When it rains, it pours The figure below plots the record-high yearly precipitation in each state against that state’s record-high $24$-hour precipitation. Hawaii is a high outlier, with a record-high yearly record of $704.83$ inches of rain recorded at Kukui in $1982$

(a) The correlation for all $50$ states in the figure is $0.408$ If we leave out Hawaii, would the correlation increase, decrease, or stay the same? Explain.

(b) Two least-squares lines are shown on the graph. One was calculated using all $50$ states, and the other omits Hawaii. Which line is which? Explain.

(c) Explain how each of the following would affect the correlation, s, and the least-squares line:

• Measuring record precipitation in feet instead of inches for both variables.
• Switching the explanatory and response variables.

A regression line is a straight line that shows how an explanatory variable $x$ affects a response variable $y$ You can use a regression line to forecast the value of $y$ for any value of $x$ by plugging this$x$ into the equation of the line.

The correlation coefficient for all $50$ states shown in the graph is $0.408$

The effect of eliminating Hawaii from the equation is illustrated below. Correlation analysis is a technique for determining the strength of a relationship between two variables. To put it another way, correlation describes the linear relationships between quantitative variables. Because correlation is sensitive to outliers, the data point in Hawaii will have an impact on the correlation.

Hawaii's data point is an upper-end outlier, with values that are greater than average for both the variable's Maximum $24$-hour participation and Maximum annual involvement. Because the outlier is on the higher end, deleting the higher-end data point reduces the correlation. As a result, the correlation decreases.

On the graph, there are two least-squares lines. One is based on all $50$states, while the other excludes Hawaii. Determine the least-squares line with and without Hawaii as a data point. Hawaii:

It is well known that eliminating a data point reduces the correlation. Because the correlation coefficient and regression off ancient have a proportionate relationship, deleting the data point Hawaii reduces the regression coefficient. The line without the data point Hawaii is the least-squares line with a small slope. The line with the data point Hawaii is a least-squares line with a small slope. As a result, the blue line represents the least-squares line with all $50$ states, whereas the red line represents the least-squares line without data.

The correlation, $s$and the last-squares line would be affected by the following:

• For both variables, record precipitation is measured in feet rather than inches.
• Explanatory and response variables are switched.

$s$ and the least-squares line will be affected by the correlation:

For all variables, record precipitation is measured in feet rather than inches:

• The correlation will not change if the units are changed.
• By adjusting the units, the standard error will be reduced.
• When the units of both the dependent and independent variables are modified, the slope of the least-squares line does not change, but the intercept value does.

Explanatory and response variables are switched:

• The correlation will not change if the explanatory and response variables are switched.
• By swapping the explanatory and response variables, the standard errors will be reduced.
• The least-squares line will alter if the explanatory and response variables are switched.

The least-squares line will alter if the explanatory and response variables are switched.