To calculate the standard error of the estimate (denoted as $s_e$) for this data set, we follow these steps:

Step 1: Calculate the Regression Line

The standard error of the estimate is related to the regression line equation: $y = a + bx$ where $b$ is the slope and $a$ is the y-intercept. The formula for $b$ (slope) and $a$ (intercept) are:

$b = \frac{n \sum{x_iy_i} - \sum{x_i}\sum{y_i}}{n \sum{x_i^2} - (\sum{x_i})^2}$ $a = \frac{\sum{y_i} - b \sum{x_i}}{n}$

Where:

• $x_i$ and $y_i$ are the data points for $x$ and $y$
• $n$ is the number of data points

Step 2: Compute the Predicted $y$-values

Once we have the regression equation $y = a + bx$, we can compute the predicted values of $y$ (denoted as $\hat{y}_i$) for each $x_i$.

Step 3: Compute the Residuals

The residuals are the differences between the observed and predicted values of $y$: $e_i = y_i - \hat{y}_i$

Step 4: Calculate $s_e$ (Standard Error of the Estimate)

The formula for $s_e$ is: $s_e = \sqrt{\frac{\sum{e_i^2}}{n - 2}}$ where $\sum{e_i^2}$ is the sum of the squared residuals and $n$ is the number of data points.

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I'll now perform these calculations step by step using the provided data to find $s_e$. Let me compute the necessary values. numpy as np

Data points

x = np.array([51, 55, 49, 79, 45, 37, 69, 45, 48]) y = np.array([150, 48, 24, 35, 36, 171, 15, 185, 27])

Number of data points

n = len(x)

Step 1: Calculate the sums needed for the regression coefficients

sum_x = np.sum(x) sum_y = np.sum(y) sum_x_squared = np.sum(x**2) sum_xy = np.sum(x * y)

Step 2: Calculate the slope (b) and intercept (a)

b = (n * sum_xy - sum_x * sum_y) / (n * sum_x_squared - sum_x**2) a = (sum_y - b * sum_x) / n

Step 3: Calculate the predicted y values (y-hat) and residuals (e_i)

y_pred = a + b * x residuals = y - y_pred

Step 4: Calculate the standard error of the estimate (s_e)

sum_squared_residuals = np.sum(residuals**2) s_e = np.sqrt(sum_squared_residuals / (n - 2))

s_eThe standard error of the estimate (denoted as $s_e$) for this data set is approximately 63.60425 (rounded to five decimal places).

Would you like further details on the calculations or any other aspect of the problem?

Here are some related questions you might find useful:

1. How do you calculate the correlation coefficient $r$ for this data set?
2. What does the slope $b$ represent in a regression analysis?
3. How can you interpret the residuals in a regression model?
4. What is the coefficient of determination ($R^2$) and how is it calculated?
5. How does the standard error of the estimate help assess the fit of the regression line?

Tip: To improve the fit of a regression model, you can try transforming the variables (e.g., log transformation) if the relationship between $x$ and $y$ seems non-linear.