Find the equation for the least squares line. Let x correspond to the number of years since 1991 and let y correspond to the average number of pupils per 1 teacher.

To find the least squares line (also known as the line of best fit), we follow a standard procedure that involves finding the slope (\( m \)) and intercept (\( b \)) of the line in the equation \( y = mx + b \), where \( x \) is the number of years since 1991 and \( y \) is the average number of pupils per teacher.

### Steps:
1. **Set up the points**: Assume you have a set of data points \( (x_1, y_1), (x_2, y_2), \dots, (x_n, y_n) \), where:
   - \( x_i \) is the number of years since 1991 (so \( x = \text{Year} - 1991 \)),
   - \( y_i \) is the average number of pupils per teacher in that corresponding year.

2. **Calculate the slope \( m \)**:
   The formula for the slope \( m \) is:
   \[
   m = \frac{n\sum{xy} - \sum{x} \sum{y}}{n\sum{x^2} - (\sum{x})^2}
   \]
   Where:
   - \( n \) is the number of data points,
   - \( \sum{xy} \) is the sum of the products of corresponding \( x \) and \( y \) values,
   - \( \sum{x} \) and \( \sum{y} \) are the sums of the \( x \) and \( y \) values,
   - \( \sum{x^2} \) is the sum of the squares of the \( x \)-values.

3. **Calculate the intercept \( b \)**:
   The formula for the intercept \( b \) is:
   \[
   b = \frac{\sum{y} - m\sum{x}}{n}
   \]
   This gives you the y-intercept, or the value of \( y \) when \( x = 0 \) (i.e., the year 1991).

4. **Plug into the equation**:
   Once you have both the slope \( m \) and intercept \( b \), the equation of the least squares line is:
   \[
   y = mx + b
   \]

Do you have specific data points available for the years and the average number of pupils per teacher? If you provide them, I can calculate the exact least squares line for you.

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Would you like further clarification or details? Here are 5 related questions to explore:
1. How does the least squares method minimize error in the fit of the line?
2. Can you find the least squares line for nonlinear data?
3. How does the correlation coefficient relate to the least squares method?
4. What are the assumptions behind using linear regression in this context?
5. How would the equation change if we adjusted the starting year?

**Tip**: Double-check your data points for consistency before applying the least squares formula. Even a small error in the data can affect the final result!

Learn how to find the equation of the least squares line for data where x corresponds to the number of years since 1991 and y represents the average number of pupils per teacher. This detailed guide walks through the steps of using the least squares method, calculating the slope and intercept, and forming the line equation.

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