A large bicycle retailer collects data on the number of bicycles in each store compared to floor space of each store. The data is given in the table below.

Number of Bicycles

60

56

208

52

70

55
Floor Space (sq ft)

1400

1140

3250

1100

1500

1280

a) Determine the equation of the linear regression function.
b) What do the slope and x-intercept represent in this context?
c) Estimate the number of bicycles in a store with 2500 sq ft of floor space.

wheres the answer

a) To determine the equation of the linear regression function, we need to find the values of the slope and y-intercept.

First, let's calculate the mean of both the number of bicycles and the floor space:

Mean of Number of Bicycles (y-bar) = (60 + 56 + 208 + 52 + 70 + 55) / 6 = 401 / 6 = 66.83 (rounded to two decimal places)

Mean of Floor Space (x-bar) = (1400 + 1140 + 3250 + 1100 + 1500 + 1280) / 6 = 8670 / 6 = 1445 (rounded to the nearest whole number)

Next, calculate the sum of the products of each pair of data points:

Σ(x*y) = (1400 * 60) + (1140 * 56) + (3250 * 208) + (1100 * 52) + (1500 * 70) + (1280 * 55)
= 84,000 + 63,840 + 676,000 + 57,200 + 105,000 + 70,400
= 1,056,440

Next, calculate the sum of the squares of each floor space measurement:

Σ(x^2) = (1400^2) + (1140^2) + (3250^2) + (1100^2) + (1500^2) + (1280^2)
= 1,960,000 + 1,299,600 + 10,562,500 + 1,210,000 + 2,250,000 + 1,638,400
= 18,920,500

Now, calculate the slope (b) using the formula:

b = (Σ(x*y) - (n * x-bar * y-bar)) / (Σ(x^2) - (n * x-bar^2))

n = number of data points = 6

b = (1,056,440 - (6 * 1445 * 66.83)) / (18,920,500 - (6 * 1445^2))

b = (1,056,440 - 597,939.9) / (18,920,500 - 1,576,790)

b = 458,500.1 / 17,343,710.1

b ≈ 0.026457

Now, calculate the y-intercept (a) using the formula:

a = y-bar - (b * x-bar)

a = 66.83 - (0.026457 * 1445)

a ≈ 28.876

Therefore, the equation of the linear regression function is:

Number of Bicycles = 28.876 + 0.026457 * Floor Space

b) In this context:
- The slope (0.026457) represents the average increase in the number of bicycles for each additional square foot of floor space. It shows the rate at which the number of bicycles increases with an increase in floor space.
- The y-intercept (28.876) represents the estimated number of bicycles when the floor space is zero. However, since floor space cannot be zero in this context, the y-intercept has no practical meaning in this specific case.

c) To estimate the number of bicycles in a store with 2500 sq ft of floor space, we can substitute the value of Floor Space (x) into the linear regression equation and solve for the Number of Bicycles (y).

Number of Bicycles = 28.876 + 0.026457 * 2500
= 28.876 + 66.142125
≈ 95.02

Therefore, the estimated number of bicycles in a store with 2500 sq ft of floor space is approximately 95.

To determine the equation of the linear regression function, we need to find the equation of the line that best fits the given data points. We can do this using the least squares method. Here are the steps to calculate the linear regression function:

Step 1: Calculate the mean of the number of bicycles and the mean of the floor space. Let's call these means x̄ and ȳ.

x̄ = (60 + 56 + 208 + 52 + 70 + 55) / 6 = 100.17
ȳ = (1400 + 1140 + 3250 + 1100 + 1500 + 1280) / 6 = 1573.33

Step 2: Calculate the deviations from the means for both variables. Let's call these deviations dx and dy.

dx = (60 - 100.17, 56 - 100.17, 208 - 100.17, 52 - 100.17, 70 - 100.17, 55 - 100.17) = (-40.17, -44.17, 107.83, -48.17, -30.17, -45.17)
dy = (1400 - 1573.33, 1140 - 1573.33, 3250 - 1573.33, 1100 - 1573.33, 1500 - 1573.33, 1280 - 1573.33) = (-173.33, -433.33, 1676.67, -473.33, -73.33, -293.33)

Step 3: Calculate the product of the deviations dx * dy. Let's call this product dxy.

dxy = (-40.17 * -173.33, -44.17 * -433.33, 107.83 * 1676.67, -48.17 * -473.33, -30.17 * -73.33, -45.17 * -293.33)
= (6966.461, 19121.119, 180681.361, -22820.258, 2211.961, 13250.868)

Step 4: Calculate the square of the deviations dx^2. Let's call this square dx^2.

dx^2 = (-40.17)^2, (-44.17)^2, (107.83)^2, (-48.17)^2, (-30.17)^2, (-45.17)^2
= (1613.2289, 1950.7089, 11627.5289, 2318.0489, 910.5289, 2041.8489)

Step 5: Calculate the slope of the linear regression line, b.

b = Σ(dxy) / Σ(dx^2)
= (6966.461 + 19121.119 + 180681.361 + -22820.258 + 2211.961 + 13250.868) / (1613.2289 + 1950.7089 + 11627.5289 + 2318.0489 + 910.5289 + 2041.8489)
= 1953.1229 / 19861.8845
= 0.0983 (approx.)

Step 6: Calculate the y-intercept of the linear regression line, a.

a = ȳ - b * x̄
= 1573.33 - 0.0983 * 100.17
= 1563.54

Therefore, the equation of the linear regression function is:

Number of Bicycles = 1563.54 + 0.0983 * Floor Space (sq ft)

Now let's move on to the next questions:

b) The slope of the linear regression line (0.0983) represents the average increase in the number of bicycles corresponding to a one-unit increase in the floor space. In other words, for every additional square foot of floor space, the expected increase in the number of bicycles is 0.0983. The y-intercept (1563.54) represents the estimated number of bicycles when the floor space is zero, although in this context, it may not have practical significance as a store with a floor space of zero is not possible.

c) To estimate the number of bicycles in a store with 2500 sq ft of floor space, substitute 2500 into the equation:

Number of Bicycles = 1563.54 + 0.0983 * 2500
= 1563.54 + 245.75
= 1809.29

Therefore, the estimated number of bicycles in a store with 2500 sq ft of floor space is approximately 1809.29.