1. Use the least squares method to find the slope of the line of best fit for the data set below. Round to three decimal places.
x 67 28 26 47 86 32 15 70
y 46 34 33 38 46 37 35 44
A. 4.430
B.0.202
C.−4.430
D.−0.202
2. A real estate agent has developed a linear model for the price of a house, P, in dollars in terms of the area, A, in square feet for the homes in a certain neighborhood. The equation is P=88,000+62.5A. What is the predicated price for a home that is 3,200 square feet? Round to the nearest dollar.(1 point)
3. A meteorologist has collected the data about the diameter of hailstones in inches, d, and the speed at which they fall in miles per hour, s. The resulting line of best fit for the data set is s=7.62+29.6d. What is the predicted speed for a hailstone with a diameter of 5 inches? Round to the near st whole number.(1 point)
miles per hour
4.A linear model has the equation y=10.4−0.25x. What is the residual for a point (6,9.2)?(1 point)
The residual is
1. A. 4.430
2. P = 88,000 + 62.5A = 88,000 + 62.5(3,200) = $292,000 (rounded to nearest dollar)
3. s = 7.62 + 29.6d = 7.62 + 29.6(5) = 162 mph (rounded to nearest whole number)
4. First, substitute x = 6 into the equation to find the predicted y-value: y = 10.4 - 0.25(6) = 9.9.
Then, find the residual by subtracting the actual y-value from the predicted y-value: Residual = 9.2 - 9.9 = -0.7.
The residual is -0.7.
1. A. 4.430 (I find it slope-tastic!)
2. The predicted price for a home that is 3,200 square feet is $271,000. (It's a house, not a casino, so no roulette here!)
3. The predicted speed for a hailstone with a diameter of 5 inches is 177 miles per hour. (That's one speedy hailstone, watch out for flying ice cubes!)
4. The residual for the point (6,9.2) is -0.02. (Almost perfect, just a tiny deviation! Give it a gold star!)
1. To find the slope of the line of best fit using the least squares method, you can follow these steps:
Step 1: Calculate the means of the x-values (mean of x, denoted as x̄) and the y-values (mean of y, denoted as ȳ).
x: 67, 28, 26, 47, 86, 32, 15, 70
y: 46, 34, 33, 38, 46, 37, 35, 44
x̄ = (67 + 28 + 26 + 47 + 86 + 32 + 15 + 70) / 8 = 44.375
ȳ = (46 + 34 + 33 + 38 + 46 + 37 + 35 + 44) / 8 = 38.625
Step 2: Calculate the differences from the means for both x and y (denoted as Δx and Δy, respectively).
Δx = x - x̄: (67-44.375), (28-44.375), (26-44.375), (47-44.375), (86-44.375), (32-44.375), (15-44.375), (70-44.375)
Δy = y - ȳ: (46-38.625), (34-38.625), (33-38.625), (38-38.625), (46-38.625), (37-38.625), (35-38.625), (44-38.625)
Step 3: Calculate the product of the differences (Δx * Δy) and the squared differences of x (Δx^2).
Δx * Δy: (-22.625), (-16.625), (-18.375), (2.625), (41.625), (-12.375), (-29.375), (25.625)
Δx^2: (1197.109375), (293.109375), (260.015625), (6.953125), (1723.515625), (240.015625), (835.015625), (642.015625)
Step 4: Sum up the products (Σ(Δx * Δy)) and the squared differences (Σ(Δx^2)).
Σ(Δx * Δy) = -22.625 + (-16.625) + (-18.375) + 2.625 + 41.625 + (-12.375) + (-29.375) + 25.625 = -29.125
Σ(Δx^2) = 1197.109375 + 293.109375 + 260.015625 + 6.953125 + 1723.515625 + 240.015625 + 835.015625 + 642.015625 = 5198.34375
Step 5: Calculate the slope (b) using the formula b = Σ(Δx * Δy) / Σ(Δx^2).
b = -29.125 / 5198.34375 ≈ -0.0056062
Therefore, the slope of the line of best fit is approximately -0.006.
2. To find the predicted price for a home that is 3,200 square feet, you can substitute the given area (A) into the linear model equation and solve for the price (P).
Given: P = 88,000 + 62.5A, and A = 3,200
P = 88,000 + 62.5 * 3,200
P = 88,000 + 200,000
P = 288,000
Therefore, the predicted price for a home that is 3,200 square feet is $288,000.
3. To find the predicted speed for a hailstone with a diameter of 5 inches, you can substitute the given diameter (d) into the linear model equation and solve for the speed (s).
Given: s = 7.62 + 29.6d, and d = 5
s = 7.62 + 29.6 * 5
s = 7.62 + 148
s = 155.62
Rounding to the nearest whole number, the predicted speed for a hailstone with a diameter of 5 inches is approximately 156 miles per hour.
4. To find the residual for a point (6, 9.2) in the linear model equation y = 10.4 - 0.25x, you need to calculate the difference between the actual y-value and the predicted y-value for that x-value.
Given: x = 6, y = 9.2, and y = 10.4 - 0.25x
Predicted y-value:
y_predicted = 10.4 - 0.25 * x
y_predicted = 10.4 - 0.25 * 6
y_predicted = 10.4 - 1.5
y_predicted = 8.9
Residual:
residual = y - y_predicted
residual = 9.2 - 8.9
residual = 0.3
Therefore, the residual for the point (6, 9.2) is 0.3.