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
Step 1 of 6 : Find the estimated slope. Round your answer to three decimal places.
1. **State the problem:** We want to find the estimated slope $$b_1$$ of the regression line $$\hat{y} = b_0 + b_1 x$$ that predicts bone density based on age.
2. **Recall the formula for slope:**
$$
b_1 = \frac{S_{xy}}{S_{xx}} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
$$
where $$x_i$$ are ages, $$y_i$$ are bone densities, $$\bar{x}$$ is the mean age, and $$\bar{y}$$ is the mean bone density.
3. **Calculate means:**
$$
\bar{x} = \frac{35 + 42 + 50 + 54 + 65}{5} = \frac{246}{5} = 49.2
$$
$$
\bar{y} = \frac{358 + 354 + 340 + 319 + 311}{5} = \frac{1682}{5} = 336.4
$$
4. **Calculate each $$x_i - \bar{x}$$ and $$y_i - \bar{y}$$:**
- For $$x$$: $$35 - 49.2 = -14.2$$, $$42 - 49.2 = -7.2$$, $$50 - 49.2 = 0.8$$, $$54 - 49.2 = 4.8$$, $$65 - 49.2 = 15.8$$
- For $$y$$: $$358 - 336.4 = 21.6$$, $$354 - 336.4 = 17.6$$, $$340 - 336.4 = 3.6$$, $$319 - 336.4 = -17.4$$, $$311 - 336.4 = -25.4$$
5. **Calculate products $$(x_i - \bar{x})(y_i - \bar{y})$$ and squares $$(x_i - \bar{x})^2$$:**
- Products: $$-14.2 \times 21.6 = -306.72$$, $$-7.2 \times 17.6 = -126.72$$, $$0.8 \times 3.6 = 2.88$$, $$4.8 \times (-17.4) = -83.52$$, $$15.8 \times (-25.4) = -401.32$$
- Squares: $$(-14.2)^2 = 201.64$$, $$(-7.2)^2 = 51.84$$, $$0.8^2 = 0.64$$, $$4.8^2 = 23.04$$, $$15.8^2 = 249.64$$
6. **Sum the products and squares:**
$$
\sum (x_i - \bar{x})(y_i - \bar{y}) = -306.72 - 126.72 + 2.88 - 83.52 - 401.32 = -915.4
$$
$$
\sum (x_i - \bar{x})^2 = 201.64 + 51.84 + 0.64 + 23.04 + 249.64 = 526.8
$$
7. **Calculate slope:**
$$
b_1 = \frac{-915.4}{526.8}
$$
Show cancellation:
$$
b_1 = \frac{\cancel{-915.4}}{\cancel{526.8}} = -1.738
$$
8. **Round to three decimal places:**
$$
b_1 \approx -1.738
$$
**Final answer:** The estimated slope $$b_1$$ is approximately $$-1.738$$.