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Lab Activity #13:  Scatterplots, Correlation and Regression

Spring 2023, STAT 1350

Please read carefully through this handout and answer each question as completely as you can. This material relates to what you have learned from Chapters 14 and 15.  We encourage you to first read this content in the textbook before you start to work through this lab activity and to review accompanying lecture material as you complete the activity.         

There are 18 questions in this activity.  You will need to submit the answers to these questions no later than 11:59 p.m. on Friday, April 21^st.  You will share the answers via a Word or PDF file that you submit through the course website (either by going to the “Submit” link within the Week 14 Overview, next to the place where you downloaded this lab activity handout, or by going to the Assignments link on the left side of the course page and clicking on Lab Activity #13).  

Your answers to each question do not have to be long, but they should be as complete as possible.  Aim to be concise but thorough.  Remember that 50% of your grade is based on effort/completeness, and 50% is based on a selection of problems that we will choose to grade for correctness.  For this reason, it’s important that you attempt to answer all questions and ask for help if you are struggling!

If any problems require calculations, please attempt to write out how you arrived at your answer so we can see your thought process.  Further, there are three scatterplots included in this activity that will be important for you to examine.  If the scatterplots are not showing up properly in the Word copy of this assignment, please open the PDF copy to review them.  

Part 1: What is the wind speed of that hurricane?

For years, scientists have studied the factors that affect the wind speed of hurricanes, and they have focused specifically on central pressure as one variable that is related to wind speed.  It turns out that hurricanes develop low pressure at their centers.  This leads to moist air being pulled in, and the moist air affects the rotation of the hurricanes.  Ultimately, this rotation will generate high winds.  Central pressure is measured in millibars and wind speed is measured in knots.   The scatterplot below shows how these variables are related for a sample of 233 hurricanes.  For this data set, the correlation between maximum wind speed and central pressure is r = -0.9, and the regression equation to predict the maximum wind speed of a hurricane based on central pressure is given below:

Predicted maximum wind speed = 1028.1 – 0.97 (central pressure)

Please use the above information to answer Questions 1 through 7.

1. Since we are attempting to predict wind speed based on central pressure, we’d call wind speed a(n) __________________ variable and central pressure a(n) __________________ variable.

2.   From the scatterplot and the value of r, we would describe the relationship between wind speed and central pressure as having a strength that is __________________ and a direction that is __________________.

3. What is the intercept in the regression equation, and how should this number be interpreted in the context of hurricane wind speed and central pressure?  

4. What is the slope in the regression equation, and how should this number be interpreted in the context of hurricane wind speed and central pressure?

5. Use the regression equation to predict the wind speed of a hurricane that has a central pressure of 956 mb. Be sure to show your work in the space below.

6. The correlation between wind speed and central pressure is r = -0.9.  This means that ________% of the variability in wind speed can be explained by the regression equation.    It would also mean that ________% of the variability in wind speed cannot be explained by the regression equation.

7. True or False?  It would be considered extrapolation to try to predict the wind speed of a hurricane that has a central pressure of 1000 mb.

Part 2:  What can help us solve a crime?

Crime scene investigators look for patterns and relationships among variables in order to better understand how a crime was committed and who may have committed that crime.  As an example, imagine that a bloody shoe print is left at a crime scene.  By understanding how shoe size relates to height, an investigator can attempt to predict the height of the individual who left the shoe print, and this piece of information can possibly identify or eliminate potential suspects.

Just how are shoe size and height related to each other?  The heights (measured in inches) and shoe lengths (also measured in inches) were obtained from a large random sample of individuals.  This data is shown in the scatterplot below.  When a regression equation was constructed in order to predict height based on shoe length, the equation was as follows:

Predicted height = 49.91 + 1.80(shoe length)

Use the above information to answer Questions 8 through 13.

8. What is the predicted height for an individual with a shoe length of 9.5 inches?  Please compute this value using the regression equation and show your work in the space below.

9. It turns out that 76% of the variability in height can be explained by the regression equation.  This means the correlation between shoe length and height must be equal to what value?  In other words, what is r?  

10.  If we decided to change the units of measurement, by measuring height in meters (instead of inches) and measuring shoe length in centimeters (instead of inches), would the correlation between height and shoe length change?  

11. If we decided to predict shoe length based on height, rather than predict height based on shoe length, would this change the correlation between height and shoe length?

12. Which one of the following statements is a correct interpretation of the regression equation?  

A. As shoe length goes up by one inch, we predict height to increase by 49.91 inches.

B. As shoe length goes down by one inch, we predict height to increase by 1.80 inches.

C. As shoe length goes up one inch, we predict height to decrease by 1.80 inches.

D. As shoe length goes down by one inch, we predict height to increase by 49.91 inches.

E. As shoe length goes up by one inch, we predict height to increase by 1.80 inches.

13.  Casey’s shoe length is 16 inches.  Should we use the regression equation to predict Casey’s height?  Please explain why or why not.

Part 3:  More reasoning through the big ideas

Questions 14 through 18, below, each present a statement.  For each statement, indicate whether you believe the statement is true or false.  No explanation of your answers is necessary.

14. Outliers have no impact on the correlation coefficient or the regression equation.

15. When you are predicting one variable based on another, the variable you are trying to predict should go on the y-axis in the scatterplot.

16. The correlation coefficient has units.

17. It’s impossible for the intercept in a regression equation to ever be negative.

18. A correlation coefficient can only range in value from -1 to 1.