Question

An agent for a residential real estate company has the business objective of developing more accurate estimates of the monthly rental cost of apartments. Toward that goal, the agent would like to use the size of the apartment as defined by square footage to predict monthly rental cost. The agent selects a sample of 100 one-bedroom apartments and collects the data

-Predict the mean monthly rent for an apartment that has 850 square feet.

- Construct a 95% confidence interval on the mean monthly rent for an apartment that has 850 square feet.

-Construct a 95% prediction interval on monthly rent for an apartment that has 850 square feet.

Result Details & Calculation

X Values
∑ = 84272.71
Mean = 842.727
∑(X - M_x)² = SS_x = 871329.292

Y Values
∑ = 126112.36
Mean = 1261.124

ŷ = 1.49683X - 0.2926
∑(Y - M_y)² = SS_y = 2785756.171

X and Y Combined
N = 100
∑(X - M_x)(Y - M_y) = 1304228.717

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 1304228.717 / √((871329.292)(2785756.171)) = 0.8371

Meta Numerics (cross-check)
r = 0.8371

Answer 1

X = 84272.71, Y = 126112.36, N=100, x = 842.727, Y = 1261. 124 SSX = E(X-Mx)^2 = 871329.292 SSy = E(Y-My)^2 = 2785756.171 SSxy = (X-Mx) (Y-My) = 1304228.717 Use following equations to fine y-intercept(Bo) and slope(B1) B1 = SSxy/SSxx = 1304228.717/871329.292 = 1.4968 Bo = ï - B1*X = 1261.124 -(1.4968)*(842.727) = -0.2698 Regression equation is: y = -0.2698 + 1.4968 х Total sum of square: SST = ssyy = 2785756.171 Regression sum of square: SSR = slope*sSxy = (1.4968)*(1304228.717) = 1304230.214 we know that SST = SSR + SSE Sum of squared error: SSE = SST - SSR = 2785756.171 - 1304230.214 = 1481525.957 Mean squared error: MSE = SSE/ (N-2) = 1481525.957/(100-2) = 15117.6118 Standard Error of estimate:01 = VMSE = V15117.6118 = 122.9537 predicted value for x = 850 y^ = -0.2698 + 1.4968 850 = 1272.0102

95% Confidence interval for predicted value y^ = 1272.0102: Margin of error of confidence interval(MEC): MEC = t(a/2, N-2)* sqrt(MSE*((1/N)+((xo-x)^2/SSx))) = 1.9845 * sqrt(15117.6118 ((1/100)+(( 850-842.727)^2/871329.292))) = 24.4741 y^ + MEC = 1272.0102 + 24.7441 (1247.2661 1296.4843) 95% Prediction interval for predicted value y^ = 1272.0102: Margin of error of prediction interval (MEC): MEP = t(a/2, N-2)* sqrt(MSE* (1+(1/n)+((xo-x)^2/55x))) = 1.9845 * sqrt(15117.6118 * (1+(1/100)+((850-842.727)^2/871329.292))) = 245.2259 y + MEP = 1272.0102 + 245.2259 (1026.7843, 1517.2361)