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. A scatterplot shows a pronounced linear pattern. With \(\sum(x-\bar{x})=1000\) and \(\sum(x-\bar{x})(y-\bar{y})=8577\), the least-squares line is \(\hat{y}=-936.22+8.577 x\). \(\begin{array}{llllll}x & 200.1 & 210.1 & 220.1 & 230.1 & 240.0\end{array}\) \(\begin{array}{llllll}y & 813.7 & 785.3 & 960.4 & 1118.0 & 1076.2\end{array}\) a. Because \(1 \mathrm{lb}=0.4536 \mathrm{~kg}\), strength observations can be re-expressed in kilograms through multiplication by this conversion factor: new \(y=0.4536(\) old \(y) .\) What is the equation of the least-squares line when \(y\) is expressed in kilograms? b. More generally, suppose that each \(y\) value in a data set consisting of \(n(x, y)\) pairs is multiplied by a conversion factor \(c\) (which changes the units of measurement for \(y\) ). What effect does this have on the slope \(b\) (i.e., how does the new value of \(b\) compare to the value before conversion), on the intercept \(a\), and on the equation of the least-squares line? Verify your conjectures by using the given formulas for \(b\) and \(a\). (Hint: Replace \(y\) with \(c y\), and see what happens \- and remember, this conversion will affect \(\bar{y} .\) )

Short answer

The equation of the least squares line when \(y\) is expressed in kilograms is \(\hat{y'} = -424.60272 + 3.8894692x\). Conversion of \(y\) values by a constant \(c\) results in the equation \(\hat{y'} = c*a + c*b*x\), implying that both the slope and the intercept get multiplied by the conversion factor.

Step-by-step solution

Convert y-values into Kilograms

This can be done by multiplying each of the original \(y\) values by the conversion factor \(0.4536\). The transformed \(y\) will be referred to as \(y'\). Therefore, the new \(y\) values would be: \(y' = 0.4536 * y\)

Find the New Equation

The original equation of the line is \(\hat{y}=-936.22+8.577 x\). When the \(y\) variables are converted to \(y'\), the new equation becomes:\(\hat{y'} = -936.22*0.4536 + 8.577*0.4536*x\)

General case interpretation

In the general case, suppose a conversion factor \(c\) is applied on each \(y\) value. This will alter the slope \(b'\) and the intercept \(a'\) of the equation. The new slope \(b'\) is \(c*b\) and the new intercept \(a'\) is \(c*a\), with the new equation of the line becoming \(\hat{y'} = a'*c + b'*c*x\). This is because the conversion factor is a constant applied uniformly on all y-values, and does not affect the relationship between \(x\) and \(y\).

Key concepts

Scatterplot

Imagine you're at an archery range, and each arrow you shoot lands on a target. A scatterplot is akin to that target, where every point is like an arrow's position, representing the relationship between two different things—in our example, weld diameter (which we can call x) and shear strength (y) for spot welds on steel.

A scatterplot is extremely useful because it gives us a visual way to see if there's a pattern to how these two variables relate. In the exercise given, the patterning suggests that as weld diameter increases, there might be a change in the shear strength. This makes a scatterplot an indispensable first step in identifying if a linear relationship exists and helps in formulating a hypothesis about the kind of correlation present.

Upon observing a linear pattern in the scatterplot, which means the points seem to form a straight-line trend, you could make an educated guess: there's likely a cause-and-effect situation happening between diameter and strength. Hence, there's good reason to delve deeper, which leads us to linear regression.

Linear Regression

When your scatterplot suggests a straight-line relationship, like in the case of our welded materials, you turn to linear regression to find the 'best fit' line. Linear regression is a way to summarize the relationship between the independent variable (x) and the dependent variable (y) numerically. It's like finding the line that best summarizes where most of the 'arrows' or data points have landed.

The equation you often hear of, y = mx + b, is the simplified version of the least-squares regression line equation, where y predicts the value of the dependent variable, m is the slope, x is the independent variable, and b is the y-intercept. A least-squares line is mathematically calculated so that the total distance between the data points and the line itself is minimized, hence its name. In practice, this line can prove to be a predictive tool, helping you estimate shear strength based on weld diameters you've not even tested yet!

Slope and Intercept

Diving into the components of the linear regression equation, the slope (b) represents how much the dependent variable (y) changes for every one unit change in the independent variable (x). If the slope is positive, as we see in our exercise with a slope of 8.577, it means that as x increases, y generally increases as well.

The intercept (a), specifically the y-intercept, is the expected value of y when x is zero—essentially where the line crosses the y-axis. Our given line has a y-intercept of -936.22, which tells us that if we had a weld diameter of 0—which isn't feasible in our context but is valuable for the sake of the equation—our model predicts the shear strength would be -936.22, which again, isn't practical but mathematically relevant.

In practical terms, you might not find a situation where x equals zero, especially in real-life scenarios such as measuring strength based on diameter. However, understanding the slope and intercept provides insight into the rate of change and initial value, which are essential in predicting outcomes and understanding data trends.

Data Conversion

Imagine you're baking and your recipe calls for ounces, but you only have a scale that measures in grams. You need to convert measurements to proceed. Similarly, converting data from one unit of measurement to another, like pounds to kilograms in our welding exercise, is straightforward math. You multiply each data value by a conversion factor.

Here's what's interesting: When you change the unit of measurement for your dependent variable (y), you're scaling every value up or down by a constant amount. But the relationship between x and y doesn't change. The slope and intercept of the regression line will be scaled by the conversion factor as well, but the 'shape' of the data remains consistent—after all, converting currency doesn't change the value of your money, it just expresses it in a different way.

The concept becomes clearer with the given exercise. When you multiply the y values by a conversion factor to get from pounds to kilograms, both your slope and intercept are multiplied by that same factor. Understanding this principle ensures that when data conversion is needed—for instance, when you need to present data to an international audience—you can make the necessary adjustments without altering the intrinsic relationships within the data set.