To find the estimated linear regression model, we follow these steps:

Step 1: Compute the Means

We calculate the mean of the age ($x$) and the mean of the bone density ($y$):

$\bar{x} = \frac{42 + 51 + 55 + 64 + 65}{5} = \frac{277}{5} = 55.4$

$\bar{y} = \frac{355 + 329 + 328 + 324 + 312}{5} = \frac{1648}{5} = 329.6$

Step 2: Compute the Slope $b_1$

The formula for the slope is:

$b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

We compute each $(x_i - \bar{x})$ and $(y_i - \bar{y})$:

$x_i$	$y_i$	$x_i - \bar{x}$	$y_i - \bar{y}$	$(x_i - \bar{x})(y_i - \bar{y})$	$(x_i - \bar{x})^2$
42	355	-13.4	25.4	-340.36	179.56
51	329	-4.4	-0.6	2.64	19.36
55	328	-0.4	-1.6	0.64	0.16
64	324	8.6	-5.6	-48.16	73.96
65	312	9.6	-17.6	-168.96	92.16

$\sum (x_i - \bar{x})(y_i - \bar{y}) = -340.36 + 2.64 + 0.64 - 48.16 - 168.96 = -553.2$

$\sum (x_i - \bar{x})^2 = 179.56 + 19.36 + 0.16 + 73.96 + 92.16 = 365.2$

$b_1 = \frac{-553.2}{365.2} = -1.515$

Step 3: Compute the Intercept $b_0$

The intercept formula is:

$b_0 = \bar{y} - b_1 \bar{x}$

$b_0 = 329.6 - (-1.515 \times 55.4)$

$b_0 = 329.6 + 83.92 = 413.52$

Step 4: Regression Equation

$\hat{y} = 413.52 - 1.515x$

Step 5: Change in $\hat{y}$ when $x$ increases by 1

The coefficient $b_1 = -1.515$ represents the change in $\hat{y}$ for every one-unit increase in $x$.

Thus, when age increases by 1 year, bone density decreases by 1.515.

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Would you like further details on this calculation?

Here are five related questions for further exploration:

1. How do you determine if the correlation coefficient is statistically significant?
2. What does a negative slope imply in real-world scenarios?
3. How can we use this model to predict bone density at age 70?
4. What assumptions are made when using linear regression for predictions?
5. What are possible limitations of using this model for medical predictions?

Tip: Always check if the correlation coefficient is significant before using the regression equation for predictions!