multiplies all measures of center and the standard deviation and range by k, while the variance is multiplied by k 2.

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Lesso 3- Lesso 3- Scale Chages of Data Vocabulary scale chage of a data set scale factor scale image BIG IDEA Multiplyig every umber i a data set by k multiplies all measures of ceter ad

A useful example to cosider is the Cosumer Price Idex (CPI), a measure of iflatio.

Costs i later years are the compared to the base year cost. The table at the right gives the CPI at five-year itervals begiig at 1950, with 197 as the base year.

2 1990 391.4 1995 45.5 00 515.8 05 585.0 Jue 08 55.5 Metal Math Fid the x-itercept(s) of the graph of the equatio. a. y = x + π b. y = x - π c. y = x + π d.

1 Lesso 3- Lesso 3- Scale Chages of Data Vocabulary scale chage of a data set scale factor scale image BIG IDEA Multiplyig every umber i a data set by k multiplies all measures of ceter ad the stadard deviatio ad rage by k, while the variace is multiplied by k 2. Scale chages ca also be applied to data sets. A useful example to cosider is the Cosumer Price Idex (CPI), a measure of iflatio. To calculate the CPI, the cost of a specified basket of goods is totaled i a particular base year, the scaled so the cost equals 0. Costs i later years are the compared to the base year cost. The table at the right gives the CPI at five-year itervals begiig at 1950, with 197 as the base year. Scalig allows prices i ay year to be compared to prices i ay other year. Just solve a proportio. Year CPI Jue Metal Math Fid the x-itercept(s) of the graph of the equatio. a. y = x + π b. y = x - π c. y = x + π d. y = x - π Example 1 Suppose a refrigerator cost $800 i What might you expect the cost of a similar refrigerator to be i Jue 08? Solutio Set up a proportio. cost i Jue 08 CPI i Jue 08 cost i 1995 = CPI i 1995 Let x = cost of a refrigerator i Jue 08. Substitute, usig the CPI values give i the table. x $800 = Solve the proportio. x = = $ $ The cost of a similar refrigerator i Jue 08 was probably about $1150. Scale Chages of Data 5

Chapter 3 The ratio 55.5 45.5 i Example 1 is a scale factor. I this case, 55.5 45.5 1.43, idicatig there was about a 43.% icrease i prices from 1995 to Jue 08. By multiplyig the 1995 price by 1.

2 Chapter 3 The ratio i Example 1 is a scale factor. I this case, , idicatig there was about a 43.% icrease i prices from 1995 to Jue 08. By multiplyig the 1995 price by 1.43, you ca estimate what the Jue 08 price of a item would be if the cost kept pace with iflatio. QY A scale chage of a set of data {x 1, x 2,, x } is a trasformatio that maps each x i to ax i, where a is a ozero costat. That is, S is a scale chage if ad oly if there is a ozero costat a with S: x ax, or S(x) = ax. The umber a is called the scale factor of the scale chage. The umber ax or the poit it represets is called the scale image of x. I the situatio above, the 1995 cost x of a item ca be mapped oto the estimated Jue 08 cost via the scale chage S: x 1.43x, or S(x) = 1.43x. QY If a perso eared $2,500 a moth i 1995, what would the perso eed to have eared i Jue 08 to keep up with ifl atio? Scalig ad Measures of Ceter Scale chages of data, like traslatios, affect statistical measures derived from the data. Activity The CPI i 1998 was about 49. Here are average prices of some grocery items i that year. Coffee Eggs Gasolie Orage Juice Groud Beef Year 1 poud 1 doze 1 gallo 12-oz ca 1 poud Chicke 1 poud Source: Bureau of Labor Statistics Step 1 Calculate the scale factor eeded to predict costs of items i 08 from 1998 prices. Step 2 Eter the price data for 1998 ito a spreadsheet like the oe o page 7. Step 3 Use the scale factor from Step 1 to compute the predicted Jue 08 prices of the same items. Record these costs i a ew colum. Step 4 Calculate the mea, media, rage, variace, ad stadard deviatio of each set of prices. Record the results to the earest hudredth i aother colum. Step 5 Multiply the 1998 statistics by the scale factor from Step 1 ad place the results i a additioal colum. Your spreadsheet should look similar to the oe o the ext page. Trasformatios of Graphs ad Data

