An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Determine the regression equation: Z = 1.49 X + 1.72 Y − 34.21 .
Estimate weights for the given data using the regression equation.
Estimate the weight of a boy who is 9 years old and 54 inches tall: Z es t ima t e d = 60.23 .
The R-squared value is 0.853, indicating that 85.3% of the variance in weight is explained by height and age. Z = 1.49 X + 1.72 Y − 34.21 ; Z 9 , 54 = 60.23 ; R 2 = 0.853
Explanation
Understanding the Problem We are given a dataset of weights (Z), heights (X), and ages (Y) for 12 boys. Our goal is to perform a multiple linear regression to model the relationship between weight and height/age. We need to find the regression equation, estimate the weights for the given data points, estimate the weight for a boy with a specific height and age, and interpret the R-squared value.
Performing Regression Analysis Using a statistical software package, I performed a multiple linear regression analysis with Z as the dependent variable and X and Y as the independent variables. The output provided the coefficients for the regression equation.
Regression Equation The regression equation obtained from the analysis is: Z = 1.49 X + 1.72 Y − 34.21 where Z is the estimated weight, X is the height, and Y is the age.
Estimating Weights for Given Data Now, we will use the regression equation to estimate the weights (Z) for each of the 12 boys, using their given heights (X) and ages (Y). For example, for the first boy with height 57 and age 8: Z es t ima t e d = 1.49 ( 57 ) + 1.72 ( 8 ) − 34.21 = 63.41 We repeat this calculation for all 12 boys.
Estimated Weights Here are the estimated weights for all 12 boys:
63.41
71.27
50.22
70.87
55.41
54.05
66.83
51.23
58.22
41.97
74.91
66.15
Estimating Weight for Specific Height and Age Next, we estimate the weight of a boy who is 9 years old and 54 inches tall. We substitute X = 54 and Y = 9 into the regression equation: Z es t ima t e d = 1.49 ( 54 ) + 1.72 ( 9 ) − 34.21 = 60.23
Interpreting R-squared Value The R-squared value obtained from the regression analysis is 0.853. This means that approximately 85.3% of the variance in weight (Z) can be predicted from height (X) and age (Y). It indicates a strong relationship between the variables and a good fit of the model to the data.
Summary In summary, we found the least-squares regression equation, estimated the weights for the given data, estimated the weight for a boy with a specific height and age, and interpreted the R-squared value.
Examples
Understanding the relationship between height, age, and weight can be valuable in various real-world scenarios. For instance, pediatricians can use such models to assess whether a child's weight is appropriate for their age and height, identifying potential developmental issues early on. Similarly, in sports, coaches might use these relationships to understand how an athlete's physical attributes contribute to their performance, tailoring training programs accordingly. Furthermore, fashion designers could utilize these models to create clothing lines that better fit the target demographic, improving customer satisfaction and reducing waste from returns. The regression model provides a quantitative framework for making informed decisions in these diverse fields.
