Unit 3 Review

A family would like to build a linear regression equation to predict the amount of grain harvested per acre of land on their farm. They subdivide their land into several smaller plots of land for testing and would like to select an explanatory variable they can control. Which of the following is an appropriate explanatory variable that the family could use to create a linear regression equation?

• The total amount of rainfall recorded at their farm

• The type of crop planted in the plot the previous year

• The average daily temperature at their farm

• The variety of grain planted in the plot

• The amount of fertilizer applied to each plot of land

Answer explanation

Rain and temperature can not be controlled on a farm outside. These can not be explanatory variables.

The type of crop and variety of grain can be controlled. However, this will no longer be two-variable data we are looking at where we could use a scatterplot to plot our data. This would require us to use something like a bar graph.

On-base percentage plus slugging (OPS) is a statistic used in baseball to measure a team’s batting success. The number of runs scored and OPS for 30 baseball teams was used to conduct a linear regression analysis. The scatterplot and computer output for the regression analysis is shown.

Which of the following is the most appropriate interpretation of the statistic 93.47% in the regression output?

• • There is a strong, positive, linear relationship between number of runs scored and OPS.

• The typical deviation between observed and predicted number of runs scored is 0.9347.

• For each one-unit increase in OPS, the regression model predicts an increase of 93.47 runs scored.

• 
  93.47%⁢ of the observed number of runs scored are close to the regression line.

• 
  93.47% of the variation in number of runs scored can be explained by the linear regression with OPS.

Answer explanation

Lesson 3.2 learning target #2 notes have the sentence stem for the interpretation of R-squared.

"The percent of the variation in y (response variable) explained by the linear relationship with x (explanatory variable)".

This is worth writing down on your graphic organizer.

Which of the following scatterplots could represent a data set with a correlation coefficient of r = -1?

• 
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• 
• 

Answer explanation

A roadrunner is a desert bird that tends to run instead of fly. While running, the roadrunner uses its tail as a balance. A sample of 10 roadrunners was taken, and the birds’ total length, in centimeters (cm), and tail length, in cm, were recorded. The output shown in the table is from a least-squares regression to predict tail length given total length.

Suppose a roadrunner has a total length of 59.0 cm and tail length of 31.1 cm. Based on the residual, does the regression model overestimate or underestimate the tail length of the roadrunner?

• Underestimate, because the residual is positive.

• Underestimate, because the residual is negative.

• Overestimate, because the residual is positive.

• Overestimate, because the residual is negative.

• Neither, because the residual is 0.

Answer explanation

A certain state law requires residential properties to receive an appraisal for tax purposes every six years. A random sample of 25 appraised properties was selected. The following scatterplot shows the value of the properties, in thousands of dollars, before and after the appraisal. Also shown is the least-squares regression line and corresponding computer output.

Which of the following is not an appropriate description of the data in the sample?

• New property values tend to be greater than old property values.

• As old property values increase, new property values tend to increase.

• The relationship between old and new property value is strong and positive.

• For each additional $1,000 in old property value, the new property value increases by $955.97, on average.

• The observed new property values typically deviate from the predicted new property values by about $7,539.

Answer explanation

New property and old property values tend to be about the same. Look at $250 old property value and you will see that at $250 there are dots that intersect. This is only one example, look for more on your own. :)

An engineering student collected data on the maximum height, in feet, and maximum speed, in miles per hour, of thirteen roller coasters. A scatterplot of the data and the least-squares regression line are shown. The maximum heights of five of the roller coasters are 60, 105, 150, 200, and 215 feet.

If the least-squares regression line is used to predict the maximum speed for the five roller coasters, for which maximum height, in feet, would the absolute value of the residual be largest?

• 60

• 105

• 150

• 200

• 215

Answer explanation

60 feet (height) has a maximum speed (mph) at about 35 which is about 5 mph below the LSRL.

150 feet (height) has a maximum speed (mph) at about 65 which is about 5 mph above the LSRL.

200 feet (height) has a maximum speed (mph) at about 70 which is sitting ON the LSRL.

215 feet (height) has a maximum speed (mph) at about 65 which is about 5 mph below the LSRL.

A sample of 15 golfers who played a golf course on a certain day was selected. For each golfer, the average driving distance (x), in yards, and the percent of fairways hit on the drive (y) were recorded.

The scatterplot displays the percent of fairways hit versus the average driving distance. Also shown is the least-squares regression line, y^=66.228+0.0002x.

The point circled on the scatterplot is considered an influential point. A new least-squares regression line will be calculated with the influential point removed. How will the removal of the influential point affect the new least-squares regression line for the remaining 14 points?

• The y-intercept will remain the same, and the slope will be negative.

• The y-intercept will decrease, and the slope will be negative.

• The y-intercept will decrease, and the slope will be positive.

• The y-intercept will increase, and the slope will be negative.

• The y-intercept will increase, and the slope will be positive.

Answer explanation

Remember that influential points like the dot circled can be treated like a magnet to the LSRL.

Right now there is a POSITIVE slope from the equation y^=66.228+0.0002x.

However, if we REMOVE the point there will now be a seesaw effect where the LSRL will drop down and