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	47
49
51
58
63
Bone Density	360
353
336
333
332


Find the estimated y-intercept. Round your answer to three decimal places.

To find the estimated \( y \)-intercept (\( b_0 \)) of the regression line, we need to follow these steps:

The regression equation is of the form:

\[
\hat{y} = b_0 + b_1 x
\]

Where:
- \( \hat{y} \) is the predicted value of bone density.
- \( x \) is the age.
- \( b_1 \) is the slope of the regression line.
- \( b_0 \) is the y-intercept of the regression line.

To find the y-intercept, we first need to compute the slope \( b_1 \), and then we can use it to find \( b_0 \).

### 1. Compute the Slope \( b_1 \)

The formula for \( b_1 \) is:

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

Where:
- \( x_i \) are the ages.
- \( y_i \) are the bone densities.
- \( \bar{x} \) is the mean of the ages.
- \( \bar{y} \) is the mean of the bone densities.

### 2. Compute the y-intercept \( b_0 \)

The formula for \( b_0 \) is:

\[
b_0 = \bar{y} - b_1 \bar{x}
\]

Let's calculate this step by step.

#### Given dataTo find the estimated \( y \)-intercept (\( b_0 \)) of the regression line, we need to follow these steps:

The regression equation is of the form:

\[
\hat{y} = b_0 + b_1 x
\]

Where:
- \( \hat{y} \) is the predicted value of bone density.
- \( x \) is the age.
- \( b_1 \) is the slope of the regression line.
- \( b_0 \) is the y-intercept of the regression line.

To find the y-intercept, we first need to compute the slope \( b_1 \), and then we can use it to find \( b_0 \).

### 1. Compute the Slope \( b_1 \)

The formula for \( b_1 \) is:

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

Where:
- \( x_i \) are the ages.
- \( y_i \) are the bone densities.
- \( \bar{x} \) is the mean of the ages.
- \( \bar{y} \) is the mean of the bone densities.

### 2. Compute the y-intercept \( b_0 \)

The formula for \( b_0 \) is:

\[
b_0 = \bar{y} - b_1 \bar{x}
\]

Let's calculate this step by step.

#### Given 
- Ages: 47, 49, 51, 58, 63
- Bone Densities: 360, 353, 336, 333, 332

We will calculate the mean of both the ages and the bone densities, then the slope, and finally the y-intercept.

I'll compute these values now.The estimated y-intercept (\(b_0\)) of the regression line is \( 428.763 \), rounded to three decimal places.

Would you like a detailed explanation of the steps, or do you have any further questions?

Here are five related questions:
1. How is the slope (\(b_1\)) of the regression line calculated in detail?
2. What does the y-intercept represent in this context of bone density prediction?
3. How can we check if the correlation coefficient is statistically significant?
4. How is the total variation in the data explained by the regression model measured?
5. Can the regression line predict bone density for ages outside the given range?

**Tip:** Always check if the assumptions of linear regression are met before using the regression line for predictions.

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: 47, 49, 51, 58, 63
Bone Density: 360, 353, 336, 333, 332

Find the estimated y-intercept. Round your answer to three decimal places.

Learn how to calculate the estimated y-intercept of a regression line using age and bone density data for five randomly selected women. This problem covers key statistical concepts such as linear regression, slope calculation, and significance of correlation. Suitable for Grades 11-12.

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