To calculate the estimated y-intercept $b_0$ for the regression line equation $\hat{y} = b_0 + b_1 x$, we need to use the following formulas for linear regression:

1. The formula for $b_1$ (the slope) is:

$b_1 = \frac{n \sum x y - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$

2. The formula for $b_0$ (the y-intercept) is:

$b_0 = \frac{\sum y - b_1 \sum x}{n}$

Where:

• $x$ represents age
• $y$ represents bone density
• $n$ is the number of data points (in this case, $n = 5$)

Let's first calculate the necessary sums.

Data:

• Age (x): 40, 45, 53, 57, 63
• Bone Density (y): 350, 348, 341, 327, 324

Step 1: Calculate the sums

• $\sum x = 40 + 45 + 53 + 57 + 63 = 258$
• $\sum y = 350 + 348 + 341 + 327 + 324 = 1690$
• $\sum x^2 = 40^2 + 45^2 + 53^2 + 57^2 + 63^2 = 1600 + 2025 + 2809 + 3249 + 3969 = 14652$
• $\sum x y = (40 \times 350) + (45 \times 348) + (53 \times 341) + (57 \times 327) + (63 \times 324) = 14000 + 15660 + 18073 + 18639 + 20412 = 86784$

Step 2: Calculate the slope $b_1$

$b_1 = \frac{5 \times 86784 - (258 \times 1690)}{5 \times 14652 - (258)^2}$ $b_1 = \frac{433920 - 435720}{73260 - 66564}$ $b_1 = \frac{-1800}{6696}$ $b_1 \approx -0.269$

Step 3: Calculate the y-intercept $b_0$

$b_0 = \frac{1690 - (-0.269 \times 258)}{5}$ $b_0 = \frac{1690 + 69.462}{5}$ $b_0 = \frac{1759.462}{5}$ $b_0 \approx 351.892$

Final Answer:

The estimated y-intercept $b_0$ is approximately 351.892.

Would you like to go over the next steps, or have any questions regarding this calculation?

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Related Questions:

1. How do we interpret the meaning of the slope $b_1$ in the context of this data?
2. What is the significance of the y-intercept $b_0$ in the regression line equation?
3. How do we calculate the correlation coefficient for this data, and what does it tell us?
4. What would be the appropriate steps to check if the regression line is statistically significant?
5. How can we use the regression line to make predictions for women of different ages?

Tip:

When calculating regression coefficients by hand, it's easy to make mistakes in summing up values. Double-check your sums to avoid errors in the final result!