The following data represnt x = boat sales and y = boat trailer sales from 1995-2000.

Year
1995
1996
1997
1998
1999
2000

Boat Sales
649
619
596
576
585
574

Boat Trailer Sales
207
194
181
174
168
159

a. Determine the least squares regression line and interpret its slope.

b. Estimate, for a year during which 500,000 boats are sold, the number of boat trailers that would be sold.

c. What reasons might explain why the number of boat trailers sold per year is less than the number of boats sold per year?

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a. .3

a. To determine the least squares regression line, you need to calculate the slope and the y-intercept.

Step 1: Calculate the means of both the x and y values:
- Calculate the mean of the boat sales (x) values: (649 + 619 + 596 + 576 + 585 + 574) / 6 = 601.5
- Calculate the mean of the boat trailer sales (y) values: (207 + 194 + 181 + 174 + 168 + 159) / 6 = 182.17

Step 2: Calculate the Covariance (Cov) of x and y:
- Cov(x, y) = Σ((xᵢ - mean(x)) * (yᵢ - mean(y))) / (n - 1)
- Cov(x, y) = ((649-601.5) * (207-182.17) + (619-601.5) * (194-182.17) + (596-601.5) * (181-182.17) + (576-601.5) * (174-182.17) + (585-601.5) * (168-182.17) + (574-601.5) * (159-182.17)) / (6-1)
- Cov(x, y) = -715.17

Step 3: Calculate the variance (Var) of x:
- Var(x) = Σ((xᵢ - mean(x))²) / (n - 1)
- Var(x) = ((649-601.5)² + (619-601.5)² + (596-601.5)² + (576-601.5)² + (585-601.5)² + (574-601.5)²) / (6-1)
- Var(x) = 1502.5

Step 4: Calculate the slope (β₁) of the regression line:
- β₁ = Cov(x, y) / Var(x)
- β₁ = -715.17 / 1502.5
- β₁ ≈ -0.476

Step 5: Calculate the y-intercept (β₀) of the regression line:
- β₀ = mean(y) - β₁ * mean(x)
- β₀ = 182.17 - (-0.476) * 601.5
- β₀ ≈ 435.33

Therefore, the least squares regression line can be expressed as: y = 435.33 - 0.476x.

Interpretation of slope (β₁): The slope of the regression line represents the change in the number of boat trailer sales (y) for a one-unit change in boat sales (x). In this case, for every additional boat sold, the number of boat trailer sales decreases by approximately 0.476.

b. To estimate the number of boat trailers that would be sold for a year during which 500,000 boats are sold, you can plug in the value of x (boat sales) into the equation of the regression line.

y = 435.33 - 0.476x
y = 435.33 - 0.476 * 500,000

Calculate the value of y:
y ≈ 435.33 - 238,000
y ≈ 197.33

Therefore, for a year with 500,000 boat sales, it is estimated that around 197 boat trailers would be sold.

c. There can be several reasons why the number of boat trailers sold per year is less than the number of boats sold per year:

1. Boats may be sold without the need for a boat trailer: Some boat buyers may already have boat trailers or may choose not to purchase a boat trailer separately. They may already have their own means of transportation or storage for the boats.

2. Different usage patterns: Not all boats require trailers, particularly small boats or those used primarily on lakes or other nearby water bodies. These boats may be launched directly from the shore without needing a trailer.

3. Limited storage space: Boat trailers require additional storage space, which may be limited or expensive for some boat owners. They may opt for alternative storage methods, such as marinas or boatyards, where trailers are not necessary.

4. Cost considerations: Boat trailers can be an additional cost for boat owners, including purchase, maintenance, and registration fees. Some boat owners may choose not to incur the extra expense and use other means for transportation and storage.

5. Accessibility to water bodies: In some cases, boat owners may not need a trailer because they have easy access to water bodies through marinas or other facilities. They can simply launch their boats from these locations without the need for a trailer.

It's important to note that these reasons may vary based on individual preferences, geographical location, boat size, and specific usage requirements.