Question

The accompanying data resulted from an experiment in which weld diameter \(x\) and shear strength \(y\) (in pounds) were determined for five different spot welds on steel. \(x\) \(\begin{array}{lllll}200.1 & 210.1 & 220.1 & 230.1 & 240.0 \\ 813.7 & 785.3 & 960.4 & 1118.0 & 1076.2\end{array}\) \(y\) a. With \(x=\) weld diameter and \(y=\) shear strength, construct a scatterplot. Does the pattern in the scatterplot look linear? b. Find the equation of the least squares regression line. c. Calculate the five residuals and construct a residual plot. Are there any unusual features in the residual plot?

Short answer

The short answer to this problem will require visual analysis of both the scatterplot to determine linearity and the residual plot for detecting any unusual features. The equation of the regression line and the calculated residuals will depend on the specific schematics of the above procedure.

Step-by-step solution

Construct the Scatterplot

First, plot the given pairs of values \((x, y)\) of weld diameter and shear strength onto a graph. Each pair represents one point in the scatter plot. Look at the pattern in the scatter plot to determine whether the relationship looks linear or not.

Calculate the Regression Line

Use the formulas for the slope and y-intercept of the line to find the equation of the least squares regression line. For the slope calculation, find: \(m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\)And for the y-intercept calculation, find:\(b = \frac{\sum y - m\sum{x}}{n}\)where \(\sum x\) and \(\sum y\) is the sum of all x and y values respectively, \(\sum xy\) is the sum of the product of x and y values, \(\sum x^2\) is the sum of square of each x value and n is the number of pairs of points, in this case, 5. Substitute your calculated values of m and b into the regression line equation \(y = m*x + b\)

Calculate and Plot Residuals

Next, calculate the residuals, which is the difference between the observed values of y and the values of y predicted by the regression line. To calculate each residual, use the formula:\(Residual = observed y - predicted y\)Substitute the x value of each pair into the regression line, calculate the predicted y and subtract this from the actual y. Repeat this process for all pairs of points. Once all residuals are calculated, plot these residuals on a residual plot. Analyze the plot for any unusual features.