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StatisticsQ&A LibraryWe 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 μx and μy, variances σ2x and σ2y, 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. μw = aμx + bμyσ2w = a2σ2x + b2σ2y + 2abσxσyρ In this formula, r is the population correlation coefficient, theoretically computed using the population of all (x, y) data pairs. The expression σxσyρ is called the covariance of x and y. If x and y are independent, then ρ = 0 and the formula for σ2w 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. μx ≈ 7.34, σx ≈ 6.55, μy ≈ 13.19, σy ≈ 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 μw and risk σw.μw = σw = (c) Repeat part (b) if w = 0.25x + 0.75y.μw = σw = (d) Compare your results in parts (b) and (c). Which investment has the higher expected return? Which has the greater risk as measured by σw? w = 0.75x + 0.25y produces higher return with lower risk as measured by σw. w = 0.25x + 0.75y produces higher return with lower risk as measured by σw. w = 0.25x + 0.75y produces higher return with greater risk as measured by σw. Both investments produce the same return with the same risk as measured by σw. w = 0.75x + 0.25y produces higher return with greater risk as measured by σw.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.

(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*r*^{2}. 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?

%

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

(d) What percentage of the variation in

%

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 σ^{2}_{x} and σ^{2}_{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} = *a**μ*_{x} + *b**μ*_{y}

*σ*^{2}_{w} = *a*^{2}*σ*^{2}_{x} + *b*^{2}*σ*^{2}_{y} + 2*ab**σ*_{x}*σ*_{y}*ρ*
In this formula, *r* is the population correlation coefficient, theoretically computed using the population of all (*x*, *y*) data pairs. The expression σ_{x}σ_{y}ρ is called the *covariance* of *x* and *y*. If *x* and *y* are independent, then ρ = 0 and the formula for σ^{2}_{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.

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

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.

(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* = 0.75*x* + 0.25*y*. Estimate your expected percentage return μ_{w} and risk σ_{w}.

μ_{w} =

σ_{w} =

(c) Repeat part (b) if*w* = 0.25*x* + 0.75*y*.

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}.
*w* = 0.75x + 0.25y produces higher return with greater risk as measured by σ_{w}.

(a) Do you think the variables

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

μ

σ

(c) Repeat part (b) if

μ

σ

(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

Both investments produce the same return with the same risk as measured by σ_{w}.

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