Pre-Calculusimages.pcmac.org/SiSFiles/Schools/AL/StClairCounty/...Show all of your work. Be sure to state the domain and range for the new function. Remember that the domain and range

Module 2

Pre-Calculus

2nd Nine Weeks Table of Contents Precalculus Module 2 Unit 4 Conics Transformations of Graphs of Conic Sections (1‐3) Conics in Parametric Form (4‐6) Planets, Parametric Curves, and Ellipses (7‐10) Unit 5 Probability and Statistics Comparing Boxplots (11‐15) Empirical Rule and Normal Distributions (16‐21) Applying the Binomial Expansion to Probabilities (22‐25) Let’s Take a quiz (26‐29)

How is my Driving (30‐34) The Jury (35‐37) I Want Candy (38‐46) Take a Sample Please (47‐54)

Student Activity

Transformations of the Graphs of Conic Sections Parabolas

1. Given ( ) 2f x x= , graph ( )y f x= .

a) Graph the following transformations of ( )f x then write an equation for each graph. i. ( )f x− ii. ( )f x− iii. ( )2f x + iv. ( ) 1f x − v. vi. ( )3 2f x − + ( )2 f x

Answer questions b - e below for ( )f x and each transformation given in part i - vi. b) Find the domain and range for the function. c) Find the minimum or maximum value of the function. d) State the intervals of x where the function is increasing or decreasing. e) Is the function an even function, an odd function, or neither?

2. a) Given ( ) 2f x x= , reflect the graph of ( )y f x= over the line and graph the reflection. Write an equation for the graph of the reflection then solve for y. Is the relation a function?

y x=

b) Reflect each of the transformations in problem 2a over the line and graph each reflection. Write an equation for the graph of each reflection then solve the equation for y.

y x=

3. Given ( )f x = x , graph ( )y f x= . a) Find the domain and range for the function. b) Find the minimum or maximum value of the function. c) State the intervals of x where the function is increasing or decreasing.

4. Region R is the region in the first quadrant bounded by the graphs of 2y x= , , and the y-axis.

4y =

a) Sketch the graph of region R. b) Sketch the graph of the solid formed by revolving region R about the y-axis.

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Student Activity

Circles

1. Graph the circle then solve the equation for y. 2 2 9x y+ =

2. Given ( ) 29f x = − x , graph ( )y f x= .

a) Find ( )1f and . ( )15f

b) Find the value of x where ( ) 2f x = .

c) Find the value of x where ( ) 2f x = − . d) Find the domain and range for the function. e) Find the maximum and minimum value(s) of the function. f) State the intervals of x where the function is increasing. State the intervals of x where the

function is decreasing.

g) Is ( )f x an even function, an odd function, or neither? Explain your answer.

h) Graph the following transformations of ( )f x then write an equation for each graph.

i. ( )f x− ii. ( )f x− iii. ( )2f x + iv. ( ) 1f x −

v. 12

f x⎛⎜⎝ ⎠

⎞⎟ vi. ( )

12

f x vii. ( )1 2f x − +

3. Region R is the region in the first quadrant bounded by the graphs of 29y = − x , the x-axis, and the y-axis.

a) Sketch a graph of region R and find the area of the region. b) Sketch a graph of the solid formed by revolving region R about the y-axis and find the

volume of the solid.

4. Given ( ) 21 4f x x= + − , graph ( )y f x= a) Find the domain and range for the function. b) Find the maximum and minimum value(s) of the function. c) State the intervals of x where the function is increasing. State the intervals of x where the



Student Activity

Hyperbolas

1. Graph the hyperbola then solve the equation for y. 2 2 9y x− =

2. Given ( ) 29f x = + x , graph ( )y f x= . a) Find x and y intercepts of the graph. b) Find the domain and range for the function. c) Find the maximum or minimum value of the function. d) State the intervals of x where the function is increasing. State the intervals of x where the


e) Is ( )f x an even function, an odd function, or neither? Explain your answer.

f) Graph the following transformations of ( )f x then write an equation for each graph.

i. ( )f x− ii. ( )f x− iii. ( )3f x −

iv. v. ( ) 2f x − 12

f x⎛⎜⎝ ⎠

⎞⎟ vi. ( )

13

f x

3. Graph the hyperbola then solve the equation for y. 2 2 4x y− =

4. Given ( ) 24f x x= − + , graph ( )y f x= . a) Find x and y intercepts of the graph. b) Find the domain and range for the function. c) Find the maximum or minimum value of the function. d) State the intervals of x where the function is increasing. State the intervals of x where the


e) Find the intervals where the function is continuous.

f) Is ( )f x an even function, an odd function, or neither? Explain your answer.

5. Given ( ) 22 1f x x= − + − + , graph ( )y f x= . Find the x and y intercepts of the graph and the domain and range for the function.


Student Activity

Conics in Parametric Form A graphing calculator should be used for most of the questions in this activity. The window should be set so that the scale is the same on both axes. Try using [–7.58, 7.58] as the interval for x and [–5, 5] as the interval for y. When changing the window, use Zoom Square to keep the scales the same in both directions

1. With the calculator in rectangular mode, graph 21 2y x= − . What conic section is represented by the graph? Sketch the graph on paper making sure that the vertex and the x-intercepts are marked correctly.

2. Change the calculator mode to parametric and set the interval for t to [–5, 5] and the t-step to 0.05. Graph t= 21 2 and x y t= − . How does this graph compare to the one drawn in question 1?

3. On your calculator set the graph style to –0 and draw the graph again. Assume that the graph represents the path that a particular point (a particle) follows. Explain the motion of the particle. (To make the calculator redraw the graph, press Draw then select ClrDraw.)

4. Change the interval for t to [0, 5] and explain how this changes the graph.

For questions 5 – 13, graph sin( )x t= and cos(2 )y t= with the interval for t as [–5, 5].

5. Compare the result to the graph draw in question 2.

6. Explain why the x-values and the y-values of this graph remain between –1 and 1?

7. What is the location for the particle at t = –5? What is the location for the particle at t = 5?

8. Set the graph style to –0 and redraw the graph. Assume that the equation defines the path of a particle and explain the behavior of the particle. (To improve the graph, set the window to [–1, 1] for both the x and y and then press zoom square.)

9. Change the interval for t several times and explain how the changes affect the graph.

10. Find an interval for t that allows the particle to travel the path exactly once beginning on the right and moving to the left.

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4

Student Activity

11. Using the double angle trig identity, , convert the parametric equations 2cos(2 ) 1 2sin ( )t = − tsin( )x t= and into a rectangular function. Show all of your work. Be sure to

state the domain and range for the new function. Remember that the domain and range will be determined by the original form of the equation.

cos(2 )y = t

12. What is lost by converting the parametric equations sin( )x t= and to rectangular form?

cos(2 )y = t

13. Put the calculator into rectangular mode and graph 21 2x y= − . What equation(s) must you enter into your calculator? Is this a function? How does this graph compare to the one drawn in question 1?

