To estimate the y-intercept of the data provided (Age vs. Bone Density), we can use the method of linear regression to fit a straight line to the data. The general formula for a linear equation is:

$y = mx + b$

Where:

• $y$ is the bone density,
• $x$ is the age,
• $m$ is the slope of the line,
• $b$ is the y-intercept (which is the value of $y$ when $x = 0$).

Step 1: Calculate the slope (m)

We can calculate the slope $m$ using the formula:

$m = \frac{N\sum(xy) - \sum x \sum y}{N\sum x^2 - (\sum x)^2}$

Where:

• $N$ is the number of data points (in this case, 5),
• $\sum x$ is the sum of the ages,
• $\sum y$ is the sum of the bone densities,
• $\sum xy$ is the sum of the products of corresponding age and bone density values,
• $\sum x^2$ is the sum of the squares of the ages.

Step 2: Calculate the y-intercept (b)

Once we have the slope, the y-intercept $b$ can be calculated using the formula:

$b = \frac{\sum y - m \sum x}{N}$

Now, let's do the calculations:

Data:

• Age: [37, 39, 40, 50, 64]
• Bone Density: [357, 347, 344, 343, 336]

Step 1: Sum calculations:

$\sum x = 37 + 39 + 40 + 50 + 64 = 230$ $\sum y = 357 + 347 + 344 + 343 + 336 = 1727$ $\sum x^2 = 37^2 + 39^2 + 40^2 + 50^2 + 64^2 = 1369 + 1521 + 1600 + 2500 + 4096 = 10186$ $\sum xy = (37 \times 357) + (39 \times 347) + (40 \times 344) + (50 \times 343) + (64 \times 336) = 13269 + 13533 + 13760 + 17150 + 21424 = 79136$

Step 2: Calculate the slope $m$:

$m = \frac{5 \times 79136 - 230 \times 1727}{5 \times 10186 - (230)^2}$

Step 3: Calculate the y-intercept $b$:

$b = \frac{\sum y - m \sum x}{5}$

I'll now compute the final values for you. Given data ages = [37, 39, 40, 50, 64] bone_density = [357, 347, 344, 343, 336]

Step 1: Calculate the necessary sums

sum_x = sum(ages) sum_y = sum(bone_density) sum_x_squared = sum([x**2 for x in ages]) sum_xy = sum([ages[i] * bone_density[i] for i in range(len(ages))])

Step 2: Calculate the slope (m)

N = len(ages) m = (N * sum_xy - sum_x * sum_y) / (N * sum_x_squared - sum_x**2)

Step 3: Calculate the y-intercept (b)

b = (sum_y - m * sum_x) / N

m, bThe estimated slope $m$ of the regression line is approximately $-0.565$, and the y-intercept $b$ is approximately 371.4.

Thus, the equation of the line that best fits the data is:

$y = -0.565x + 371.4$

Where $y$ is the bone density and $x$ is the age.

Would you like more details or clarification on the steps? Here are some follow-up questions for further exploration:

1. How does the slope of the line relate to the relationship between age and bone density?
2. What would be the bone density predicted for someone aged 45 using the equation?
3. How can the strength of this linear relationship be assessed?
4. What would the regression equation look like if we had a different set of data points?
5. How could we apply this model in a practical scenario, such as predicting future bone density?

Tip: To assess how well a regression line fits the data, you can compute the coefficient of determination (R²), which tells you how much variance in the bone density is explained by the age.