Modeling Two-variable Data

Vocab

100

A measure of how far a prediction is from what is actually observed.

What is a residual?

200

A unique line that has the smallest possible value for the sum of the squares of the residuals

What is the Least Squares Regression Line (LSRL)?

300

The measure of the strength of the association between two variables.

What is the correlation coefficient (r)?

400

The four categories that are used to describe association.

What is form, direction, strength, and unusual features?

500

Predicting values that are outside the data set

What is Extrapolation?

r and R-squared

100

Interpret what a correlation coefficient of r = 0.9 means.

What is strong, positive association?

200

If the correlation coefficient is zero, what does that mean about the association?

There is no association.

300

What must be true for the correlation coefficient to be -1?

All of the points are on the LSRL and the slope is negative.

400

Given an LSRL of y = -0.5x + 17 and an R-squared value of .68, determine and interpret the correlation coefficient. Round to the nearest hundredth.

What is r = -0.82? This means there is a moderately strong, negative association.

500

y = 1.25x + 10, where x is the amount of fertilizer in mL and y is the height of the plant (cm). r = 0.92 Determine and interpret R-squared.

What is R-squared = 0.8464 About 85% of the variability in plant height can be explained by a linear relationship with the amount of fertilizer used.

Interpreting Slope & Y-intercept

100

y = 1.25x + 10, where x is the amount of fertilizer in mL and y is the height of the plant (cm). Interpret the y-intercept.

When zero mL of fertilizer are used, the plant is expected to be 10 cm tall.

200

y = 1.25x + 10, where x is the amount of fertilizer in mL and y is the height of the plant (cm). Interpret the slope.

For every additional mL of fertilizer used, the height of the plant increases by 1.25 cm.

300

Given the interpretations of slope and y-intercept, write the equation of the LSRL: When the length of the paint on a pencil is 0 cm, the weight of the pencil is 2 grams. For every additional cm of paint, the weight of the pencil increases by 1.4 grams.

y = 1.4x + 2

400

y = 0.41x -14 where x is temperature (degrees F) and y is the number of people at the park (in 1000s). Interpret the y-intercept.

When the temperature is 0 degrees Fahrenheit, there are -14000 people in the park. (Doesn't make sense)

500

y = 0.41x -14 where x is temperature (degrees F) and y is the number of people at the park (in 1000s). Interpret the slope.

For every additional degree F in temperature, the number of people at the park increases by 410. When the temperature increases by 1 degree F, the number of people at the park increases by 410.

Miscellaneous

100

Describe the association in the graph comparing temperature and ice cream sales. https://www.mathsisfun.com/data/images/scatter-ice-cream1.gif

positive, linear, strong association with no apparent outliers

200

What is Mrs. Jordan's son's name? 

Andy!

300

A reporter was interested in the relationship between the size of a dining party and the amount of time it takes for the dining party to be seated at a restaurant. The reporter selected a random sample of dining parties from a certain region. The resulting data were used to complete a linear regression analysis of the time, in minutes, it takes for a dining party to be seated versus the number of people in the dining party. The linear regression analysis produced the residual plot shown.

Would the linear model be appropriate for the data? 

No, there is a pattern in the residual plot.

400

Given the interpretations of slope and y-intercept, write the equation of the LSRL: When the length of the paint on a pencil is 0 cm, the weight of the pencil is 2 grams. For every additional cm of paint, the weight of the pencil increases by 1.4 grams.

y hat = 2+1.4x

500

When comparing age and height, Eva calculated R-squared to be 0.76. Write a sentence to explain what R-squared means in this context and provide at least one other factor that can explain the variability in height.

76% of the variability in height can be explained by a linear relationship with age. The other 24% could be explained by genetics, general health, environmental factors, etc.