14. Many different procedures can be used to convert an equation in rectangular form to one in parametric form; however, one simple procedure is often used in order to graph equations that are not functions of x. Set one of the variables equal to t then substitute t for that variable in the rectangular equation. To do this for the equation in question 13, let y t= and replace y with t in the x-equation. Put the calculator back into parametric mode using an interval for t of [–5, 5] and [–7.58, 7.58] for x and [–5, 5] for y and graph the results.

15. Use the technique developed in question 14 to graph 22 3x y y 2= − − Discuss the steps that are necessary to graph the equation in rectangular form. Be specific.

16. Try using the technique from question 14 to convert 2 2 1x y+ = to parametric form. Explain why this is not a particularly helpful procedure in this case. Note: When both variables are squared, the conversion to parametric form is often made with trig functions using the Pythagorean Identities.

17. Use zoom square to make sure that the scales on the x- and the y-axes are the same, then graph x cos( )t= sin( )y t= and Which conic section is drawn? Convert the equations to rectangular form to confirm your answer. Show your work. Hint: Remember the Pythagorean Identity.

18. Find an interval that can be used for t that will draw the figure in question 17 without retracing a portion of the path. Explain your answer.

19. Graph sin( )x t= and How does the motion change? cos( )y = t

20. Graph cos(2 )x t= and . Explain how the motion changes from that of the equations in question 16. Check the results algebraically to make sure that the shape of the path has not been changed.

sin(2 )y = t


5

Student Activity

21. Vary the equations cos( )x t= and sin( )y t= to change the radius of the figure. Explain your process.

22. Vary these equations to change the center of the figure. Explain your process.

23. Write a set of parametric equations for the circle ( ) ( )2 23 2x y 4− + + = . Explain your reasoning.

24. Modify the equations in question 21 to create the graph of an ellipse. Explain your reasoning.

25. Given the ellipse defined by ( ) ( )2 23 2

19 16

x y− −+ = , write a set of parametric equations that

will produce the same ellipse. Hint: Rewrite the equation in the form 2 2

2 2( ) sin ( ) 1t t

3 2 13 4

x y− −⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

and relate it to the identity, cos + =

5cos( ) 3x t

. Show your work and discuss the domain for t.

26. Without using your calculator or doing any calculations, describe the graph that will be drawn by the parametric equations + 6sin( ) 1y t and = = −[0, 2 ]

Set the domain for t as π then graph the equations with a calculator.

27. With the domain for t set as [0, 2 ]π , graph the parametric equations 5cos 34

x t π⎛ ⎞= −⎜ ⎟⎝ ⎠

+ and

6sin 14

y t= −⎜ ⎟⎝ ⎠

π⎛ ⎞ −

=

Explain how this graph differs from the one in question 26.

28. Use the identity to convert the hyperbola 2 2sec ( ) tan ( ) 1t t−2 22 1 1

3 4x y− +⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ to

parametric form. Remember that on the calculator, sec must be entered as t 1cos t

. Show

your algebraic calculations then graph the hyperbola in parametric mode. (Using dot mode may eliminate the extra lines that the calculator draws to connect the two branches of the hyperbola. If this does not work, restrictions must be added to the equation to limit the domain.)


6

Student Activity

Planets, Parametric Curves, and Ellipses From February 7, 1979 to February 11, 1999, Pluto’s distance from the Sun was less than that of Neptune and made it the 8th planet instead of the 9th. This occurrence repeats every 248 years. The tilt of Pluto’s orbit is much greater than that of Neptune so in actuality, their paths never cross. However, ignoring reality, if the orbits of Pluto and Neptune are in the same plane, will they ever crash into each other? Three assumptions must be made in this activity.

(1) The planets revolve on the same plane. This assumption allows the equations to be written in two dimensions rather than three and permits the use of a graphing calculator to visualize the motion of the planets as they orbit the Sun.

(2) At time t = 0, Pluto and Neptune are aligned with the Sun at their perihelions (the closest distance from the Sun).

(3) The physical size of the planet is ignored.

Astronomy Definitions: Astronomical Unit (AU): The average distance from the Earth to the Sun. One AU = 93 million

miles or 149.6 million km. (This avoids having to use really big numbers in the calculations!)

Aphelion: The point on the orbit of a planet where the planet is furthest from the Sun.

Perihelion: The point on the orbit of a planet where the planet is closest to the Sun. Eccentricity: The eccentricity of an orbit is a measure of how much the orbit

deviates from that of a circle. The eccentricity, e, of any ellipse is 0. The eccentricity of a circle is 0. The closer the value of e

is to zero, the more circular the orbit appears. The value of e is the ratio between the distance from the center of the ellipse to a focal point, divided by the distance from the center of the ellipse to the

endpoint of the major-axis or

0 e<

Student Activity

Parametric Equations of an ellipse:

Center of the ellipse

b

a

focal point - Sun

c

Aphelion Perihelion

( )cossin( )

x a wty b wt=

=

where, a is the length of the semi-major axis (distance from the center of the ellipse to a vertex on the major axis) b is the length of the semi-minor axis (distance from the center of the ellipse to a vertex on the minor axis) c is the distance from the center of the ellipse to a focal point

2period in Earth years

w π=

Planet Aphelion in

astronomical units (AU)

Perihelion in astronomical units (AU)

Eccentricity Orbital Period in Earth years

Mercury 0.466697 0.307499 0.205630 0.240846 Venus 0.72823128 0.71843270 0.0068 0.6151970 Earth 1.0167103335 0.9832898912 0.016710219 1.0000175 Mars 1.665861 1.381497 0.093315 1.8808 yrs Jupiter 5.458104 4.950429 0.048775 11.85920 Saturn 10.11596804 9.04807635 0.055723218 29.657296 Uranus 20.08330526 18.37551863 0.044405586 84.323326 Neptune 30.44125206 29.76607095 0.011214269 164.79 *Pluto 49.30503287 29.65834067 0.24880766 248.09

* Pluto is now classified as a dwarf planet instead of a planet.


Student Activity

For Neptune: 1. Examine the table of planet values and calculate the distance between the aphelion and the

perihelion then calculate the value of a, the length of the semi-major axis. 2. Calculate the value of c, the distance from the center of the ellipse to the focal point. The

distance from a focal point to the perihelion is given in the table. Also remember the eccentricity

equals the ratio of c to a. cea

⎛ =⎜⎝ ⎠

⎞⎟

2

Use the value of the eccentricity to check your calculations

of c and a. 3. Calculate the value of b, the length of the semi-minor axis. This one comes from the equation

relating the constants in the equation of the ellipse, 2 2a b c= + . 4. The orbital period for Neptune is 164.79 Earth years. Complete the parametric equations. 5. Translate the x-equation so that the focal point is moved c units to the left in order to place the

Sun at the origin of the coordinate plane. 6. Set the calculator to parametric mode. Arrange the window so that the graph will fit, then use

zoom square to make the scale the same for the x- and the y-axes. Turn on the graph type –0 so that the motion of the planet can be viewed. Set the domain for t so that the planet will make at least one complete orbit, and set the t-step to 1 so that the position is plotted once for each Earth year.

For Pluto: 7. Repeat the procedure given above, and write the parametric equations for Pluto. Graph the new

equations on the same graph as Neptune. Make sure that at least one cycle for Pluto is graphed.

