Questions: Correct 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 + 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 53 56 60 68 Bone Density 358 352 331 327 321 Table Step 4 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 ŷ.

Correct 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 + 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 53 56 60 68 Bone Density 358 352 331 327 321 Table Step 4 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 ŷ.
Transcript text: Correct 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 & 42 & 53 & 56 & 60 & 68 \\ \hline Bone Density & 358 & 352 & 331 & 327 & 321 \\ \hline \end{tabular} Table Step 4 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 $\hat{y}$.
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Solution

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Determine the regression line for predicting a woman's bone density based on her age.

Calculate the means of \( x \) and \( y \).

The mean of \( x \) (ages) is given by \( \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 55.8 \). The mean of \( y \) (bone densities) is \( \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 337.8 \).

Calculate the correlation coefficient \( r \).

The correlation coefficient is calculated as \( r = -0.9192 \).

Calculate the slope \( \beta \).

The numerator for \( \beta \) is \( \sum_{i=1}^{n} x_i y_i - n \bar{x} \bar{y} = 93676 - 5 \cdot 55.8 \cdot 337.8 = -570.2 \). The denominator for \( \beta \) is \( \sum_{i=1}^{n} x_i^2 - n \bar{x}^2 = 15933 - 5 \cdot 55.8^2 = 364.8 \). Thus, the slope is \( \beta = \frac{-570.2}{364.8} = -1.563 \).

Calculate the intercept \( \alpha \).

The intercept is calculated as \( \alpha = \bar{y} - \beta \bar{x} = 337.8 - (-1.563) \cdot 55.8 = 425.0181 \).

Formulate the regression line equation.

The line of best fit is given by \( y = 425.0181 - 1.563x \).

The regression line is \( y = 425.0181 - 1.563x \).

Determine the change in the dependent variable \( \hat{y} \) when the independent variable \( x \) is increased by one unit.

Identify the change in \( \hat{y} \).

According to the model, if the value of the independent variable \( x \) is increased by one unit, the change in the dependent variable \( \hat{y} \) is equal to the slope \( \beta \), which is \( -1.563 \).

The change in \( \hat{y} \) is \( -1.563 \).

The regression line is \( y = 425.0181 - 1.563x \).

The change in \( \hat{y} \) when \( x \) is increased by one unit is \( -1.563 \).

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