The table below gives the age and bone density for five randomly selected women. Using this data, consider the equation of the regression line, yˆ=b0+b1x, 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

40

61

62

68

69

Bone Density

357

350

343

340

315

Table

Step 5 of 6 :  

Substitute the values you found in steps 1 and 2 into the equation for the regression line to find the estimated linear model. According to this model, if the value of the independent variable is increased by one unit, then find the change in the dependent variable yˆ.

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.

The table below gives the age and bone density for five randomly selected women. Using this data, consider the equation of the regression line, yˆ=b0+b1x, 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.

Explore how to calculate the linear regression equation to predict bone density based on age. This problem involves finding the slope and y-intercept using the least squares method. We calculate the regression line for five women's ages and bone densities, demonstrating the correlation between age and bone density. Suitable for high school students in Grades 11-12.

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