For the Crash: 8. Zoom in on the graphs so that the orbit of Pluto can be seen inside that of Neptune. Approximate

the x- and y-coordinates of the two points of intersection. Check to see if the two planets are at these points at the same time. Explain your answer.

9. Return the graph to the original window and increase the domain for t so that both planets make

numerous complete orbits. Set the t-step to 2 in order to speed up the process. Try letting t-min equal 250. What is a rough estimate of the first time that they appear to possibly collide after the first orbits?


Student Activity

10. If the two planets collide, the distance between them must be zero. Graphing the distance function will give a better picture of the positions of the planets with respect to each other. Let x = t and let y = distance. Write a set of parametric equations for the calculator that will graph the distance between the two planets with respect to time then enter them in the calculator without erasing the orbit equations for the planets.

( ) ( )2 2N P N P

x t

y x x y y

=⎧⎪⎨

= − + −⎪⎩

11. Turn off the equations for the orbits of the planets. The window for this graph is very different.

Since x = t, the intervals for x and t must be the same. Adjust the interval for y until the graph appears to fill the window. To speed up the process try a t-step of 5. Approximately what is the minimum AU distance in between the two planets the second time that they are close together? If 1 AU = 93,000,000 miles approximately what is the distance in miles between the two planets?

12. For , is the value of y ever equal to zero? Increase the domain for t and x and

continue the investigation. Vary the value of x-min in order to more efficiently examine larger time values.

0 30000t≤ ≤

Extension: 13. One of the assumptions made for the activity was that at t = 0, Neptune and Pluto would be at the

perihelion of their orbits. Turn off the distance graph and return to the graphs of the orbits. Translate the equations for Neptune by translating its time approximately 25% of its orbital period so that a right angle is formed between the two orbiting bodies and the Sun. (This translation is done by shifting the arguments of both trig functions.) This is closer to the current position between the two planets.) Examine the distance between the planets and make a statement about the possibility of a crash.

14. Explain how the equations could be altered to insure that a crash will occur. 15. Since an approximate position for Pluto is known in 1979 and in 1999, locate Pluto’s

approximate position on the plane (x, y) in 2009. Calculate the distance between the Pluto and the Sun in 2009.

16. Compare the distances between other planets in Astronomical Units (AU).


Comparing Boxplots

1. On the boxplot provided, estimate the values of the maximum, the minimum, the median, the first quartile and the third quartile.

a) Label the values on the diagram.

10 20 30 40 50 60 70 80 90 100 b) How do the quartiles divide the data? c) What is the range of the middle 50% of the data (IQR)? d) What is the range for the lower 75% of the data? e) What is the shape of the boxplot? f) Does the boxplot display any outliers? Show your work to support your answer. g) How do you expect the value of the mean to compare to the value of the median? Justify

your answer.

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2. Below is a table of selected American and Japanese vehicles and their weights in pounds.

Japanese Vehicles Weight

Japanese Vehicles Wt.

American Vehicles Weight

American Vehicles Weight

1 3195 14 4280 1 4180 15 4520 2 2610 15 3500 2 3500 16 5810 3 3375 16 3240 3 7270 17 4225 4 3305 17 3285 4 5900 18 2770 5 1875 18 2570 5 4515 19 3255 6 4315 19 2595 6 4410 20 3495 7 4450 20 2150 7 5295 21 3050 8 2790 21 3915 8 2760 22 3341 9 4060 22 5435 9 3270 23 4935

10 3875 23 2985 10 3870 24 4145 11 4710 24 2235 11 3340 25 3990 12 3070 25 2750 12 3905 26 5590 13 5280 13 4200 27 5505

14 3055 28 4660 Source: Consumers Report

, April 2003

The following are the boxplots representing the previous table of values.

a) Complete the following sentence to compare the centers of the two boxplots using comparative words such as greater than or less than in the context of the situation.

The approximate median weight of an American vehicle is _______ (give the value) which is ___________ (use a comparative word – lighter or heavier) than the approximate median weight of a Japanese vehicle of ______(give the value).

Type

_of_

Car

Amer

ican

Japa

nese

Weight1000 2000 3000 4000 5000 6000 7000 8000

Vehicle Weights Box PlotWeights of Selected Vehicles

Weight in Pounds

12

b) Complete the following sentences to compare the spread of the data shown in the boxplot.

The middle 50% of the Japanese vehicles lie between approximately ______ pounds and

______ pounds compared to the middle 50% of the American vehicles that lie between

approximately______ pounds and _____ pounds.

We see that ____ of the Japanese vehicles weigh less than all of the American vehicles.

c) Complete the following sentence to compare the shape of the two boxplots.

Both the ____________ and the ______________ boxplots appear to be slightly

__________ toward the _________________ vehicles.

For both the American and the Japanese vehicles, the _________ 25% of the vehicles has a

larger range than the ____________25%.

d) Complete the following sentence to compare any unusual features of the data.

There is one ____________ vehicle that weighs noticeably more than all vehicles.

e) What information can be gained from the table of values that is not available from the boxplot?

f) Using the information developed in parts (a) through (d), write a short article for your school newspaper comparing the weights of the Japanese and the American cars.

13

1. 3. Below are boxplots for the number of floors in the 50 tallest buildings in New York and

Chicago.

Source: skyscraperpage.com

In each of the following sentences, include a specific reference to the situation being described by the boxplots. a) Write a sentence based on the boxplots that compares the center of each distribution. b) Write a sentence based on the boxplots that compares the spread of each distribution. c) Write a sentence based on the boxplots that compares the shape of each distribution. d) Write a sentence addressing the issue of outliers. e) Write a concluding

City

Chica

goNe

w Yo

rk

Number of floors30 40 50 60 70 80 90 100 110 120

Collection 1 Box Plot

summary about the difference between the number of floors in the Chicago and New York buildings that is based on the boxplots.

Number of Floors in the 50 Tallest Buildings

14

4. The Highway Loss Data Institute rated four-door cars based on the number of personal injury insurance claims. Lower numbers mean a better safety record.

In each of the following sentences, include a specific reference to the situation being described by the boxplots. a) Write a sentence based on the boxplots that compares the center of each distribution. b) Write a sentence based on the boxplots that compares the spread of each distribution. c) Write a sentence based on the boxplots that compares the shape of each distribution. d) Write a sentence addressing the issue of outliers. e) Write a concluding summary about the safety rating of small, midsize, and large vehicles

based on the boxplots.

Size

Larg

eM

idsiz

eSm

all

InjuryRatings50 60 70 80 90 100 110 120 130 140 150 160

Four-Door Models Box PlotPersonal Injury Insurance Claims

15

1Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.

Mathematics NATIONALMATH + SCIENCEINITIATIVE

Empirical Rule and Normal Distributions

Data distributions that have a symmetrical mound shape can be described as having an “approximately normal” distribution. Distributions of this shape have certain properties that allow estimates to be made about the total population percentage having particular values.

1. Consider the distribution of SAT math scores for a group of students. 504,542,544,568,568,573,575,577,578,585,599,603,609,610,628,645,655,670,679a. Draw a histogram of the SAT scores on the graph provided.

b. Describe the shape of the distribution of scores.

c. Determine the mean and standard deviation of this distribution.

d. What percent of the SAT scores are less than one standard deviation from the mean? Within two standard deviations?

