Question: The table below gives the age and bone density for five randomly selected women. Using this data, consider the equation of the regression line, $$\hat{y} = b_0 + b_1 x$$, for predicting a woman's bone density based on her age. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant.
Age 35 42 50 54 65
Bone Density 358 354 340 319 311
1. **State the problem:** We want to find the regression line equation $$\hat{y} = b_0 + b_1 x$$ to predict bone density ($y$) from age ($x$) using the given data points:
$$x: 35, 42, 50, 54, 65$$
$$y: 358, 354, 340, 319, 311$$
2. **Calculate the means:**
$$\bar{x} = \frac{35 + 42 + 50 + 54 + 65}{5} = \frac{246}{5} = 49.2$$
$$\bar{y} = \frac{358 + 354 + 340 + 319 + 311}{5} = \frac{1682}{5} = 336.4$$
3. **Calculate the slope $$b_1$$:**
Formula:
$$b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$$
Calculate each term:
| $x_i$ | $y_i$ | $x_i - \bar{x}$ | $y_i - \bar{y}$ | $(x_i - \bar{x})(y_i - \bar{y})$ | $(x_i - \bar{x})^2$ |
|-------|-------|-----------------|-----------------|-------------------------------|-------------------|
| 35 | 358 | 35 - 49.2 = -14.2 | 358 - 336.4 = 21.6 | $-14.2 \times 21.6 = -306.72$ | $(-14.2)^2 = 201.64$ |
| 42 | 354 | -7.2 | 17.6 | $-7.2 \times 17.6 = -126.72$ | 51.84 |
| 50 | 340 | 0.8 | 3.6 | $0.8 \times 3.6 = 2.88$ | 0.64 |
| 54 | 319 | 4.8 | -17.4 | $4.8 \times -17.4 = -83.52$ | 23.04 |
| 65 | 311 | 15.8 | -25.4 | $15.8 \times -25.4 = -401.32$ | 249.64 |
Sum these:
$$\sum (x_i - \bar{x})(y_i - \bar{y}) = -306.72 -126.72 + 2.88 -83.52 -401.32 = -915.4$$
$$\sum (x_i - \bar{x})^2 = 201.64 + 51.84 + 0.64 + 23.04 + 249.64 = 526.8$$
Calculate slope:
$$b_1 = \frac{-915.4}{526.8} \approx -1.738$$
4. **Calculate intercept $$b_0$$:**
Formula:
$$b_0 = \bar{y} - b_1 \bar{x}$$
Calculate:
$$b_0 = 336.4 - (-1.738)(49.2) = 336.4 + 85.53 = 421.93$$
5. **Write the regression line equation:**
$$\hat{y} = 421.93 - 1.738 x$$
6. **Interpretation:**
- The slope $$b_1 = -1.738$$ means bone density decreases by about 1.738 units for each additional year of age.
- The intercept $$b_0 = 421.93$$ is the predicted bone density at age 0 (not meaningful here but part of the model).
7. **Note on correlation significance:**
- The correlation coefficient should be checked for statistical significance before using this model for prediction.
**Final answer:**
$$\boxed{\hat{y} = 421.93 - 1.738 x}$$