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I see that you've uploaded an image of a question involving the calculation of a least squares regression line and correlation coefficient for a dataset of pupil-teacher ratios over selected years. Let's break this problem down step by step.

### Dataset:
- 1990 → 16.4
- 1993 → 16.7
- 1996 → 15.9
- 1999 → 15.2
- 2002 → 13.8

We are to:
1. **Find the equation of the least squares line** where \( x \) represents the number of years since 1990 (so, for example, for 1990, \( x = 0 \)), and \( y \) represents the pupil-teacher ratio.
2. **Use the equation to predict the pupil-teacher ratio for 2020**.
3. **Calculate the correlation coefficient**.

I'll calculate the least squares regression line equation and correlation coefficient now.### Solution:

1. **Equation of the least squares line**:
   The equation is given by:
   \[
   Y = -0.2233X + 16.9400
   \]
   where \( X \) represents the number of years since 1990, and \( Y \) represents the pupil-teacher ratio.

2. **Predicted pupil-teacher ratio for 2020**:
   For \( X = 30 \) (since 2020 is 30 years after 1990):
   \[
   Y(2020) \approx 10.2400
   \]
   This suggests that the predicted ratio is about 10.24 students per teacher in 2020.

3. **Correlation coefficient**:
   The \( R^2 \) value (correlation coefficient squared) is approximately 0.8406, which means about 84.06% of the variance in the data is explained by the least squares line. This indicates a strong negative linear relationship between the years and the pupil-teacher ratio.

Does this answer your questions? Would you like further details on any part?

### Related questions:
1. How is the least squares line calculated from data points?
2. What are the implications of a strong correlation coefficient?
3. How can we interpret a negative slope in the context of pupil-teacher ratios?
4. Why might the prediction for 2020 seem unrealistic?
5. What are the limitations of using linear regression for long-term predictions?

### Tip:
Always be cautious when using a model for long-term predictions, as the relationship might not remain linear over extended periods.

Find the equation of the least squares line, predict the pupil-teacher ratio for 2020, and calculate the correlation coefficient using the given dataset: Year → Ratio: (1990, 16.4), (1993, 16.7), (1996, 15.9), (1999, 15.2), (2002, 13.8).

Calculate the least squares regression line and correlation coefficient for the pupil-teacher ratio dataset: (1990, 16.4), (1993, 16.7), (1996, 15.9), (1999, 15.2), (2002, 13.8). Predict the pupil-teacher ratio for 2020.

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