16

Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.2

Mathematics—Empirical Rule and Normal Distributions

In an approximately normal distribution, the empirical rule states that about 68% of the values fall within one standard deviation of the mean, 95% of the values will fall within two standard deviations of the mean, and 99.7% will fall within three standard deviations of the mean.

2. Consider the distribution of defensive lineman in a professional football league. The mean weight for defensivelinemenintheleagueis284.6lbswithastandarddeviationof5.3lbs,andthedistributionisapproximately normal. a. Draw vertical lines on the sketch of the distribution showing the mean and standard deviations out

to ±3 standard deviations. Label the percent of the population within one, two, and three standard deviations.

b. Aboutwhatpercentofdefensivelinemenintheleagueweighbetween274and295.2pounds?

c. Since the distribution of weights is symmetric, the mean and median are equal. What percent of the defensivelinemenweighlessthanthemean,284.6pounds?

d. About what percent of the defensive linemen weigh less than 279.3 pounds? Use the sketch of the distribution to support your answer.

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e. About what percent of defensive linemen in the league have a weight above 295.2 pounds? Fill in the sketch of the distribution to support your answer.

f. Michaelweighs293.4pounds.Atleast________percentandnomorethan_________percentofthe defensive linemen in this league weigh less than Michael.

g. ToapproximatethepercentoflinementhatweighlessthanMichael,firstdeterminethestandardized

score or “z-score” of his weight. The z-score is computed using where xᵢ represents

the value under consideration, μ represents the mean of the data set, and σ represents the standard deviation of the data set. What is Michael’s z-score?

h. A table of population percentages below particular z-scores is provided at the end of the lesson. The z-table gives the population percent below a particular z-score if the population distribution is normally distributed. Use the table to determine the approximate percentage of defensive linemen that weigh less than Michael.

i. What is the z-score for Marcus who weighs 275 pounds?

j. Considering that 100% of the linemen are included in the distribution, what percent of linemen weigh more than Marcus? Fill in the sketch of the distribution to support your answer.

k. Whatpercentofthedefensivelinemenweighbetween280and293.4 pounds? Fill in the sketch of the distribution to support your answer.

l. If a certain golf cart used to transport injured players is only rated to hold players up to 300 pounds, whatpercentofthedefensivelinemanwouldneedadifferentcartiftheywereinjured?

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3. AtFill-er-upBottlingCompany,themachinesthatfillthesodacansaresettoput12ouncesofsodainacanlabeledtocontain12ounces.Duetovariabilityinthefillingmachines,theactualvolumeofsodain the 12 ounce cans has an approximately normal distribution with a mean of 12 ounces and a standard deviation of 0.2 ounces. a. What percent of soda cans produced by Fill-er-up contain less than the labeled 12 ounces?

Use a sketch of the distribution to support your answer.

b. Whatpercentofthesodacansproducedwillcontaingreaterthan12.45ouncesofsoda? Use a sketch of the distribution to support your answer.

c. What percent of the soda cans produced will contain between 11.75 and 12.25 ounces of soda? Use a sketch of the distribution to support your answer.

d. Assuming that the can has a true capacity of 12.6 ounces of soda, how likely is it that a can will beoverfilled?

e. Fill-er-up Bottling Company produces and ships 10,000 cans of soda every day. About how many cans per day should they expect to have at least 12.55 ounces of soda?

f. Luke is the production manager at Fill-er-up and has just found three cans in a row containing at least 12.55 ounces. Assuming the machine is functioning correctly, what is the probability of 3 cans in a row containing at least 12.55 ounces? Is this outcome likely to happen if the machine is working properly?(Hint:assumethebottlesarefilledindependentlyofoneanother)

g. Based on your answers to part (f), do you think the machine is working properly? Explain your answer.

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z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09-3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010-2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014-2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019-2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026-2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036

-2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048-2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064-2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084-2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110-2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143

-2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183-1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233-1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294-1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367-1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455

-1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559-1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681-1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823-1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170

-1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379-0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611-0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867-0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148-0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451

-0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776-0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121-0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483-0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859-0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.42470.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641

20



z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.53590.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.68790.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224

0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.78520.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.83891.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88301.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.90151.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.91771.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.93191.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441

1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.96331.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.97061.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.97672.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817

2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.98572.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.98902.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.99162.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.99362.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952

2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.99642.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.99742.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.99812.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.99863.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

21



Applying the Binomial Expansion to Probabilities

1. Suppose Jim Brown historically makes 90% of the free throws he attempts. In one game, he attempts 6 free throws. a. Writethebinomialexpansionthatcanbeusedtodeterminetheprobabilityofeachpossibleoutcome

when Jim attempts 6 free throws.

b. WhatistheprobabilitythatJimwillmakeexactly4outofthe6freethrows?

c. Create the probability distribution (make a table or a chart) to determine the most likely occurrence when Jim attempts 6 free throws.

d. Usingyourdistributioninpart(c),whatisthemostlikelyoutcome?Whatistheprobability thisoccurs?

e. Expected value is the theoretical mean number of successful free throws and is the product of the number of attempted free throws and the probability of making a single free throw. On average, how manyfreethrowsdoyouexpectJimtomakewhenheattempts6freethrows?

f. Graph the probability distribution of Jim making 0 through 6 free throws as a histogram.

22


Mathematics—Applying the Binomial Expansion to Probabilities

g. Jimmakes90%ofhisfreethrows.Whenheshoots6freethrows,theprobabilitythathemakesall6shots is a little more than 50%. Explain how both statements can be true.

2. Dan Nguyen historically makes 40% of the free throws he attempts and plays in a game against Jim’s team. Assume Dan also attempts 6 free throws in that game.a. Create the probability distribution (make a table or a chart) for when Dan attempts 6 free throws.

Display the distribution as a histogram.

b. WhatistheexpectednumberoffreethrowsDanwillmake?

c. The graph of the probability distribution for Dan differs from Jim’s graph in part 1f. Use your answers in parts (a) and (b) to explain why the distribution looks different.

d. Suppose that, in the game, Jim and Dan both make 4 out of 6 free throws. The sports reporter that follows Jim’s team describes his free throw shooting as an “off night,” while the sports reporter that follows Dan’s team records his performance as a “great night.” Explain how, although both men made the same number of free throws, Jim gets a bad review and Dan receives praise. Justify your answer using your answers to question 1c and question 2a.

23



3. Suppose that a quarterback historically completes 77% of his passes. In the next quarter, he plans to pass four times.a. Use binomial expansion to write out the probability distribution for the number of incomplete passes

in that quarter.

b. Graph the probability distribution for 0 to 4 incomplete passes in that quarter as a histogram.

c. Whatistheprobabilitythatexactlyonepasswillbeincomplete?

d. Whatistheprobabilitythatnomorethanonepasswillbeincomplete?

e. Explain why the probability in part (d) is high.

24



f. Use the options in the cells of the table to complete the sentences provided. Some may be used more than once while others may not be used at all.

4 2 symmetric3.08 0.92 skewed right

3 uniform skewed left

The probability that he throws at least __________ incomplete passes is 4.0273%.

