To find the estimated y-intercept ($b_0$) of the regression line, we need to follow these steps:

1. Calculate the regression coefficients

The equation of the regression line is given as:

$\hat{y} = b_0 + b_1x$

Where:

• $b_0$ is the y-intercept (what we're solving for),
• $b_1$ is the slope of the regression line,
• $\hat{y}$ is the predicted bone density,
• $x$ is the age.

Step 1: Compute the slope $b_1$ using the formula:

$b_1 = \frac{n\sum{x_iy_i} - \sum{x_i}\sum{y_i}}{n\sum{x_i^2} - (\sum{x_i})^2}$

Where:

• $n$ is the number of data points (5 in this case),
• $x_i$ and $y_i$ are the values of age and bone density for each data point.

Step 2: Calculate the y-intercept $b_0$ using the formula:

$b_0 = \frac{\sum{y_i} - b_1 \sum{x_i}}{n}$

Now, let's break down the data:

• $x = [42, 51, 55, 64, 65]$
• $y = [355, 329, 328, 324, 312]$

We need to compute the following sums:

• $\sum{x_i}$
• $\sum{y_i}$
• $\sum{x_iy_i}$
• $\sum{x_i^2}$

Let me calculate all of this now. numpy as np

Data

x = np.array([42, 51, 55, 64, 65]) y = np.array([355, 329, 328, 324, 312])

Calculations

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

Calculate b1 (slope)

b1 = (n * sum_xy - sum_x * sum_y) / (n * sum_x_squared - sum_x ** 2)

Calculate b0 (y-intercept)

b0 = (sum_y - b1 * sum_x) / n

b0, b1The estimated y-intercept ($b_0$) of the regression line is approximately 413.671 (rounded to three decimal places).

If you need more details or further calculations, feel free to ask!

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Here are some related questions that may help you dive deeper into this concept:

1. How do you interpret the slope ($b_1$) in the context of this problem?
2. What does the correlation coefficient tell us about the relationship between age and bone density?
3. How do you calculate the standard error of the regression line?
4. Why is it important to check the statistical significance of the correlation coefficient before using a regression model for prediction?
5. What assumptions must be met for linear regression to be appropriate in this scenario?

Tip: To interpret the y-intercept meaningfully, think about what the predicted bone density would be if the woman’s age were zero. However, since age zero doesn’t make sense in this context, it's mainly useful for calculations.