You provided the equation $y = ax + b$ with specific values for $a$ and $b$:

• $a = -21.6285714286$
• $b = 858.904761905$

This represents a linear equation where:

• $y$ is the dependent variable (in this case, the number of new cases),
• $x$ is the independent variable (the number of years since 2018).

Interpretation of the Equation

The equation $y = -21.6285714286x + 858.904761905$ suggests:

• Slope ($a$): The slope $a = -21.6285714286$ indicates that the number of new cases is decreasing by approximately 21.63 cases each year.
• Intercept ($b$): The intercept $b = 858.904761905$ represents the estimated number of new cases in the year when $x = 0$, which corresponds to the year 2018.

Application

Given this equation, you can calculate the predicted number of new cases for any year since 2018 by substituting the corresponding $x$ value into the equation.

Example Calculations:

1. For 2020 (2 years since 2018): $y = -21.6285714286(2) + 858.904761905 = -43.2571428572 + 858.904761905 = 815.6476190478$ Approximate number of cases = 816 (though this seems inconsistent with the trend in the data provided, which may indicate another model could be more appropriate).

2. For 2023 (5 years since 2018): $y = -21.6285714286(5) + 858.904761905 = -108.142857143 + 858.904761905 = 750.761904762$ Approximate number of cases = 751 (which is close to the actual data of 731).

Analysis

• The negative slope suggests a decreasing trend in the number of new cases over time. However, the original data seemed to show an increase, particularly between the 4th and 5th year. This discrepancy implies that a linear model might not perfectly fit the data, especially if the real-world trend is non-linear.

Would you like further analysis, such as evaluating the goodness of fit for this model, or perhaps fitting a different type of curve to the data?

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Here are some questions to consider:

1. How well does this linear model fit the actual data points?
2. What might explain the difference between the model's prediction and the actual data for the 5th year?
3. Would a quadratic or exponential model better capture the trend?
4. How can the rate of decrease (slope) be interpreted in a real-world context?
5. What other factors might be influencing the trend in crime cases that are not captured by this linear model?

Tip: Always consider plotting the data and the fitted model to visually inspect how well the model represents the actual data before drawing conclusions.