A study to investigate how the propagation of an ultrasonic stress wave through
ID: 3305207 • Letter: A
Question
A study to investigate how the propagation of an ultrasonic stress wave through a substance depends on the properties of the substance was completed, and a full linear regression analysis is to be performed on the data set. The explanatory variable is fracture strength (as a percentage of ultimate tensile strength), and the response variable is the attenuation (the decrease in amplitude of the stress wave, in neper/cm). A scatterplot of the data set used in the study is presented below.
Identify the independent variable, x.
Identify the dependent variable, y.
By looking at the scatter plot, describe the strength and direction of the linear relationship between fracture strength and attenuation.
The correlation coefficient between the variables is -0.9451. Describe the strength and direction of the linear relationship between fracture strength and attenuation.
Calculate r2.
Interpret r2 within the context of the problem.
The regression equation is: yhat = 3.67 0.03x. Identify the slope.
Interpret the slope within the context of the problem.
Identify the y-intercept.
Interpret the y-intercept within the context of the problem.
Use the regression line to predict the attenuation for a fracture strength of 75%. What is the predicted value? Is this a reliable prediction? Why or why not? Explain your answer.
Explanation / Answer
Independent variable: A fracture strength
Dependent variable: The attenuation
Scatter plot shows a strong negative relationship between the variables a fracture strength and the attenuation . As the fracture strength increases, the attenuation decreases.
The correlation coefficient between the variables is -0.9451. It shows a strong negative relationship between the variables.
The r-square is:
r = (-0.9451) * (-0.9451) = 0.89321401
It shows that approximately 89.3% of variation in attenuation is accounted for by fracture strength .
The slope is -0.03
For each unit increase in fracture strength, attenuation is decreased by 0.03 units.
The intercept is: 3.67
When fracture strength is zero, attenuation is 3.67.
Use the regression line to predict the attenuation for a fracture strength of 75%. What is the predicted value? Is this a reliable prediction? Why or why not? Explain your answer.
The required predicted value is
yhat = 3.67- 0.03*75 = 1.42
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