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Regression Analysis & Significance of Correlation

I need the full answers to the questions in the attached file. They are similar to exercises from Statistics for Engineers and Scientists, William Navidi.

Exercise 1

The following table presents shear strengths (in kN/mm) and weld diameters (in mm) for a sample of spot welds.

Diameter Strength
4.2 51
4.4 54
4.6 69
4.8 81
5.0 75
5.2 79
5.4 89
5.6 101
5.8 98
6.0 102

a) Construct a scatterplot of strength (y) versus diameter (x). Verify that a linear model is appropriate. (done by hand)
b) Compute the least-squares line for predicting strength from diameter. (show all your calculations)
c) Compute the fitted values (i.e. the predicted values) and the residual for each point. Which point has the residual with the largest magnitude? (can be done in excel)
d) If the diameter is increased by 0.3 mm, by how much would you predict the strength to increase or decrease?
e) Predict the strength for a diameter of 5.5 mm.
f) Can the least-squares line be used to predict the strength for a diameter of 8mm? If so, predict the strength. If not, explain why not.
g) For what diameter would you predict a strength of 95 kN/mm?
h) Compute the regression sum of squares, the error sum of squares, and the total sum of squares.
i) Divide the regression sum of squares by the total sum of squares. What is the relationship between this quantity and the correlation coefficient?

Exercise 2

A chemical company wishes to study the effect of extraction time on the efficiency of an extraction operation. Here are the descriptive statistics that were generated in SAS based on the extraction time (in minutes) and the extraction efficiency (y):
The SAS System
The CORR Procedure
2 Variables: time efficiency
Covariance Matrix, DF = 9
time efficiency
time 138.8888889 106.1111111
efficiency 106.1111111 119.1666667

Simple Statistics

Variable N Mean Std Dev Sum Minimum Maximum

time 10 32.00000 11.78511 320.00000 15.00000 49.00000
efficiency 10 63.50000 10.91635 635.00000 46.00000 80.00000

(a) Compute the sample correlation coefficient.
(b) Is there evidence of a significant correlation between extraction time and extraction efficiency? Run the appropriate test at .
(c) Compute the equation of the line that best describes the relationship between extraction time and extraction efficiency. [Note: Extraction efficiency is the dependent variable.]

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Solution Summary

The solution provides step by step method for the calculation of regression analysis and test statistic for significance of correlation coefficient. Formula for the calculation and Interpretations of the results are also included.

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