We use the form

ŷ = a + bx

for the least-squares line. In some computer printouts, the least-squares equation is not given directly. Instead, the value of the constant a is given, and the coefficient b of the explanatory or predictor variable is displayed. Sometimes a is referred to as the constant, and sometimes as the intercept. Data from a report showed the following relationship between elevation (in thousands of feet) and average number of frost-free days per year in a state.

A Minitab printout provides the following information.

Predictor Coef SE Coef T P

Constant 315.27 28.31 11.24 0.002

Elevation -31.812 3.511 -8.79 0.003

S = 11.8603 R-Sq = 96.8%

Notice that "Elevation" is listed under "Predictor." This means that elevation is the explanatory variable x. Its coefficient is the slope b. "Constant" refers to a in the equation

ŷ = a + bx.

(a) Use the printout to write the least-squares equation.

ŷ =   +   x

(b) For each 1000-foot increase in elevation, how many fewer frost-free days are predicted? (Round your answer to three decimal places.)


(c) The printout gives the value of the coefficient of determination r². What is the value of r? Be sure to give the correct sign for r based on the sign of b. (Round your answer to four decimal places.)


(d) What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line?
  %

What percentage is unexplained?
  %

2. Previously, you studied linear combinations of independent random variables. What happens if the variables are not independent? A lot of mathematics can be used to prove the following: Let x and y be random variables with means μ_x and μ_y, variances σ²_x and σ²_y, and population correlation coefficient ρ (the Greek letter rho). Let a and b be any constants and let w = ax + by for the following formula.
μ_w = aμ_x + bμ_y
σ²_w = a²σ²_x + b²σ²_y + 2abσ_xσ_yρ
In this formula, r is the population correlation coefficient, theoretically computed using the population of all (x, y) data pairs. The expression σ_xσ_yρ is called the covariance of x and y. If x and y are independent, then ρ = 0 and the formula for σ²_w reduces to the appropriate formula for independent variables. In most real-world applications the population parameters are not known, so we use sample estimates with the understanding that our conclusions are also estimates.

Do you have to be rich to invest in bonds and real estate? No, mutual fund shares are available to you even if you aren't rich. Let x represent annual percentage return (after expenses) on the Vanguard Total Bond Index Fund, and let y represent annual percentage return on the Fidelity Real Estate Investment Fund. Over a long period of time, we have the following population estimates.
μ_x ≈ 7.34,      σ_x ≈ 6.55,      μ_y ≈ 13.19,      σ_y ≈ 18.57,      ρ ≈ 0.424

(a) Do you think the variables x and y are independent? Explain your answer.

Yes. Interest rates probably has no effect on the investment returns.

No. Interest rates probably has no effect on the investment returns.

No. Interest rate probably affects both investment returns.

Yes. Interest rate probably affects both investment returns.

(b) Suppose you decide to put 75% of your investment in bonds and 25% in real estate. This means you will use a weighted average w = 0.75x + 0.25y. Estimate your expected percentage return μ_w and risk σ_w.
μ_w =
σ_w =

(c) Repeat part (b) if w = 0.25x + 0.75y.
μ_w =
σ_w =
(d) Compare your results in parts (b) and (c). Which investment has the higher expected return? Which has the greater risk as measured by σ_w?

w = 0.75x + 0.25y produces higher return with lower risk as measured by σ_w.

w = 0.25x + 0.75y produces higher return with lower risk as measured by σ_w.

w = 0.25x + 0.75y produces higher return with greater risk as measured by σ_w.

Both investments produce the same return with the same risk as measured by σ_w.

w = 0.75x + 0.25y produces higher return with greater risk as measured by σ_w.

Question

We use the form

ŷ = a + bx

for the least-squares line. In some computer printouts, the least-squares equation is not given directly. Instead, the value of the constant a is given, and the coefficient b of the explanatory or predictor variable is displayed. Sometimes a is referred to as the constant, and sometimes as the intercept. Data from a report showed the following relationship between elevation (in thousands of feet) and average number of frost-free days per year in a state.

A Minitab printout provides the following information.

Predictor	Coef	SE Coef	T	P
Constant	315.27	28.31	11.24	0.002
Elevation	-31.812	3.511	-8.79	0.003
S = 11.8603	R-Sq = 96.8%

Notice that "Elevation" is listed under "Predictor." This means that elevation is the explanatory variable x. Its coefficient is the slope b. "Constant" refers to a in the equation

ŷ = a + bx.

(a) Use the printout to write the least-squares equation.

ŷ =

+ x

(b) For each 1000-foot increase in elevation, how many fewer frost-free days are predicted? (Round your answer to three decimal places.)


(c) The printout gives the value of the coefficient of determination r². What is the value of r? Be sure to give the correct sign for r based on the sign of b. (Round your answer to four decimal places.)


(d) What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line?
  %

What percentage is unexplained?
  %

2. Previously, you studied linear combinations of independent random variables. What happens if the variables are not independent? A lot of mathematics can be used to prove the following: Let x and y be random variables with means μ_x and μ_y, variances σ²_x and σ²_y, and population correlation coefficient ρ (the Greek letter rho). Let a and b be any constants and let w = ax + by for the following formula.

μ_w = aμ_x + bμ_y
σ²_w = a²σ²_x + b²σ²_y + 2abσ_xσ_yρ

In this formula, r is the population correlation coefficient, theoretically computed using the population of all (x, y) data pairs. The expression σ_xσ_yρ is called the covariance of x and y. If x and y are independent, then ρ = 0 and the formula for σ²_w reduces to the appropriate formula for independent variables. In most real-world applications the population parameters are not known, so we use sample estimates with the understanding that our conclusions are also estimates.

Do you have to be rich to invest in bonds and real estate? No, mutual fund shares are available to you even if you aren't rich. Let x represent annual percentage return (after expenses) on the Vanguard Total Bond Index Fund, and let y represent annual percentage return on the Fidelity Real Estate Investment Fund. Over a long period of time, we have the following population estimates.

μ_x ≈ 7.34, σ_x ≈ 6.55, μ_y ≈ 13.19, σ_y ≈ 18.57, ρ ≈ 0.424

(a) Do you think the variables x and y are independent? Explain your answer.

Yes. Interest rates probably has no effect on the investment returns.

No. Interest rates probably has no effect on the investment returns.

No. Interest rate probably affects both investment returns.

Yes. Interest rate probably affects both investment returns.

(b) Suppose you decide to put 75% of your investment in bonds and 25% in real estate. This means you will use a weighted average w = 0.75x + 0.25y. Estimate your expected percentage return μ_w and risk σ_w.
μ_w =
σ_w =

(c) Repeat part (b) if w = 0.25x + 0.75y.
μ_w =
σ_w =
(d) Compare your results in parts (b) and (c). Which investment has the higher expected return? Which has the greater risk as measured by σ_w?

w = 0.75x + 0.25y produces higher return with lower risk as measured by σ_w.

w = 0.25x + 0.75y produces higher return with lower risk as measured by σ_w.

w = 0.25x + 0.75y produces higher return with greater risk as measured by σ_w.

Both investments produce the same return with the same risk as measured by σ_w.

w = 0.75x + 0.25y produces higher return with greater risk as measured by σ_w.

Accepted Answer

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