Year :1981 to 1992
No engage: 10,15,12,20,16,8,30,15,12,4,9,8
Using regression method find trend

To find the trend using a regression method, we'll use **linear regression**, which models the relationship between a dependent variable (in this case, "No engage") and an independent variable (the years, from 1981 to 1992).

### Step 1: Set up the data
We treat the years as numerical values starting from 1 (1981) to 12 (1992). The "No engage" values for the years will be used as the dependent variable.

- **Years** (independent variable \( x \)): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
- **No engage** (dependent variable \( y \)): 10, 15, 12, 20, 16, 8, 30, 15, 12, 4, 9, 8

### Step 2: Calculate the regression line
The regression line has the form:

\[
y = a + bx
\]

Where:
- \( y \) is the predicted value (trend),
- \( x \) is the independent variable (year),
- \( a \) is the intercept,
- \( b \) is the slope (trend per year).

To find \( a \) and \( b \), we use the formulas:

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

Where \( n \) is the number of data points, \( \sum x \) is the sum of the years, \( \sum y \) is the sum of "No engage", \( \sum xy \) is the sum of the product of corresponding years and "No engage" values, and \( \sum x^2 \) is the sum of squares of the years.

Let me calculate the values of \( a \) and \( b \).The equation of the trend line is:

\[
y = 16.73 - 0.53x
\]

Where:
- The intercept \( a = 16.73 \) represents the starting value when \( x = 0 \),
- The slope \( b = -0.53 \) indicates that the "No engage" decreases by approximately 0.53 units per year.

Would you like to explore further details or any questions on this result?

Here are some related questions for further understanding:
1. How is the slope related to the trend in data?
2. Can we use this regression model to predict values beyond 1992?
3. How do we interpret a negative slope in this context?
4. How would the trend change if some data points were modified?
5. What are the limitations of linear regression for this type of data?

**Tip**: Always plot your data to visually verify if a linear model is appropriate.

Learn how to find the trend in 'No engage' data from 1981 to 1992 using linear regression. This problem covers key statistical concepts, including calculating the regression line and interpreting the slope and intercept. Suitable for students in Grades 10-12.

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