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Answer :
We use a linear regression model to predict the height of the woman based on her weight. The predicted height for a woman weighing 164 pounds is 71.1 inches. The root-mean-square error of the regression line is 2.76 inches, and the regression equation to predict height is height = 66 + 0.06*(weight-130).
We are using a linear regression model to predict the height of a woman based on her weight. The given data includes the averages and standard deviations for both weight and height, as well as the correlation coefficient ('r').
1. To predict the height of a woman who weighs 164 pounds, we first need to find the standardized score (Z-score) for her weight. This is calculated as Z = (X - μ) / σ, where X is the value (164 pounds), μ is the mean (130 pounds), and σ is the standard deviation (20 pounds). So, Z = (164-130)/20 = 1.7.
Now, we predict her height by multiplying this Z-score with the standard deviation of the height and adding it to the average height. This gives us predicted height = (1.7 * 3) + 66 = 71.1 inches.
2. The root-mean-square error (RMSE) of the regression line is calculated as SD*√(1-r²). Here, SD is the standard deviation of the dependent variable (height in inches, which is 3) and r is the correlation coefficient (0.4). Therefore, RMSE = 3*√(1-0.4²) = 2.76 inches.
3. The regression equation can be found by Y = a + bX where 'a' is the y-intercept and 'b' is the slope of the line (b = r*(SD_Y/SD_X)). Here, X is weight and Y is height. So, b = 0.4 * (3/20) = 0.06.
Therefore, the regression equation becomes: height = 66 + 0.06*(weight - 130).
For 130 pounds, the predicted height = 66 + 0.06*(130 - 130) = 66 inches.
For 164 pounds, the predicted height = 66 + 0.06*(164 - 130) = 68.04 inches.
Learn more about the topic of Linear Regression here:
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