The expected number of incomplete passes thrown in a quarter where he throws 4 passes is __________.

The probability distribution for 4 passes is _________________in shape.

25



Let’s Take a Quiz

1. You have to take a ten-question quiz next period, and you completely forgot to study for it. Not only that, but you have no idea how to answer any of the questions. The teacher said the quiz was going to be all True/False. You decide to take your chances and are going to randomly guess all the answers. To pass the quiz, you must guess correctly on at least six of the questions. Use the following model to simulate the probability of guessing correctly on at least six questions.a. Assume the probability for answering correctly remains the same throughout the simulation and that

the probability of selecting a correct answer does not affect the outcome of the next answer. What is the probability of “guessing” a correct answer?

b. Select a randomization device to use for a simulation to determine the probability that you get at least sixquestionscorrectjustbyguessing.Describehowyouwillusethisdevicetomodeltheprobabilityof a correct answer.

c. Run 10 trials of your simulation and record your results in the table.

Trial 1 2 3 4 5 6 7 8 9 10

# Correct

# Incorrect

d. Combine your results with the entire class and graph the distribution of the number of correct answers for this ten-question quiz using a dotplot.

e. What is the approximate probability that you will get at least six questions correct simply by guessing? Base your answer on the data collected by the entire class. The theoretical probability is approximately0.377.Howclosewastheexperimentalprobabilitytothetheoreticalprobability?

f. Based on your simulation, explain to a classmate why guessing would not be a good method to use to pass this quiz.

26


Mathematics—Let’s Take a Quiz

2. Today, you cannot believe it, but as you walk into class, your teacher announces another pop quiz. The teachersaystherearegoingtobetenquestionswitheachquestionhavingfivechoices.Sinceyouknownothing about the subject being tested, you again decide to randomly guess on all 10 questions.a. Assume the probability for answering correctly remains the same throughout the simulation and that

the probability of selecting a correct answer does not affect the outcome of the next answer. What is the probability of “guessing” a correct answer?

b. Construct a spinner to use as your randomization device to run your simulation. A spinner template is providedonthelastpageoftheStudentActivity.Describehowyouwillcalculatethedegreesforthesections that you will use in your spinner.

c. Describehowyouwillusethespinnertodeterminetheprobabilitythatyougetatleastsixquestionscorrect just by “guessing.” How many times will you spin? Explain.

d. Run 10 trials of your simulation and record your results in the table.

Trial 1 2 3 4 5 6 7 8 9 10

# Correct

# Incorrect

e. Combine your results with the entire class and graph the distribution of the number of correct answers for this ten-question quiz using a histogram.

27



f. Describetheshapeofthegraph.

g. Should the distribution of correct answers for this ten-question quiz look different than the distribution for the ten-question True/False quiz? Explain why or why not.

h. What is the approximate probability that you will get at least six questions correct simply by guessing? Base your answer on the data collected from the entire class. The theoretical probability is approximately 0.0064. How close was your experimental probability?

i. Expected value is the theoretical mean number of correct answers and is the product of the number of questions in the quiz and the probability of answering a question correctly. How many questions should you “expect” to answer correctly if you guess the answer for every question? What is the difference between the class arithmetic mean of correct answers and the expected value?

j. Based on your simulation, explain to a classmate why guessing would not be a good method to use to pass this quiz.

28



SPINNER

29

Student Activity

How Is My Driving? California, Connecticut, New Jersey, New York, and Washington have enacted laws that make talking on a cell phone illegal while driving. Based on several experiments, researchers have concluded that people who talk on their cell phones while driving are prone to have more collisions because of a lack of attention to traffic conditions. In a particular experiment to test the effects of talking on a cell phone, 48 people, 24 males and 24 females, were randomly assigned to two driving simulators. In one simulator the drivers talked on a cell phone. In the other simulator the drivers talked to a passenger. The simulators set up basic navigational tasks, such as changing lanes, exiting at a rest area, and maintaining an appropriate speed. The results of successfully completing the tasks were recorded. Twelve of the 24 drivers were unsuccessful with basic driving tasks while talking on their cell phone while only 3 out of 24 drivers who were talking with passengers did not successfully complete these basic tasks. 1. Complete the following table using the data from the experiment.

Talking on Cell Phone Talking to a Passenger TotalSuccessful at Navigational Tasks Unsuccessful at Navigational Tasks Total

2. Complete the bar graph that compares the counts of the drivers who completed the tasks to the counts of the drivers who did not for both groups, talking on a cell phone and talking to a passenger. The graph displays the data for the drivers talking on a cell phone. Construct the bars for the drivers who were talking to a passenger. Title your graph.

Drivers Talking on a Cell Phone

5

10

15

20

25

0

Freq

uenc

y

Unsuccessful

Successful

Drivers Talking to a Passenger

Drivers Talking on a Cell Phone


30

Student Activity

3. Answer the following probability questions based on the chart and the double bar graph. a) What is the probability that a randomly selected driver was successful in completing basic

driving tasks? b) What is the probability that a randomly selected driver was unsuccessful in completing basic

driving tasks? c) What is the probability that a randomly selected driver was unsuccessful while talking on a

cell phone?

d) What is the probability that a randomly selected driver was unsuccessful while talking to a passenger?

e) What is the difference between the probability a driver failed to navigate appropriately while talking on a cell phone and the probability a driver failed while talking to a passenger? Interpret the significance of this difference in the context of the situation.

f) If a randomly selected driver is talking on a cell phone, what is the probability that he/she

will be unsuccessful?


31

Student Activity

In statistics, a hypothesis is used to propose a model based on data. The researchers’ hypothesis was that drivers are more distracted while talking on a cell phone than when talking to a passenger. Through our simulation, we are going to determine if our data is consistent with the model. The purpose of this simulation is to determine whether or not there is enough evidence to support the findings of the research. We are not trying to prove or disprove the results. Through simulation, we are trying to model the situation to determine what happens over the long run. In order to perform the simulation, we must assume that the drivers who were unsuccessful in completing the basic driving tasks would be distracted whether they were talking on the phone or talking to a passenger. 4. Do you agree with this “hypothesis” or do you think the proportion of drivers who did not

successfully complete a navigational task will be the same for either distraction? Explain your answer.

5. We are going to model the situation using part of a deck of cards. a) How many cards should we use to simulate the number of drivers who did not successfully

complete navigational tasks? b) How many cards do we need to represent the drivers who successfully completed

navigational tasks? c) From a standard deck of cards, remove 2 cards each of spades, diamonds, and clubs and add

the two jokers. You should have a deck of 48 cards, 13 of which are hearts, and 2 jokers. Which cards should represent the drivers who did not successfully complete navigational tasks?

6. Conduct the simulation working in pairs and using the following steps:

• Deal two stacks of 24 cards. • The cards the dealer receives will represent the cell phone users. • The cards the player receives will represent the drivers who are talking to a passenger. • Record the number of drivers who were unsuccessful in each group in the following table. • Repeat this process a total of 5 times without changing dealers.

