Question 2 (8 marks)
The following shows the sales for item X for year 2020 to 2024
Calculate the trend line.
year Sales, Y Code, x  𝑥2   xy
2020   2.1
2021   2.5
2022  2.7
2023  3.1
2024  3.7

To calculate the trend line for the sales data from 2020 to 2024, we'll use the least squares method to fit a linear regression line of the form:

\[ Y = a + bX \]

Where:
- \( Y \) is the sales
- \( X \) is the coded year value

### Steps to Calculate the Trend Line:

1. **Code the Years**: We'll code the years to simplify the calculations. Let \( X \) be the coded year such that 2020 is coded as 1, 2021 as 2, etc.
2. **Calculate the Sums**:
   - \(\sum X\) (sum of the coded years)
   - \(\sum Y\) (sum of the sales)
   - \(\sum XY\) (sum of the product of coded years and sales)
   - \(\sum X^2\) (sum of the squares of coded years)

3. **Calculate the coefficients**:
   - Slope \( b = \frac{n \sum XY - (\sum X)(\sum Y)}{n \sum X^2 - (\sum X)^2} \)
   - Intercept \( a = \frac{\sum Y - b \sum X}{n} \)

### Given Data:
\[
\begin{array}{|c|c|c|}
\hline
\text{Year} & \text{Sales (Y)} & \text{Code (X)} \\
\hline
2020 & 2.1 & 1 \\
2021 & 2.5 & 2 \\
2022 & 2.7 & 3 \\
2023 & 3.1 & 4 \\
2024 & 3.7 & 5 \\
\hline
\end{array}
\]

### Calculations:

\[
\begin{array}{|c|c|c|c|c|}
\hline
\text{Year} & \text{Sales (Y)} & \text{Code (X)} & X^2 & XY \\
\hline
2020 & 2.1 & 1 & 1 & 2.1 \\
2021 & 2.5 & 2 & 4 & 5.0 \\
2022 & 2.7 & 3 & 9 & 8.1 \\
2023 & 3.1 & 4 & 16 & 12.4 \\
2024 & 3.7 & 5 & 25 & 18.5 \\
\hline
\end{array}
\]

\[
\begin{align*}
\sum X & = 1 + 2 + 3 + 4 + 5 = 15 \\
\sum Y & = 2.1 + 2.5 + 2.7 + 3.1 + 3.7 = 14.1 \\
\sum X^2 & = 1 + 4 + 9 + 16 + 25 = 55 \\
\sum XY & = 2.1 + 5.0 + 8.1 + 12.4 + 18.5 = 46.1 \\
n & = 5 \\
\end{align*}
\]

### Calculate the slope \( b \):

\[
b = \frac{n \sum XY - (\sum X)(\sum Y)}{n \sum X^2 - (\sum X)^2} = \frac{5 \times 46.1 - 15 \times 14.1}{5 \times 55 - 15^2}
\]

\[
b = \frac{230.5 - 211.5}{275 - 225} = \frac{19}{50} = 0.38
\]

### Calculate the intercept \( a \):

\[
a = \frac{\sum Y - b \sum X}{n} = \frac{14.1 - 0.38 \times 15}{5}
\]

\[
a = \frac{14.1 - 5.7}{5} = \frac{8.4}{5} = 1.68
\]

### Trend Line Equation:

\[ Y = 1.68 + 0.38X \]

This is the trend line that represents the sales for item X from 2020 to 2024.

Would you like more details or have any questions?

### Related Questions:

1. How would you interpret the slope of the trend line in this context?
2. What would be the estimated sales in 2025 based on this trend line?
3. How would you calculate the residuals for each year?
4. What does the intercept represent in this trend line?
5. How can you determine if this trend line is a good fit for the data?
6. What other statistical methods can be used to analyze this data?
7. How would you handle outliers if they were present in the data?
8. Can this trend line be used to forecast future sales accurately?

### Tip:
Always check the goodness-of-fit measures, such as R-squared, to assess how well your trend line fits the data.

Learn how to calculate the trend line for sales data from 2020 to 2024 using the least squares method and linear regression. This problem involves statistical analysis of sales figures over multiple years, determining coefficients, and interpreting trend lines.

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