To solve this problem, we need to follow a few steps to determine the regression line equation, $\hat{y} = b_0 + b_1x$, where $\hat{y}$ represents the predicted bone density based on age, $x$ is the age, $b_1$ is the slope of the regression line, and $b_0$ is the y-intercept.

Step 1: Find the Slope $b_1$

The formula for the slope of the regression line $b_1$ is:

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

Where:

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

We first need the following calculations:

• $\sum x$ = 40 + 61 + 62 + 68 + 69 = 300
• $\sum y$ = 357 + 350 + 343 + 340 + 315 = 1705
• $\sum xy$ = (40 \times 357) + (61 \times 350) + (62 \times 343) + (68 \times 340) + (69 \times 315) = 14280 + 21350 + 21266 + 23120 + 21735 = 101751
• $\sum x^2$ = 40^2 + 61^2 + 62^2 + 68^2 + 69^2 = 1600 + 3721 + 3844 + 4624 + 4761 = 18550

Now, substitute these values into the formula for $b_1$:

$b_1 = \frac{5(101751) - (300)(1705)}{5(18550) - (300)^2}$

$b_1 = \frac{508755 - 511500}{92750 - 90000}$

$b_1 = \frac{-2745}{2750} \approx -0.9982$

Thus, the slope $b_1 \approx -0.9982$.

Step 2: Find the Y-Intercept $b_0$

The formula for the y-intercept is:

$b_0 = \frac{\sum y}{n} - b_1 \left( \frac{\sum x}{n} \right)$

Substitute the values:

$b_0 = \frac{1705}{5} - (-0.9982) \left( \frac{300}{5} \right)$

$b_0 = 341 - (-0.9982)(60)$

$b_0 = 341 + 59.892 = 400.892$

Thus, the y-intercept $b_0 \approx 400.892$.

Step 3: Write the Regression Equation

Now, substitute $b_0$ and $b_1$ into the regression equation:

$\hat{y} = 400.892 - 0.9982x$

Step 4: Change in $y$ When $x$ Increases by 1

The slope $b_1$ represents the change in the predicted bone density for every 1-year increase in age. Since $b_1 = -0.9982$, we conclude that for every additional year in age, the predicted bone density decreases by approximately 0.9982 units.

Final Result:

The regression equation is:

$\hat{y} = 400.892 - 0.9982x$

If the value of the independent variable (age) increases by 1 year, the dependent variable (bone density) decreases by approximately 0.9982 units.

Do you want more details or have any questions about this process?

Here are 5 related questions to explore further:

1. How do we test the significance of the correlation coefficient?
2. What are the assumptions behind linear regression?
3. How do residuals influence the accuracy of the regression model?
4. Can we predict bone density for ages outside of the given range?
5. How can we interpret the y-intercept in this context?

Tip: Always check the statistical significance of the slope (using hypothesis testing) before making predictions with a regression model.