Trial Number of drivers Number of drivers distracted by talking on the cell phone Number of drivers distracted

by talking to a passenger 1 48 2 48 3 48 4 48 5 48

Total


32

Student Activity

7. Answer the following probability questions based on your five simulations. a) What is the probability that a randomly selected driver is unsuccessful in completing basic

driving tasks while talking on the cell phone? Compare this answer to your answer in question 3c.

b) What is the probability that a randomly selected driver was distracted by talking to a

passenger? Compare this answer to your answer in question 3d. c) If a randomly selected driver is talking on a cell phone, what is the probability that he/she is

unsuccessful in completing basic driving tasks? Compare this answer to your answer in question 3f.

8. Combine your data with that of your classmates for the number of drivers who were distracted

while talking on the cell phone on the dotplot at the front of the room. Record the pooled results on the following graph.

Drivers Distracted while Talking on a Cell Phone

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

Freq

uenc

y


33

Student Activity

9. Answer the following questions based on the class results. a) How many trials resulted in 12 drivers who failed to drive safely while talking on the phone?

What percent of the class data does this represent? b) According to the class data, what is the mean number of drivers who were unsuccessful in

completing navigational tasks while talking on the cell phone? c) How does the mean of the class data compare to the research data? Explain the difference

between these two numbers in the context of the situation. d) Do you think the difference between these two values is significant enough to challenge the

results of the research? Explain your reasoning. e) Could there be other reasons why the drivers might not successfully complete the driving

tasks other than talking on a cell phone or talking to a passenger while driving that could account for this behavior? If so, list at least two examples.

10. The school newspaper has asked you to write an article about the dangers of driving while

talking on a cell phone and to a passenger. Write a paragraph summarizing your findings from the simulation.


34



The Jury

In a certain city, women and men each make up 50% of the adult population. A jury was selected that contained nine men and three women. The defendant in a trial, a woman, claims bias in the jury selection. Shethinksthatthejurydoesnotreflectthenumberofmenandwomeninthecity.Theprosecutorclaimsthat the jury was selected without regard to gender. Is the defendant’s claim believable? In other words, if the population is 50% women, is selecting 3 or fewer women unlikely to happen by chance? This activity uses a spinner as a randomization device to make a simulation to estimate the probability of selecting 3 or fewer women simply by chance. The results will be used to analyze the defendant’s claim. Use the spinner model provided on the last page of the activity to make a spinner.

1. Assume the probability for selecting a woman remains the same throughout the simulation and that the probability of selecting one juror does not affect the outcome of selecting the next juror. What is the probability of selecting a woman?

2. How many times will you spin to simulate the selection of the jury? What information will be recorded after each spin?

3. Instead of a spinner, could a coin, die, or deck of cards be used to simulate the probability of selecting a woman? If so, describe how you will use this device to simulate selecting a jury.

4. Complete10runs(trials)ofyoursimulation.Recordyouroutcomesinthetable.

Trial Number of Females Number of Males

12345678910

35


Mathematics—The Jury

5. Complete the table for the combined results of the entire class.

# of Females Tally Frequency

0123456789101112

6. Estimate the probability of selecting three or fewer women for this jury based on your simulation in question 4. The theoretical probability is approximately 0.073. How close was your experimental probability to the theoretical probability?

7. Based on the class data in question 5, what is the probability of selecting three or fewer women for the jury? How does this compare to the theoretical probability of 0.073?

8. If you were the judge in this case, what would you say to this defendant about bias in the jury selection? This answer should be based on your simulation and should be several sentences long.

36


Mathematics—The Jury

37

Student Activity

I Want Candy! As a reward for great results on their test, Mrs. Copp is going to give each student a “mini-bag” of candy. Mrs. Copp purchases the candy in packages that contain 8 mini-bags. The students are excited about their reward and quickly open their bags of candy. Mrs. Copp notices that Noah and Tanner are arguing about the number of pieces of candy they received. Noah is upset because he has 17 pieces and Tanner has 19 pieces in his mini bag and claims that Mrs. Copp likes Tanner more than Noah. In order to stop the bickering, Mrs. Copp quickly informs the boys that the mini-bags weigh the same so even though Tanner has more pieces, they both have the same amount of candy based on the weight. As a result of this discussion, other students begin counting their pieces of candy. Mrs. Copp notices that the number of pieces of candy in each mini-bag varies from 16 to 20 pieces. She is sure, however, that the weight of each bag is the same since the total weight is stated on the package. She is confident that the manufacturer makes sure each bag weighs the same. The students are not convinced, so Mrs. Copp buys more candy and weighs each mini bag. 1. According to the information printed on the package, the net weight of the candy is 4.31 ounces.

If there are 8 mini-bags of candy in each package, what is the weight of the candy in each mini-bag? List all the decimals shown on your calculator.

2. After weighing 4 packages of candy (32 mini-bags) Mrs. Copp posts the data on the board.

Bag # Weight Bag # Weight Bag # Weight Bag # Weight 1 0.563 9 0.500 17 0.553 25 0.554 2 0.577 10 0.566 18 0.540 26 0.566 3 0.513 11 0.544 19 0.561 27 0.568 4 0.543 12 0.553 20 0.521 28 0.567 5 0.570 13 0.573 21 0.527 29 0.484 6 0.516 14 0.507 22 0.539 30 0.504 7 0.538 15 0.535 23 0.516 31 0.577 8 0.518 16 0.542 24 0.560 32 0.615

a) Mrs. Copp admits that she was not correct, but at the same time, she is pleasantly surprised to

see that most of the mini-bags weigh more than the mean weight calculated in question 1. What is the mean weight of a mini-bag of candy according to the data in the table? List all the decimals shown on your calculator.


38

Student Activity

b) What is the median weight of a mini-bag of candy based on the data in the table? c) What is the probability that you received a mini-bag of candy that weighed less than the

weight determined by dividing the weight of the package containing 8 mini-bags by 8? d) Do you think the probability from part (c) is significant? In other words, is there a good

chance that you received a mini-bag of candy that weighed less than the mean weight? Explain.

3. Someone in the class asks Mrs. Copp if the total weight of the candy in the package is always

4.31 ounces. Unfortunately the packages were not weighed at the same time she collected the data. The class is going to use a simulation to test the statement by the manufacturer that there are 4.31 ounces of candy in each package. A simulation is a method that models the structure of the real situation. Before conducting a simulation, certain steps must be followed.

• First, you must state what “event” you are going to repeat. For this simulation, the event is selecting 8 mini bags of candy for one package.

• Second, you need to explain how you will model the set of outcomes. In this case, you will use 32 slips of paper, labeled with the weight of each mini-bag from the table in question 2, to model the manufacturing process of producing a package of candy.

• Third, you need to explain how you will simulate each trial including a “stopping rule”. In other words, you will select 8 weights for each trial.

• Fourth, you need to run several trials and record the results. • The last step involves stating your conclusion based on the simulation.

a) How will you conduct the simulation?

b) What are we trying to determine from our simulation?


39

Student Activity

c) Select 8 slips of paper and calculate the total weight of the entire package of 8 mini-bags. Divide the total weight by 8 to calculate the mean weight of a mini-bag. Repeat this process 10 times and record your results in the table.

