To estimate the $y$-value using a linear model, we first calculate the equation of the line $y = mx + b$ based on the data provided. Here are the steps:

Step 1: Compute the slope ($m$)

The formula for the slope is:

$m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

Step 2: Compute the intercept ($b$)

The formula for the intercept is:

$b = \bar{y} - m\bar{x}$

Step 3: Predict $y$

Once $m$ and $b$ are calculated, substitute the given $x$-value into the linear equation:

$y = mx + b$

Let’s calculate these values. numpy as np

Data

x = np.array([2, 3, 4, 5, 6, 7, 8, 9]) y = np.array([21.02, 27.43, 28.74, 30.05, 31.66, 33.97, 36.48, 39.99])

Calculate mean of x and y

x_mean = np.mean(x) y_mean = np.mean(y)

Calculate slope (m)

m = np.sum((x - x_mean) * (y - y_mean)) / np.sum((x - x_mean) ** 2)

Calculate intercept (b)

b = y_mean - m * x_mean

Define the linear model

def predict_y(x_val): return m * x_val + b

Predict y values for x = 5 and x = 7 (as examples)

y_for_5 = predict_y(5) y_for_7 = predict_y(7)

m, b, y_for_5, y_for_7The equation of the linear model is:

$y = 2.3255x + 18.3774$

Using this model:

1. For $x = 5$: $y = 2.3255(5) + 18.3774 = 30.00$

2. For $x = 7$: $y = 2.3255(7) + 18.3774 = 34.66$

Let me know if you'd like a more detailed breakdown or calculations for different $x$-values.

5 Related Questions:

1. How can we verify if the linear model is a good fit for this dataset?
2. What is the interpretation of the slope and intercept in this context?
3. Can this model be used to predict values outside the provided $x$-range? Why or why not?
4. How would we calculate the residuals for this model?
5. How does the variability in $x$ and $y$ affect the accuracy of the model?

Tip:

Always check the correlation coefficient to evaluate how well a linear model fits the data!