Questions: The table below gives the age and bone density for five randomly selected women. Using this data, consider the equation of the regression line, ŷ = b0 + b1 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 47 49 50 51 58 Bone Density 360 353 336 333 310

The table below gives the age and bone density for five randomly selected women. Using this data, consider the equation of the regression line, ŷ = b0 + b1 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 47 49 50 51 58
Bone Density 360 353 336 333 310

Transcript text: 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 \begin{tabular}{|c|c|c|c|c|c|} \hline Age & 47 & 49 & 50 & 51 & 58 \\ \hline Bone Density & 360 & 353 & 336 & 333 & 310 \\ \hline \end{tabular} Table Copy Data

Solution

Solution Steps

To solve this problem, we need to perform a linear regression analysis to find the equation of the regression line $\hat{y} = b_0 + b_1 x$. The steps are as follows:

Calculate the means of the age and bone density.
Compute the slope ($b_1$) using the formula: $b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$.
Compute the intercept ($b_0$) using the formula: $b_0 = \bar{y} - b_1 \bar{x}$.
Use the slope and intercept to form the regression equation.

Step 1: Calculate Means

The mean of the ages is calculated as: \[ \bar{x} = 51.0 \] The mean of the bone densities is calculated as: \[ \bar{y} = 338.4 \]

Step 2: Calculate the Slope ($b_1$)

The slope ($b_1$) is calculated using the formula: \[ b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \] Substituting the given values: \[ b_1 = \frac{-312.0}{70.0} = -4.4571 \]

Step 3: Calculate the Intercept ($b_0$)

The intercept ($b_0$) is calculated using the formula: \[ b_0 = \bar{y} - b_1 \bar{x} \] Substituting the given values: \[ b_0 = 338.4 - (-4.4571 \times 51.0) = 565.7143 \]

Step 4: Form the Regression Equation

The regression line equation is: \[ \hat{y} = b_0 + b_1 x \] Substituting the calculated values of $b_0$ and $b_1$: \[ \hat{y} = 565.7143 - 4.4571x \]