Trial Total weight of the 8 mini-bags Mean weight of a mini-bag Trial

Total weight of the 8 mini-bags

Mean weight of a mini-bag

1 6 2 7 3 8 4 9 5 10

d) What is the probability that a randomly selected package of candy weighs less than the

manufacturer’s stated weight of 4.31 ounces? e) According to your data, what is the probability that the mean weight of a mini-bag of candy

will be more than 0.53875, the weight claimed by the manufacturer on the package divided by the eight mini-bags in the package?

f) Use the data in your table to determine the mean weight of the ten entire packages of 8 mini-

bags. Also, determine the mean weight of the mini-bags. How do these answers compare to the manufacturer’s published weight?

g) Record your answers from part (f) on the table provided at the front of the room. When all of

the data has been recorded, calculate the mean and median for the pooled data.

h) Do you think the pooled data supports the manufacturer’s published weight? Explain your

reasoning.


40

Student Activity

4. Noah and Tanner are still upset by the inconsistency in the number of pieces of candy in each mini-bag. Mrs. Copp counted the number of pieces of candy in each mini-bag and posted the results.

Bag

# Weight # of

Pieces Bag

# Weight# of

PiecesBag

# Weight# of

Pieces Bag # Weight

# of Pieces

1 0.563 19 9 0.500 16 17 0.553 19 25 0.554 18 2 0.577 19 10 0.566 19 18 0.540 18 26 0.566 19 3 0.513 17 11 0.544 18 19 0.561 19 27 0.568 19 4 0.543 18 12 0.553 18 20 0.521 17 28 0.567 19 5 0.570 19 13 0.573 19 21 0.527 18 29 0.484 16 6 0.516 17 14 0.507 17 22 0.539 18 30 0.504 17 7 0.538 18 15 0.535 18 23 0.516 17 31 0.577 19 8 0.518 18 16 0.542 18 24 0.560 19 32 0.615 20

a) How many mini-bags of candy contained 19 or more pieces? What is the probability of

randomly selecting a mini-bag that contained 19 or more pieces? b) How many mini-bags of candy contained at most 17 pieces? What is the probability that a

randomly selected mini-bag contained at most 17 pieces? c) Without counting, what is the probability that a randomly selected mini-bag contained 18

pieces? Show the work that leads to your answer. d) Without counting, what is the probability of randomly selecting a mini-bag with at least 18

pieces? e) What is the mean of the number of pieces per mini-bag? f) Do you think Noah’s complaint that he has only 17 pieces of candy in his mini-bag is valid?

Explain.


41

Student Activity

5. Tanner observes that the data in the table suggests that there may be an association between the number of pieces of candy and the weight of the mini-bag. The more pieces of candy in the mini-bag, the more it weighs. Mrs. Copp decides to ask the class to analyze both the relationship between the number of pieces of candy and the weight in a mini-bag to determine if Tanner’s observation is true.

a) Create a scatterplot of the data provided in question 4. Since the manufacturer packages by

weight rather than number of pieces, identify the weight as the independent variable and the number of pieces as the dependent variable. Label the axes and title the graph.

b) Interpret the meaning of the ordered pair (0.567, 19) in the context of the situation. c) Determine the mean weight for each number of pieces in a mini-bag based on the data from question 4. Number of Pieces

in a mini-bag Mean Weight of each mini-bag 16 17 18

19 20

d) Draw a line of “good fit” on your scatterplot that passes through at least two of your data points. Use two of those points to determine the equation for the line. Record the equation of the line below.


42

Student Activity

e) Interpret the slope and y-intercept in the context of the situation. Is the y-intercept meaningful in this scenario? Explain.

f) Based on your equation and the mean weights from part (c), how many pieces of candy will

be in a mini-bag? If each of the values were rounded to the nearest whole number, are there any values that are not equal to the mean weight for the number of pieces? If so, did the equation overestimate or underestimate the value?

g) Could you use your equation to predict the number of pieces of candy in the package when

the weight of candy in the package is 4.31 ounces? Explain. 6. You are asked to write a paper for your English class based on the results of the simulation and

scatterplot. Was there sufficient evidence, based on your results, to support the manufacturer’s printed weight of 4.31 ounces for each package of 8 mini-bags? Do you feel that Noah could have been “shorted” on the number of pieces of candy he received in his mini-bag just by chance? Include your assessment of the reliability of the weight of the package and number of pieces of candy in each bag. Do you think that the manufacturer should correct the information on its packaging? Explain.


43

Student Activity

Class Data Student Mean Weight of Entire

Package of 8 mini bags for 10 trials

Mean of the Mean weight of the mini-bags

for the 10 trials

1 2 3 4

5 6

7 8 9

10 11

12 13 14

15 16

17 18

19 20 21

22 23

24 25 26

27 28

29 30


44

Student Activity

0.563 0.577 0.513

0.543 0.570 0.516

0.538 0.518 0.500

0.566 0.544 0.553

0.573 0.507 0.535

0.542 0.553 0.540


45

Student Activity

0.561 0.521 0.527

0.539 0.516 0.560

0.554 0.566 0.568

0.567 0.484 0.504

0.577 0.615


46

Take a Sample, Please Suppose you have been given the task of computing the average age, arithmetic mean, of everyone who lives in California. Actually collecting the ages of each Californian, adding them together, and dividing by the size of the population would be virtually impossible. Even if you could collect all of the ages, by the time you actually finish, the data would have changed, births and deaths would have occurred and at least some people would have celebrated birthdays, making them another year older. Statisticians face this conundrum constantly. Thankfully, they can rely on some basic concepts of inferential statistics so that research in the social sciences and marketing, opinion polling, and the evaluation of new medicines can be conducted. The following activities are designed to demonstrate some of these basic statistical concepts. The methods for applying these concepts to market research or opinion polling will be left to future mathematics courses. The goal in these activities is to experience the development of the concepts with “populations” of a very manageable size so that you can infer their validity when used in more daunting situations. The following concepts will be explored: Concept #1: For a fixed sample size, the mean of all possible sample means is equal to the mean of

the population. Concept #2: The mean of the sample means of a randomly selected subset of all possible samples of

a fixed size provides a good approximation of the mean of the population. Concept #3: The Central Limit Theorem (in simplified terms) says that, regardless of the shape of

the distribution of the original “population”, as the sample size increases, the distribution of sample means will approach the shape of a normal distribution. When the center of the graph is located at the mean and the shape is approximately symmetrical, the shape is described as “normal”. Additionally, as the sample size increases, the spread of the distribution of the sample means will decrease, while the mean of the sample means remains remarkably close to the population mean.

47

1. Concept #1: For a fixed sample size, the mean of all possible sample means is equal to the mean of the population.

Begin with a very small “population” of four quiz scores: 72, 80, 88, 98. a) What is the mean of the four quiz scores? b) In how many ways can you select 1 score and then a 2nd

score, if you are allowed to select the same score more than once and if the same two scores listed in a different order represents a different sample? In other words, how many 2-score samples of the 4 scores, with replacement, are possible? Three of the possible samples are {72,80}, {80,72}, and {72,72}.

c) List all of the samples in the table below and calculate the mean of each sample.

d) Calculate the mean of these sample means. How does this answer confirm Concept #1?

