Question

Drilling down beneath a lake in Alaska yields chemical evidence of past changes in climate. Biological silicon, left by the skeletons of single-celled creatures called diatoms, is a measure of the abundance of life in the lake. A rather complex variable based on the ratio of certain isotopes relative to ocean water gives an indirect measure of moisture, mostly from snow. As we drill down, we look further into the past. Here is a scatterplot of data from 2300 to 12,000 years ago:

a. Identify the unusual point in the scatterplot and estimate its x and y coordinates.

b. Describe the effect this point has on

i. the correlation.

ii. the slope and y-intercept of the least-squares line.

iii. the standard deviation of the residuals.

Short answer

Part (a)$(-19.4,340)$

Part (b)

i. Correlation decreases.

ii. Slope increases and $y-$intercept increases.

iii. Standard deviation increases.

Step-by-step solution

Part (a) Step 1: Given information

Part (a) Step 1: Given information

The point in the upper right corner of the scatterplot is the uncommon point since it deviates from the overall pattern in the other points. We can see that this point roughly correlates to $19.4$ on the horizontal axis and 340 on the vertical axis, implying that the coordinates of the point are $(19.4,340)$

Part (b) Step 1: Explanation

The point in the upper right corner of the scatterplot is an uncommon point since it deviates from the overall pattern in the other points.

i. Because the uncommon point differs from the overall linear pattern in the other points, and because the correlation assesses the strength of the linear link between the variables, the unusual point reduces the correlation.

ii. Because the point is above the regression line to the right, we expect it to steepen the regression line, increasing the slope while also increasing the $y-$intercept.

iii. Because the uncommon point is the farthest away from the regression line, it will have the largest residual. When the odd point is included, the variability in the residuals grows, and the standard deviations of the residuals increase.