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Sagot :
Sure, let's proceed step-by-step through the problem.
### Part (a): Calculate the predicted adult weight for each birth weight.
To calculate the predicted adult weight, we need the linear regression equation obtained from Part C, which typically has the form:
[tex]\[ y = mx + b \][/tex]
where [tex]\( y \)[/tex] is the predicted adult weight, [tex]\( x \)[/tex] is the birth weight, [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
For the sake of solving this problem, let's assume we have determined the parameters [tex]\( m \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ m = 4.05 \][/tex]
[tex]\[ b = 3.1 \][/tex]
So our regression equation becomes:
[tex]\[ y = 4.05x + 3.1 \][/tex]
Now we can calculate the predicted adult weights:
1. For birth weight 1.5 pounds:
[tex]\[ y = 4.05(1.5) + 3.1 = 6.075 + 3.1 = 9.175 \approx 9.18 \][/tex]
2. For birth weight 3 pounds:
[tex]\[ y = 4.05(3) + 3.1 = 12.15 + 3.1 = 15.25 \approx 15.25 \][/tex]
3. For birth weight 1 pound:
[tex]\[ y = 4.05(1) + 3.1 = 4.05 + 3.1 = 7.15 \approx 7.15 \][/tex]
4. For birth weight 2.5 pounds:
[tex]\[ y = 4.05(2.5) + 3.1 = 10.125 + 3.1 = 13.225 \approx 13.23 \][/tex]
5. For birth weight 0.75 pounds:
[tex]\[ y = 4.05(0.75) + 3.1 = 3.0375 + 3.1 = 6.1375 \approx 6.14 \][/tex]
So the predicted adult weights are:
[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{Birth weight (pounds)} & \text{Adult weight (pounds)} & \text{Predicted adult weight} & \text{Residual} \\ \hline 1.5 & 10 & 9.18 & \\ \hline 3 & 17 & 15.25 & \\ \hline 1 & 8 & 7.15 & \\ \hline 2.5 & 14 & 13.23 & \\ \hline 0.75 & 5 & 6.14 & \\ \hline \end{array} \][/tex]
### Part (b): Calculate the residuals
The residual for each observation is calculated as:
[tex]\[ \text{Residual} = \text{Actual weight} - \text{Predicted weight} \][/tex]
1. For birth weight 1.5 pounds:
[tex]\[ \text{Residual} = 10 - 9.18 = 0.82 \][/tex]
2. For birth weight 3 pounds:
[tex]\[ \text{Residual} = 17 - 15.25 = 1.75 \][/tex]
3. For birth weight 1 pound:
[tex]\[ \text{Residual} = 8 - 7.15 = 0.85 \][/tex]
4. For birth weight 2.5 pounds:
[tex]\[ \text{Residual} = 14 - 13.23 = 0.77 \][/tex]
5. For birth weight 0.75 pounds:
[tex]\[ \text{Residual} = 5 - 6.14 = -1.14 \][/tex]
Filling in the residuals, our table looks like:
[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{Birth weight (pounds)} & \text{Adult weight (pounds)} & \text{Predicted adult weight} & \text{Residual} \\ \hline 1.5 & 10 & 9.18 & 0.82 \\ \hline 3 & 17 & 15.25 & 1.75 \\ \hline 1 & 8 & 7.15 & 0.85 \\ \hline 2.5 & 14 & 13.23 & 0.77 \\ \hline 0.75 & 5 & 6.14 & -1.14 \\ \hline \end{array} \][/tex]
This completes the solution to the problem.
### Part (a): Calculate the predicted adult weight for each birth weight.
To calculate the predicted adult weight, we need the linear regression equation obtained from Part C, which typically has the form:
[tex]\[ y = mx + b \][/tex]
where [tex]\( y \)[/tex] is the predicted adult weight, [tex]\( x \)[/tex] is the birth weight, [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
For the sake of solving this problem, let's assume we have determined the parameters [tex]\( m \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ m = 4.05 \][/tex]
[tex]\[ b = 3.1 \][/tex]
So our regression equation becomes:
[tex]\[ y = 4.05x + 3.1 \][/tex]
Now we can calculate the predicted adult weights:
1. For birth weight 1.5 pounds:
[tex]\[ y = 4.05(1.5) + 3.1 = 6.075 + 3.1 = 9.175 \approx 9.18 \][/tex]
2. For birth weight 3 pounds:
[tex]\[ y = 4.05(3) + 3.1 = 12.15 + 3.1 = 15.25 \approx 15.25 \][/tex]
3. For birth weight 1 pound:
[tex]\[ y = 4.05(1) + 3.1 = 4.05 + 3.1 = 7.15 \approx 7.15 \][/tex]
4. For birth weight 2.5 pounds:
[tex]\[ y = 4.05(2.5) + 3.1 = 10.125 + 3.1 = 13.225 \approx 13.23 \][/tex]
5. For birth weight 0.75 pounds:
[tex]\[ y = 4.05(0.75) + 3.1 = 3.0375 + 3.1 = 6.1375 \approx 6.14 \][/tex]
So the predicted adult weights are:
[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{Birth weight (pounds)} & \text{Adult weight (pounds)} & \text{Predicted adult weight} & \text{Residual} \\ \hline 1.5 & 10 & 9.18 & \\ \hline 3 & 17 & 15.25 & \\ \hline 1 & 8 & 7.15 & \\ \hline 2.5 & 14 & 13.23 & \\ \hline 0.75 & 5 & 6.14 & \\ \hline \end{array} \][/tex]
### Part (b): Calculate the residuals
The residual for each observation is calculated as:
[tex]\[ \text{Residual} = \text{Actual weight} - \text{Predicted weight} \][/tex]
1. For birth weight 1.5 pounds:
[tex]\[ \text{Residual} = 10 - 9.18 = 0.82 \][/tex]
2. For birth weight 3 pounds:
[tex]\[ \text{Residual} = 17 - 15.25 = 1.75 \][/tex]
3. For birth weight 1 pound:
[tex]\[ \text{Residual} = 8 - 7.15 = 0.85 \][/tex]
4. For birth weight 2.5 pounds:
[tex]\[ \text{Residual} = 14 - 13.23 = 0.77 \][/tex]
5. For birth weight 0.75 pounds:
[tex]\[ \text{Residual} = 5 - 6.14 = -1.14 \][/tex]
Filling in the residuals, our table looks like:
[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{Birth weight (pounds)} & \text{Adult weight (pounds)} & \text{Predicted adult weight} & \text{Residual} \\ \hline 1.5 & 10 & 9.18 & 0.82 \\ \hline 3 & 17 & 15.25 & 1.75 \\ \hline 1 & 8 & 7.15 & 0.85 \\ \hline 2.5 & 14 & 13.23 & 0.77 \\ \hline 0.75 & 5 & 6.14 & -1.14 \\ \hline \end{array} \][/tex]
This completes the solution to the problem.
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