The sales by Walker’s Milk and Dairy products, in millions of dollars, for the seven-year period from 2014 to 2020 are reported in the following table.

year=2014, sales=16,200,000
year=2015, sales=17,500,000
year=2016, sales=19,300,000
year=2017, sales=20,700,000
year=2018, sales=23,400,000
year=2019, sales=25,200,000
year=2020, sales=27,100,000

Determine the least squares trend equation. (Round the final answers in millions of dollars to 2 decimal places.)

To determine the least squares trend equation, we will follow these steps:

1. **Define the Variables:**
   - Let \( x \) be the number of years after 2014 (i.e., for 2014, \( x = 0 \); for 2015, \( x = 1 \), etc.).
   - Let \( y \) represent the sales in millions of dollars.

2. **Calculate the Necessary Sums:**
   - \( n \): the number of data points
   - \( \sum x \): the sum of the years (in the new scale where 2014 is \( x = 0 \))
   - \( \sum y \): the sum of sales
   - \( \sum xy \): the sum of the products of \( x \) and \( y \)
   - \( \sum x^2 \): the sum of the squares of the years

3. **Apply the Formulas for the Least Squares Regression Line:**
   - The least squares trend equation is given by \( y = a + bx \), where:
     - \( b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \)
     - \( a = \frac{\sum y - b\sum x}{n} \)

Let's proceed with the calculations:

### 1. Define the variables:

| Year | Sales (millions) | \( x \) | \( y \) | \( xy \) | \( x^2 \) |
|------|------------------|---------|---------|----------|-----------|
| 2014 | 16.20            | 0       | 16.20   | 0        | 0         |
| 2015 | 17.50            | 1       | 17.50   | 17.50    | 1         |
| 2016 | 19.30            | 2       | 19.30   | 38.60    | 4         |
| 2017 | 20.70            | 3       | 20.70   | 62.10    | 9         |
| 2018 | 23.40            | 4       | 23.40   | 93.60    | 16        |
| 2019 | 25.20            | 5       | 25.20   | 126.00   | 25        |
| 2020 | 27.10            | 6       | 27.10   | 162.60   | 36        |

### 2. Calculate the necessary sums:

- \( n = 7 \)
- \( \sum x = 0 + 1 + 2 + 3 + 4 + 5 + 6 = 21 \)
- \( \sum y = 16.20 + 17.50 + 19.30 + 20.70 + 23.40 + 25.20 + 27.10 = 149.40 \)
- \( \sum xy = 0 + 17.50 + 38.60 + 62.10 + 93.60 + 126.00 + 162.60 = 500.40 \)
- \( \sum x^2 = 0 + 1 + 4 + 9 + 16 + 25 + 36 = 91 \)

### 3. Apply the formulas for \( a \) and \( b \):

- \( b = \frac{7(500.40) - (21)(149.40)}{7(91) - (21)^2} \)
- \( a = \frac{149.40 - b(21)}{7} \)

Let's calculate \( b \) and \( a \).The least squares trend equation is:

\[
y = 15.75 + 1.86x
\]

Where:
- \( y \) is the sales in millions of dollars.
- \( x \) is the number of years after 2014.

### Do you have any questions or would you like more details?

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Here are some related questions:

1. How would you use this trend equation to predict future sales?
2. What are the limitations of using a linear trend model?
3. How would you calculate the sales for the year 2022 using this model?
4. How can you determine if this model fits the data well?
5. What is the significance of the slope (\( b \)) in the trend equation?
6. How would the trend equation change if the sales data had an outlier?
7. How does the sum of squares help in finding the best fit line?
8. How can you test the accuracy of this model with actual sales data?

**Tip:** Always plot your data and the trend line to visually check the fit of the model.

Learn how to calculate the least squares trend equation for Walker’s Milk and Dairy products sales from 2014 to 2020, using statistical methods. This problem involves finding the regression line to predict future sales based on historical data.

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