Sample Scores Sample Mean Sample Scores Sample Mean

48

2. Concept #2: The mean of the sample means of a randomly selected subset of all possible samples of a fixed size provides a good approximation of the mean of the population.

a) How many 3-score samples of the 4 test scores, with replacement, are possible? In other

words, in how many ways can you select one score, then a 2nd score and then a 3rd

score, if you are allowed to select the same score more than once and if the same three scores listed in a different order represents a different sample?

b) Rather than listing all the 3-score samples, collect a randomly selected subset of all the

possible samples. To begin, number the quiz scores.

#1: 72 #2: 80 #3: 88 #4: 98 To randomly select 3 of the 4 scores for a sample, use a calculator’s random number generator. Steps for the TI-83/84 are shown. • The random integer command is located in Math PRB 5: randInt( • The parameters for the command are: randInt(smallest integer allowed, largest integer allowed, number of integers to generate) • The command randInt (1, 4, 3) will generate three independent random integers

from 1 to 4, which will in turn identify the quiz scores for a particular sample. • For example, if the calculator returns {4, 1, 3}, the sample mean would be

98 72 88 863

+ += .

If the calculator returns {4, 2, 3}, what is the sample mean? c) Collect random 3-score samples of the quiz data. Record the scores, not the random numbers

generated by the calculator, and the mean of each sample in the table. Continue collecting samples until your teacher directs you to stop.

d) Combine your Sample Mean data with that of the other members of your class. Calculate the

mean of the combined sample means. How close is this answer to the actual mean of the four quiz scores? How does this activity confirm Concept #2? Explain how this activity could be applied to determining the average age of the population of California.

Sample Scores

Sample Mean

Sample Scores

Sample Mean

Sample Scores

Sample Mean

49

3. Concept #3: The Central Limit Theorem (in simplified terms) says that, regardless of the shape of the distribution of the original “population”, the distribution of sample means will approach the shape of a normal distribution as the sample size increases. Additionally, as the sample size increases, the spread of the distribution of the sample means will decrease, while the mean of the sample means remains remarkably close to the population mean.

To help visualize Concept #3, work with a “population” that is larger than the four quiz scores. The table below lists the salary for the highest paid player on each National Football League team as reported by the team for the 2008 season.

a) For ease in working with the large numbers, code the data in millions to the nearest tenth of a million. For instance, Dallas’ Terrell Owens would have a coded salary of $8.7 million.

Team Player 2008 Salary Coded Data Salary #

Arizona Larry Fitzgerald (WR) 6,999,574 1 Atlanta John Abraham (DE) 8,506,720 2 Baltimore Chris McAlister (CB) 10,907,082 3 Buffalo Aaron Schobel (DE) 8,729,795 4 Carolina Julius Peppers (DE) 14,137,500 5 Chicago Charles Tillman (CB) 8,216,666 6 Cincinnati Carson Palmer (QB) 13,980,000 7 Cleveland Joe Thomas (OL) 9,460,000 8 Dallas Terrell Owens (WR) 8,666,668 9 Denver Champ Bailey (CB) 12,690,050 10 Detroit Roy Williams (WR) 6,292,834 11 Green Bay Brett Favre (QB) 12,800,000 12 Houston Andre Johnson (WR) 8,704,848 13 Indianapolis Peyton Manning (QB) 18,700,000 14 Kansas City Patrick Surtain (CB) 8,380,000 15 Miami Jason Taylor (DE) 10,025,385 16 Minnesota Bernard Berrian (WR) 9,538,333 17 New England Tom Brady (QB) 14,626,720 18 New Orleans Drew Brees (QB) 9,000,000 19 New York Giants Eli Manning (QB) 12,916,666 20 New York Jets Dewayne Robertson (DT) 11,191,369 21 San Diego LaDainian Tomlinson (RB) 7,822,786 22 San Francisco Alex Smith (QB) 9,916,262 23 Seattle Matt Hasselbeck (QB) 9,950,000 24 St Louis Terry Holt (WR) 9,204,714 25 Tampa Bay Jeff Faine (OL) 7,000,000 26 Tennessee Keith Bulluck (LB) 7,864,908 27 Washington Shawn Springs (CB) 7,483,333 28

50

b) Create a histogram of the data, with bin width of 1 million, on the grid marked Population Data on the Results Page at the end of the activity.

c) Describe the shape and any unusual features of the histogram. d) Calculate the mean of the data, record it below and in the blank on the Results Page, and

mark its location with a vertical dotted line on the histogram. e) Calculate and list the 5-number summary for this data, and then construct a box and whiskers

plot above the histogram on the graph at the end of the activity. Explain how the box and whiskers plot adds additional understanding about the distribution of the population.

f) Begin to collect samples of size 5 using the calculator’s random number generator to

generate five random integers between 1 and 28 to indicate which salaries are included in the sample. Calculate the mean of each sample. Continue the process until instructed to record your Sample Mean data on the class calculator.

g) Combine your Sample Mean data with that of the other members of your class. Create a

histogram of the combined data, with bin width of 1 million, on the grid marked Samples of Size 5 on the Results Page at the end of the activity. It may be necessary to adjust the scale on the vertical axis, depending on the number of combined data items.

h) Calculate the mean of this data, record it below and in the blank on the Results Page, and

mark its location with a vertical dotted line on the histogram.

Sample Salaries Sample Mean

51

i) How does the shape of this histogram compare with the shape of the histogram of the population? Compare the mean of the sample means with the mean of the population.

j) Begin to collect samples of size 10 using the calculator’s random number generator to

generate ten random integers between 1 and 28 to indicate which salaries are included in the sample. Calculate the mean of each sample. Continue the process until instructed to record your Sample Mean data on the class calculator.

k) Create a histogram of the combined data, with bin width of 1 million, on the grid marked

Samples of Size 10 on the Results Page at the end of the activity. It may be necessary to adjust the scale on the vertical axis, depending on the number of combined data items.

l) Calculate the mean of this data, record it below and in the blank on the Results Page, and

mark its location with a vertical dotted line on the histogram. m) How does the shape of this histogram compare with shape of the histogram of the

population? Compare the mean of the sample means with the mean of the population. Describe the changes that you observe in the histograms as the sample size increases.

Sample Salaries Sample Mean

52

4. Consider the histograms shown. The data includes 50 samples of each size. Use the graphs to answer the questions.

Samples of Size 5 Samples of Size 7

Samples of Size 10 Samples of Size 12

a) Describe the effects of the increase of sample size on the “spread” or range of the data.

b) As the sample size increases, does the shape appear more symmetrical and is the mean located closer to the center of the graph? In other words, does the shape appear more “normal”? Explain.

c) Describe the effect of the increase in sample size on the mean.

d) The histogram in question 3b and boxplot in question 3e show Payton Manning’s salary as an outlier. As the sample sizes increase, why is this “outlier” not a factor in the shape and spread of the graph?

e) How does this activity confirm Concept #3? How does this activity apply to determining the

average age of the population of California?

53

8 9 10 11 13 14 12 15 16 19 18 17 7 6

8 9 10 11 13 14 12 15 16 19 18 17 7 6

8 9 10 11 13 14 12 15 16 19 18 17 7 6

Results Page

Population Data

Mean

Sample Size 5

Mean

Sample Size 10

Mean

54

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