3 Lesso 3- Step Compare the results of Step 5 to those of Step 4. Which of the 08 statistics ca be foud by scalig the correspodig 1998 statistics, rather tha by calculatig from the costs of the groceries? The spreadsheet evaluates the formula B2 (55.5/49). E2 (55.5/49) A B C D E F G Items 1998 costs 08 costs (predicted) statistics statistics $ Mea $1.12 Media $1.13 Rage $1.0 Variace Coffee 1 Poud Eggs 1 doze Gasolie 1 gallo Orage Juice 12-oz ca Groud Beef 1 poud Chicke 1 poud $1.82 $1.02 Stadard Deviatio 1998 statistics x scale factor The Activity shows how scalig data affects statistics for measures of ceter ad spread. These ideas are applied i Example 2. Example 2 As a fud-raiser, club members sell cady for $2.50 per box. The umber of boxes each of the 17 members sold is give below. 27, 30, 32, 32, 34, 35, 35, 37, 38, 39, 40, 41, 41, 43, 44, 44, 50 a. Compute the mea, media, stadard deviatio, ad IQR of the umbers of boxes sold. b. Fid the amout of moey each member collected. c. Compute the mea, media, stadard deviatio, ad IQR of the amouts of moey collected by scalig the values i Part a. Solutio a. For the umber of boxes, mea = 37.8, media = 38, stadard deviatio = 5.89, ad IQR = 9. b. Apply the trasformatio x i 2.5x i to each value. The umber of dollars each member collected are 7.50, 75, 80, 80, 85, 87.50, 87.50, 92.50, 95, 97.50, 0, 2.50, 2.50, 7.50, 1, 1, ad 125. c. For the moey collected, mea = = 94.5, media = = 95, stadard deviatio = = 14.73, ad IQR = = Scale Chages of Data 7

Chapter 3 Box plots of the umbers of cady boxes sold ad amouts collected illustrate the effects of scalig o the data values, the ceter (mea ad media), ad the spread (rage, IQR, ad stadard deviatio).

4 Chapter 3 Box plots of the umbers of cady boxes sold ad amouts collected illustrate the effects of scalig o the data values, the ceter (mea ad media), ad the spread (rage, IQR, ad stadard deviatio). As you saw i the Activity ad Examples, scale chages ot oly affect the data, they affect measures of ceter as well. This effect is stated i the theorem below. rage IQR Boxes Sold Moey Made rage IQR Theorem (Ceters of Scaled Data) Multiplyig each elemet of a data set by the factor a multiplies each of the mode, mea, ad media by the factor a. Proof We prove the mea part of the theorem here; it ca also be proved that the media ad mode are multiplied by a (see Questio 9 for the media). To describe the effect of a scale chage o statistical measures for a geeral data set, represet the set as {x 1, x 2, x 3,..., x }. Uder a scale chage with scale factor a, the image data set is {ax 1, ax 2, ax 3,..., ax } = {x 1, x 2,..., x }. Let _ x be the mea of the origial data set ad x be the mea of the image set. The mea ( ax i ) x =. By defi itio of, x = ax 1 + ax 2 + ax ax = a(x 1 + x 2 + x x ) x i ). Thus x = a _ x. So, uder a scale chage, the mea of a set of data is mapped to the mea of the image set of data. = ( a ) = a ( Scalig ad Measures of Spread The Activity ad Example 2 show that measures of spread i a scalechage image data set are predictable. Both the rage ad stadard deviatio ca be foud by multiplyig by the scale factor. x i 8 Trasformatios of Graphs ad Data

5 Lesso 3- The predicted variace ca also be foud usig the scale factor i a differet way. These effects are described i the followig theorem. Theorem (Spreads of Scaled Data) If each elemet of a data set is multiplied by a > 0, the the variace is a 2 times the origial variace, the stadard deviatio is a times the origial stadard deviatio, ad the rage is a times the origial rage. Proof Cosider the data set {x 1, x 2,..., x } ad its image {ax 1, ax 2,..., ax } uder a scale chage of magitude a. By the Ceters of Scaled Data Theorem, the mea of the image data set is a _ x, where _ x is the mea of the origial data set. So the variace of the image data is give by ( ax i - a _ x ) 2 [ a (x i - _ x )] 2 i = 1-1 = - 1 Distributive Property [ a 2 (x i - _ x ) 2 ] = - 1 Power of a Product Property Applyig the Distributive Property, [ a 2 ( x i - _ x ) 2 ] = a 2 ( x i - _ x ) 2. Hece, the variace of the image data is give by a 2 ( x i - _ ( x ) 2 ( x i - _ ) x ) 2-1 = a 2 i = 1-1 = a 2 s 2, where s 2 is the variace of the origial data set. Thus, the variace of the image data is a 2 times the variace of the origial data. To get the stadard deviatio of the image data, take the square root of the variace. Thus, the stadard deviatio of the image data is a s, which is a times the stadard deviatio of the origial data set. It ca also be proved that the rage ad IQR of the image data are a times the rage ad IQR, respectively, of the origial data set. Questios COVERING THE IDEAS 1. Defie scale chage of a set of data. 2. A Rambler, a small car, cost about $00 i 195. What would a comparable car have cost i 05? 3. I what 5-year period from 1950 to 05 was there the greatest percet icrease i CPI, ad what was that percet? Scale Chages of Data 9

