Oil and residuals The Trans-Alaska Oil Pipeline is a tube that is formed from $1/2$-inch-thick steel and carries oil across $800$miles of sensitive arctic and subarctic terrain. The pipe segments and the welds that join them were carefully examined before installation. How accurate are field measurements of the depth of small defects? The figure below compares the results of measurements on $100$defects made in the field with measurements of the same defects made in the laboratory. The line $y=x$ is drawn on the scatterplot.

(a) Describe the overall pattern you see in the scatterplot, as well as any deviations from that pattern.

(b) If field and laboratory measurements all agree, then the points should fall on the $y=x$ line drawn on the plot, except for small variations in the measurements. Is this the case? Explain.

(c) The line drawn on the scatterplot $(y=x)$ is not the least-squares regression line. How would the slope and $y-$intercept of the least-squares line compare? Justify your answer.

A regression line is a straight line that depicts the change in a response variable $y$ when an explanatory variable $x$changes. By putting this $x$into the equation of the line, you may use a regression line to predict the value of $y$for any value of $x$

The general pattern moves from bottom left to higher right, as shown in the graph. In other words, higher laboratory measurement usually translates into higher field measurement. Between the two variables, there is a positive linear relationship. The line fits quite well for small values of laboratory measurement, while measurement points depart greatly from the line for larger values of laboratory measurement.

Let's start with the scatterplot displayed here: if all field and laboratory measurements agree that each and every data point should sit exactly on this line, the field and laboratory measurements will be perfectly correlated. This is not the case in this illustration. In comparison to tiny measurements, variance is high for bigger values of measurement. As a result, the points do not fall on the plot's $y=x$ line. Because the scatterplot's points to the right all fall below the line $y=x$ (except for one), the points do not roughly fall one $y=x$ line, implying that field and laboratory data may differ.

When y = x is compared to the least-square line $y=a+bx$, the $y-$intercept is $0$and the slope is $1$When the explanatory variable $x=0$, the predicted response of $y=0$is shown as intercept=$0$Slope$=1$ indicates that as the explanatory variable $x$ varies, the expected response of$y$changes along the line. When $x$ grows by one unit, $y$ increases by one unit.