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	42
51
55
64
65
Bone Density	355
329
328
324
312
Table


Step 3 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 estimate the linear regression model for predicting a woman's bone density based on her age, we need to follow these steps:

### Step 1: Compute the necessary statistics

1. **Sum of ages (∑x)**
2. **Sum of bone density (∑y)**
3. **Sum of ages squared (∑x²)**
4. **Sum of age × bone density (∑xy)**

Given dataTo estimate the linear regression model for predicting a woman's bone density based on her age, we need to follow these steps:

### Step 1: Compute the necessary statistics

1. **Sum of ages (∑x)**
2. **Sum of bone density (∑y)**
3. **Sum of ages squared (∑x²)**
4. **Sum of age × bone density (∑xy)**

Given 
- Age \( x \): 42, 51, 55, 64, 65
- Bone Density \( y \): 355, 329, 328, 324, 312

Now calculate the sums:

- \( \sum x = 42 + 51 + 55 + 64 + 65 = 277 \)
- \( \sum y = 355 + 329 + 328 + 324 + 312 = 1648 \)
- \( \sum x^2 = 42^2 + 51^2 + 55^2 + 64^2 + 65^2 = 1764 + 2601 + 3025 + 4096 + 4225 = 15711 \)
- \( \sum xy = (42 \times 355) + (51 \times 329) + (55 \times 328) + (64 \times 324) + (65 \times 312) \)
  \[
  = 14910 + 16779 + 18040 + 20736 + 20280 = 90745
  \]

### Step 2: Calculate the slope \( b_1 \) and intercept \( b_0 \)

The formula for the slope \( b_1 \) of the regression line is:
\[
b_1 = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
where \( n = 5 \) (the number of data points).

Substitute the values:
\[
b_1 = \frac{5(90745) - (277)(1648)}{5(15711) - (277)^2}
= \frac{453725 - 456496}{78555 - 76729}
= \frac{-2771}{1826} \approx -1.517
\]

Now, calculate the intercept \( b_0 \) using the formula:
\[
b_0 = \frac{\sum y - b_1(\sum x)}{n}
\]
\[
b_0 = \frac{1648 - (-1.517 \times 277)}{5}
= \frac{1648 + 420.209}{5}
= \frac{2068.209}{5} \approx 413.642
\]

### Step 3: Write the regression equation

Now that we have both \( b_0 \) and \( b_1 \), the regression line equation is:
\[
\hat{y} = 413.642 - 1.517x
\]

### Step 4: Find the change in \( \hat{y} \) when \( x \) increases by 1

The slope \( b_1 = -1.517 \) represents the change in the predicted value of \( y \) (bone density) for every one-unit increase in \( x \) (age).

Thus, if the value of the independent variable (age) is increased by 1, the change in the dependent variable \( \hat{y} \) (bone density) is:
\[
\Delta \hat{y} = -1.517
\]

This means that for every additional year of age, the bone density decreases by approximately 1.517 units.

---

Would you like more details or have any questions about the calculations?

Here are five related questions:
1. How would you interpret the intercept \( b_0 \) in the context of this data?
2. What is the correlation coefficient for this data, and is it statistically significant?
3. How can we determine if the regression model is appropriate for prediction?
4. How do we calculate the residuals for each data point in this regression model?
5. Can this model be used for predictions outside the observed age range?

**Tip**: Always check the correlation coefficient before making predictions using the regression line, as it ensures the relationship between variables is significant.

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: 42, 51, 55, 64, 65
Bone Density: 355, 329, 328, 324, 312

Step 3 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.

Explore how to use linear regression to predict a woman's bone density based on her age. Learn the key steps for calculating the slope and intercept of the regression line and understand how age affects bone density. This problem is ideal for Grades 11-12 students studying statistical methods and correlation analysis.

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