Price as a function of Demand?

Accepted Answer

$$m = -1.9$$

Question 4

Using the Price-Demand regression from the previous question, what is the estimated demand when the price is $$50$$?

Accepted Answer

$$	ext{Demand} = 13.7$$

Question 5

The speed of sound is 1403 m/s at 0 degrees Celsius and 1481 m/s at 20 degrees Celsius. What is the linear equation for the speed of sound $$v as a $$function of temperature $$T$$?

Accepted Answer

$$v = 3.9T + 1403$$

Question 6

If an investor wants an 8% return and has the option to invest at 5% and 9%, what proportion should be invested at each rate to achieve exactly 8% return?

Accepted Answer

75% at 9%, 25% at 5%

Question 7

Given the cost function $$C(x) = 20x^2$ - 360x + 12000$$, what is the formula for the average cost $$A(x)$$?

Accepted Answer

$$A(x) = 20x - 360 + \frac{12000}{x}$$

Question 8

Find the derivative of the average cost function $$A(x) = 20x - 360 + \frac{12000}{x}.$$

Accepted Answer

$$A'(x) = 20 - \frac{12000}{x^2$}$$

Question 9

At what value of $$x$$ does the average cost $$A(x)$$ reach its minimum for $$C(x) = 20x^2$ - 360x + 12000$$?

Accepted Answer

$$x = 24$$

Question 10

If the minimum or maximum of a function occurs where its derivative is zero, what is the minimum value of the average cost $$A(x)$$ for $$C(x) = 20x^2$ - 360x + 12000$$?

Accepted Answer

$$A(24) = 140$$

Question 11

If $$cov(X, Y) = E[(X - E[X])(Y - E[Y])]$$, and $$b = \frac{cov(X, Y)}{cov(X)}$$, what does $$b$$ represent in the context of linear regression?

Accepted Answer

The slope of the regression line

Question 12

Suppose the demand for a product is modeled by a linear equation $$D = mP + b. If $$the price increases by $$1$$ unit, what happens to the demand if $$m is $$negative?

Accepted Answer

Demand decreases

Question 13

If the cost function is $$C(x) = 20x^2$ - 360x + 12000$$, what is the marginal cost at $$x = 10$$?

Accepted Answer

$$C'(10) = 40$$

Question 14

Given the regression equation $$y = mx + b$$, if $$m = 3.9$$ and $$b = 1403$$, what is the predicted value of $$y$$ when $$x = 15$$?

Accepted Answer

$$y = 1461.5$$

Question 15

If the covariance $$cov(X, Y) is $$positive, what does this indicate about the relationship between $$X$$ and $$Y$$?

Accepted Answer

As $$X$$ increases, $$Y$$ tends to increase

Mathematical Applications in Economics and Physics: Linear Equations, Demand, and Optimization

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