To forecast the sales for year 6 using the trend projection method (linear regression), we need to perform the following steps:

Step 1: Set up the linear regression equation

The linear regression equation can be written as: $Y = a + bX$ Where:

• $Y$ is the sales for a given year.
• $X$ is the year number.
• $a$ is the y-intercept.
• $b$ is the slope of the line.

Step 2: Calculate the slope (b) and the y-intercept (a)

The formulas to calculate the slope $b$ and the intercept $a$ are: $b = \frac{n(\sum XY) - (\sum X)(\sum Y)}{n(\sum X^2) - (\sum X)^2}$ $a = \frac{\sum Y - b(\sum X)}{n}$ Where:

• $n$ is the number of data points.
• $\sum X$ is the sum of the years.
• $\sum Y$ is the sum of the sales.
• $\sum XY$ is the sum of the product of the year and sales.
• $\sum X^2$ is the sum of the squares of the years.

Let's compute these values with the given data:

• Years: $X = [1, 2, 3, 4, 5]$
• Sales: $Y = [450, 495, 520, 555, 595]$

Step 3: Perform the calculations

• $\sum X = 1 + 2 + 3 + 4 + 5 = 15$
• $\sum Y = 450 + 495 + 520 + 555 + 595 = 2615$
• $\sum X^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55$
• $\sum XY = (1 \times 450) + (2 \times 495) + (3 \times 520) + (4 \times 555) + (5 \times 595) = 450 + 990 + 1560 + 2220 + 2975 = 8195$

Now, substitute these values into the formulas for $b$ and $a$:

$b = \frac{5(8195) - (15)(2615)}{5(55) - (15)^2} = \frac{40975 - 39225}{275 - 225} = \frac{1750}{50} = 35$ $a = \frac{2615 - 35(15)}{5} = \frac{2615 - 525}{5} = \frac{2090}{5} = 418$

Step 4: Forecast the sales for year 6

Now that we have the equation $Y = 418 + 35X$, we can forecast the sales for year 6 by substituting $X = 6$ into the equation: $Y = 418 + 35(6) = 418 + 210 = 628$

Conclusion:

The forecasted sales for year 6 using the trend projection method (linear regression) is 628 sales.

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Would you like further explanation or more examples? Here's a list of related questions for further learning:

1. How would you calculate the coefficient of determination (R²) for this linear regression model?
2. How would you handle outliers in a linear regression analysis?
3. What is the difference between linear and nonlinear regression methods?
4. How do you interpret the slope of the regression line in the context of this sales data?
5. Can you use polynomial regression for this data, and how would the forecast change?

Tip: It's helpful to visualize your data and the regression line on a scatter plot to check for any noticeable patterns or outliers.