Public opinion research companies are often interested in ... 4/Students/Word/lesson_4.1.1_versi… · Web viewDuring the last half of the 20th century, several sites in Nevada were

STATWAY™ STUDENT HANDOUT

Lesson 4.1.1 Investigating Patterns in Data

STUDENT NAME DATE

INTRODUCTION

During the last half of the 20th century, several sites in Nevada were used for testing nuclear devices (i.e., dangerous nuclear devices were exploded on those sites). Nuclear explosions spread dangerous radioactive particles in the air, which settle on the ground; this is dangerous to living things in that area until the radioactive material decays. Testing ended around 1985, so no new radioactive sources were added after that time.

The data in the table and graph [on the next page] give the amount of radioactivity measured at various times. Radiation quantity is measured in milliroentgens (mR). Because radiation exposure is dangerous as it accumulates, these data are measured in mR/year, which is how many mR a person staying on that site for a full year would receive.

(Note: These data are from Position N-8 in Area 2 of the Nevada Test Site of the U.S. Department of Energy.1)

1Data are based on data gathered from U.S. Department of Energy. (2009). Nevada test site environmental report 2008

© 2011 THE CARNEGIE FOUNDATION FOR THE ADVANCEMENT OF TEACHINGA PATHWAY THROUGH STATISTICS, VERSION 1.5, STATWAY™ - STUDENT HANDOUT

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Year x (Years since 1985)

Area 2 mR/yr

1988 3 1,683

1989 4 1,650

1990 5 1,534

1991 6 1,442

1992 7 1,433

1993 8 1,335

1994 9 1,241

1995 10 1,141

1996 11 1,101

1997 12 1,078

1998 13 986

1999 14 945

2000 15 921

2001 16 841

2002 17 784

2003 18 742

2004 19 786

2005 20 693

2006 21 692

2007 22 640

2008 23 600




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When is an area safe?

For a location to be safe for people, animals, and plants, the radioactivity level does not need to be zero. There is always a certain amount of background radioactivity present. Natural background radiation differs some in places, but 340 mR/year is a typical amount. A place is considered safe for people to live if the radiation level is no more than 440 mR/year.

The long-term effects of low-level radioactivity (up to 1,000 mR/year) are not clear.

A one-time radiation dose of 400,000 mR is typically fatal if untreated.

Over the course of this lesson, we will try to predict when Area 2 will be safe for people to live.




TRY THESE

1 Which of the following two questions is easier to answer? Why?

A If you consider a level of 1,000 mR/year safe, when would you say that Area 2 was safe?

B If you decide that the safety level should be no more than 440 mR/year, when will you consider Area 2 safe?

2 Answer the easier of Questions 1A or 1B, and explain how you found the answer.




NEXT STEPS

In this course, you have only worked with linear relationships. Let’s use the graph of a linear relationship to summarize the data and see how satisfactory it is to answer your questions.

Area 2: mR/year

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TRY THESE

3 Use the line, extended as needed, to answer the following questions:

A If you consider a level of 1,000 mR/year safe, when would you say that Area 2 was safe?

B If you decide that the safety level should be no more than 440 mR/year, when will you consider Area 2 safe?

4 Based on your earlier ideas and what you see here, do you think this line is a good summary to use to answer these questions? Why or why not? Do you trust your answers to the questions in Part 1?




NEXT STEPS

In Module 3, you worked with linear relationships. Those are very useful, and so are some other types of relationships between variables. Following are some graphs and examples of nonlinear relationships frequently used in modeling.

TRY THESE5 Notice the different types of patterns. Pick one pattern, and read its example(s) that follows all the

graphs. Discuss whether the example’s description suggests the type of pattern seen in the graph.

Exponential decayy=85 (0.8 )x

Quadraticy=50+42 x−x2

Exponential growthy=5 (1.5 )x

Constrained exponential (logistic)

Periodic (sinusoidal)

y=sin (3.14 x )+2

Cubic

y=x3−2x2−x+2




y= 181+2(7− x)

Exponential decay examples

Radioactive decay: The amount of radioactivity emitted from an object ( y) over time (x).

Medicine: The amount of a medication in your system ( y) over time in minutes (x) starting an hour after you took the dose.

Quadratic example

Height of an object: The height of an object thrown upward ( y) over time in seconds (x).

Exponential growth examples

Population: The population of a country ( y) over time in years (x).

Money invested: The amount of money in an account ( y) that is invested at a certain interest rate, compounded.

Constrained exponential (logistic) example

Population: The population ( y) over time in years (x) in a situation where there are constraints, such as the amount of room or food. It starts from x=0 with an exponential growth pattern, but there is a time where it is not growing as fast, and the rate of growth tapers off.




