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Find the equation of trend and predict the value of sale in the 4th quarter of 2024

To find the equation of the trend and predict the value of sales in the 4th quarter of 2024, we need to use linear regression based on the data provided. The linear equation for the trend is typically of the form:

\[
y = a + bx
\]

Where:
- \( y \) is the sales (dependent variable),
- \( x \) is the time period (independent variable),
- \( a \) is the y-intercept (the value of \( y \) when \( x = 0 \)),
- \( b \) is the slope of the line (the rate of change in sales per unit of time).

The formulas for \( a \) and \( b \) in simple linear regression are:

\[
b = \frac{n \sum xy - (\sum x)(\sum y)}{n \sum x^2 - (\sum x)^2}
\]
\[
a = \frac{\sum y - b(\sum x)}{n}
\]

From the table provided:
- \( \sum x = 36 \)
- \( \sum y = 42 \)
- \( \sum xy = 192 \)
- \( \sum x^2 = 204 \)
- \( n = 8 \) (since there are 8 data points: 4 from 2022 and 4 from 2023)

### Step 1: Calculate the slope \( b \)
\[
b = \frac{8(192) - (36)(42)}{8(204) - (36)^2}
\]

### Step 2: Calculate the intercept \( a \)
\[
a = \frac{42 - b(36)}{8}
\]

### Step 3: Predict the sales for the 4th quarter of 2024
For 2024, \( x \) will continue as follows:
- For the 1st quarter of 2024, \( x = 9 \),
- For the 2nd quarter of 2024, \( x = 10 \),
- For the 3rd quarter of 2024, \( x = 11 \),
- For the 4th quarter of 2024, \( x = 12 \).

Substitute \( x = 12 \) into the equation \( y = a + bx \) to predict the value for the 4th quarter of 2024.

I will now calculate these values.The equation of the trend is:

\[
y = 4.93 + 0.071x
\]

Using this equation, the predicted value of sales for the 4th quarter of 2024 (\( x = 12 \)) is approximately:

\[
y \approx 5.79
\]

This means the sales for the 4th quarter of 2024 are predicted to be around 5.79 units.

Would you like more details or have any questions? Here are some related questions you might find helpful:

1. How would you compute predictions for other quarters of 2024?
2. How can you interpret the slope and intercept in this context?
3. What is the significance of linear regression in sales forecasting?
4. How accurate is the trend line prediction for future values?
5. What would happen if we had more data points for 2024?

**Tip:** In linear regression, the slope tells you how much the dependent variable (sales) changes with every unit increase in the independent variable (time or quarters).

Find the equation of trend and predict the value of sale in the 4th quarter of 2024.

This problem demonstrates how to use linear regression to find the trend and predict the sales in the 4th quarter of 2024. It involves calculating the slope (b) and intercept (a) based on given sales data from 2022 and 2023. Key mathematical concepts include the least squares method and trend analysis.

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