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The table below gives the age and bone density for five randomly selected women. Using this data, consider the equation of the regression line, [tex]\hat{y} = b_0 + b_1 x[/tex], 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 & 34 & 45 & 48 & 60 & 65 \\
\hline
Bone Density & 357 & 341 & 331 & 329 & 325 \\
\hline
\end{tabular}

Step 1 of 6: Find the estimated slope. Round your answer to three decimal places.



Answer :

GinnyAnswer
To find the estimated slope [tex]\( b_1 \)[/tex] of the regression line, we first need to calculate the required intermediate values using the data provided. Here, we have the ages of five women and their corresponding bone densities.

Given data:

[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline \text{Age} & 34 & 45 & 48 & 60 & 65 \\ \hline \text{Bone Density} & 357 & 341 & 331 & 329 & 325 \\ \hline \end{array} \][/tex]

Step-by-step solution to find the estimated slope [tex]\( b_1 \)[/tex]:

1. Calculate the mean of Age ([tex]\( \bar{x} \)[/tex]):

[tex]\[ \bar{x} = \frac{34 + 45 + 48 + 60 + 65}{5} = \frac{252}{5} = 50.4 \][/tex]

2. Calculate the mean of Bone Density ([tex]\( \bar{y} \)[/tex]):

[tex]\[ \bar{y} = \frac{357 + 341 + 331 + 329 + 325}{5} = \frac{1683}{5} = 336.6 \][/tex]

3. Compute the numerator of the slope (the sum of the products of the differences from the means):

[tex]\[ \sum (x_i - \bar{x})(y_i - \bar{y}) = (34 - 50.4)(357 - 336.6) + (45 - 50.4)(341 - 336.6) + (48 - 50.4)(331 - 336.6) + (60 - 50.4)(329 - 336.6) + (65 - 50.4)(325 - 336.6) \][/tex]

From the calculations, we get:

[tex]\[ -587.2 \][/tex]

4. Compute the denominator of the slope (the sum of the squares of the differences from the mean of Age):

[tex]\[ \sum (x_i - \bar{x})^2 = (34 - 50.4)^2 + (45 - 50.4)^2 + (48 - 50.4)^2 + (60 - 50.4)^2 + (65 - 50.4)^2 \][/tex]

From the calculations, we get:

[tex]\[ 609.2 \][/tex]

5. Calculate the estimated slope [tex]\( b_1 \)[/tex]:

[tex]\[ b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} = \frac{-587.2}{609.2} \approx -0.964 \][/tex]

Thus, the estimated slope [tex]\( b_1 \)[/tex] is [tex]\(\boxed{-0.964}\)[/tex] rounded to three decimal places.