Chapter 3.2, Problem 59E

(a)

To determine

To describe: the overall pattern in the scatterplot and deviations from that pattern.

Answer to Problem 59E

There is a linear association between the two variables.

For small values of laboratory measurement, line fits very well but for higher values of laboratory measurement points deviate much from the line.

Explanation of Solution

Given:

  

The graph shows a clear direction: the overall pattern moves from lower left to upper right. That is, higher laboratory measurement tends to have higher field measurement. This is a positive linear association between the two variables. For small values of laboratory measurement, line fits very well but for higher values of laboratory measurement points deviate much from the line.

(b)

To determine

To explain: whether the points fall on the $y=x$ line drawn on the plot

Answer to Problem 59E

No, the points do not fall on the $y=x$ line drawn on the plot

Explanation of Solution

Let's start with the scatterplot shown here, if field and laboratory measurements all agree then each and every one of the data points were to lie exactly on this line, there would be a perfect correlation between the field and laboratory measurements. However, that is not seen in this figure. For higher values of measurement, variation is high as compared to small measurement.

Thus, the points do not fall on the $y=x$ line drawn on the plot. Because, the points to the right in the scatterplot all fall below the line $y=x$ (except from one point) and thus the points does not roughly fall one $y=x$ line, which indicates that field and laboratory measurements might not agree.

(c)

To determine

To compare: the slope and y intercept of the least-squares line

Answer to Problem 59E

Slope decreases

y-intercept increases.

Explanation of Solution

If we compare $y=x$ to least-square line $\widehat{y}=a+bx$ , y intercept $=0$ and slope $=1$ . $y$ Intercept $=0$ shows the predicted response of $y=0$ when the explanatory variable $x=0$ . Slope $=1$ shows that predicted response of y changes along the line as the explanatory variable x changes. Specifically, y increases by 1 unit when x increases by 1 unit.