Periodic (sinusoidal) examples

Ferris wheel: The height of a particular Ferris wheel car ( y) over time in seconds (x). If you look at that over several revolutions of the Ferris wheel, you see a periodic pattern.

Temperature: For a particular location, the average daily temperature for each month ( y) over time measured in months (x). If you look at that over several years, you see a periodic pattern.

Tide: For a particular location on the beach, the height of the tide ( y) at a given time in hours (x) after midnight.




Cubic example

U.S. natural gas consumption (y) over the years 1960 to 2000 (x). Consumption increased from 1960 to 1969, decreased from 1970 to 1989, and then increased from 1990 to 2000. A function that can change directions twice is needed here, and a cubic function does that. (Retrieved from the U.S. Energy Information Administration at www.eia.doe.gov/dnav/ng/hist/n9140us2a.htm)




TRY THESE

6 Using the graphs of the more common non-linear models you just examined, complete each task for Graphs A-F:

A As you look through the noise in the data, visualize which of the common non-linear models would best summarize the data. State the name of the model above the graph.

B Sketch a smooth curve on the scatterplots that models the data.

Graph A Graph B Graph C

Graph D Graph E Graph F




TRY THESE

7 Use the Area 2 scatterplots [on the next page] to answer the following questions

A As you look through the noise in the data, visualize which of the common non-linear models would best summarize the data.

B Sketch a smooth curve on the scatterplot that best models the data.

C What is the name of this type of relationship?

D Notice the second graph, which includes a smooth curve and extended time. Using that, estimate when the amount of radioactivity will decrease to a safe level for people to live in the area (440 mR/year or less).

E Using that smooth curve, estimate when the amount of radioactivity decreased to 1,000 mR/year.

F Look back to question 1 of this lesson where you observed the data graph that was provided without any summary line or curve and estimated the time when the amount of radioactivity decreased to 1,000 mR/year. What year did you find and what was your method for finding it?

G How close was that first estimate of the time when it decreased to 1,000 mR/year to your estimate from this smooth exponential curve? Was it pretty good or not so good?

H Compare the exponential model to the linear model and note the differences, particularly how much the linear model is off (error).




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Area 2—mR/year, with linear and exponential models and with extended time

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I If you did not have software to provide you with a graph of the best exponential smooth curve to fit the data, could you sketch something to make the predictions?




TAKE IT HOME

1 Summarize what you learned today.

On the next page you will find the radiation data and a graph for Area 4, another testing site in Nevada. Use this information for questions 2 through 4:

2 Using the data-only graph, estimate your answers to the two questions about when the area will be safe.

If you consider a level of 1,000 mR/year safe, when would you say that Area 4 was safe?

If you decide that the safety level should be no more than 440 mR/year, when will you consider Area 4 safe?

3 Is it a good idea to use the linear model graph (with the data and the summary straight line) to answer the following questions: Explain your reasoning. If it is a good idea, do it.






4 Is it a good idea to use the exponential model graph (with the data and the summary curved line) to answer the following questions? Explain your reasoning. If it is a good idea, do it.



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Area 4—mR/year with linear model and extended time

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Area 4—mR/year with exponential model and extended time

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Draw a sketch of the situation described, then using the non-linear models presented in class, identify which model fits your sketch.

5 Fuel Efficiency It was found by the EPA that most cars use their fuel most efficiently when being driven at 55 mph. An idling car (engine on, not driving gets 0 mpg) is the least efficient, but the efficiency improves as you approach 55 mph. According to the “2005 Model Year Fuel Efficiency Guide” put out by the EPA (http://www.fueleconomy.gov/feg/pdfs/guides/FEG2005.pdf) , to help improve fuel efficiency of your vehicle it states that the driver must:

Observe the speed limit—each 5 miles per hour (mph) you drive

over 60 mph is like paying an additional 10¢/gallon.Avoid idling—idling gets 0 miles per gallon.

6 Facebook Users In December 2004 Facebook had one million users. By December 2007, there were 50 million users, and by December 2010 that number grew to 650 million. The sketch should represent the growth of Facebook users over this six year period.


http://www.fueleconomy.gov/feg/pdfs/guides/FEG2005.pdf)%20



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This lesson is part of STATWAY™, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. Version 1.0, A Pathway Through Statistics, Statway™ was created by the Charles A. Dana Center at the University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version 1.5 and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is an open-resource research and development community that seeks to harvest the wisdom of its diverse participants in systematic and disciplined inquiries to improve developmental mathematics instruction. For more information on the Statway Networked Improvement Community, please visit carnegiefoundation.org. For the most recent version of instructional materials, visit Statway.org/kernel.

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Public opinion research companies are often interested in ... 4/Students/Word/lesson_4.1.1_versi… · Web viewDuring the last half of the 20th century, several sites in Nevada were