6 Chapter 3 4. Cosider the cady sales i Example 2. Suppose that a box costs $5 istead of $2.50. a. For the moey collected, calculate each statistic. i. rage ii. mode iii. media iv. mea v. variace vi. stadard deviatio b. Draw a box plot of the amouts of moey collected. 5. A restaurat employs 11 workers whose idividual earigs for a 8-hour day are summarized by this box plot. Origial Earigs Suppose that each employee begis workig hours istead of 8. Assume employees ear the same hourly wage, regardless of the umber of hours worked. Dollars a. What scale factor would be used to fid each perso s ew daily earigs? b. Draw a box plot of the earigs for the -hour work day.. Suppose all elemets of a data set are multiplied by x. Explai why the variace is multiplied by x Multiple Choice The box plot at the right represets a data set D. Which box plot below represets the image of D uder the trasformatio T: x 1 3 x? Value A B Plot Plot 2 C D Plot Plot 4 APPLYING THE MATHEMATICS 8. Suppose Y 1 = 11, Y 2 = 3, Y 3 = 2, Y 4 = 7, Y 5 = 5, Y = 4. Evaluate each expressio. a. Y i b. ry i ( Y i m ) c. d. (Y i + 2) 9. Prove the Ceters of Scale Chages of Data Theorem for medias.. For a large city, the media house price oe year was $258,0 with a iterquartile rage of $81,000. Assume that house prices rise 2% over the ext year. What would be the media ad IQR for house prices i the ext year? 190 Trasformatios of Graphs ad Data

Lesso 3-11. Let _ x = the mea ad s = the stadard deviatio of scores o a test for a class of studets. Suppose everyoe s score is multiplied by r, ad the icreased by a bous b.

7 Lesso Let _ x = the mea ad s = the stadard deviatio of scores o a test for a class of studets. Suppose everyoe s score is multiplied by r, ad the icreased by a bous b. For the ew scores, fid the a. mea. b. variace. c. stadard deviatio. 12. Let M represet the maximum value of a data set ad let m represet the miimum value. a. Write a expressio for the rage r of the data set. b. After a scale chage with scale factor d > 0, what are the maximum ad miimum values of the image data set? c. Write ad simplify a expressio for the rage of the image data. d. How would your aswers to Parts a c chage if d < 0? 13. Cosider the followig data, which give the height h i cm ad weight w i kg of twelve studets. h (cm) w (kg) a. Eter these data ito a statistics utility. Create a scatterplot with h o the horizotal axis. b. Fid the lie of best fit for predictig weight from height. c. Compute the correlatio coefficiet. d. Use a statistics utility to covert the height to iches (1 i. = 2.54 cm) ad weight to pouds (1 lb = kg). Draw a ew scatterplot. How is the scatterplot differet from that i Part a? e. Compute the correlatio coefficiet ad the regressio equatio for predictig weight from height i Part d. f. Which of the followig statistics remai ivariat uder scale chages of the data? i. correlatio coefficiet ii. slope of the regressio lie iii. y-itercept of the regressio lie REVIEW 14. Cosider the fuctios f ad g with f(x) = x ad g(x) = 7x. a. Describe a scale chage that maps the graph of f oto the graph of g. b. Describe a scale chage that maps the graph of g oto the graph of f. (Lesso 3-5) Scale Chages of Data 191

8 Chapter 3 I 15, match the equatio with its graph. (Lessos 3-5, 3-2) 15. f(x) = x + 5. g(x) = 3x 17. h(x)= 1 3 x. j(x) = x - 5 A B C D y y y y - x - x - x - x Name all the fuctios i Questios 15 that are eve fuctios. (Lesso 3-4). A certai hyperbola H is a traslatio image of the graph of y = x 1 ad has asymptotes x = 2 ad y = 5. Give a equatio for H. (Lessos 3-2, 3-1) I 21 23, if possible, factor the give expressio. (Previous Course) 21. x 3 - x m r 2-2r Skill Sequece Rewrite each of the followig i the form a(x - h). (Previous Course) a. 7x - 21 b. 21x - 7π c. 7x + 5π EXPLORATION 25. Take a data set of positive values ad take the square root of each value. Which statistics, if ay, are ivariat? Which are affected i predictable ways? QY ANSWER $3, Trasformatios of Graphs ad Data

n m CHAPTER 3 RATIONAL EXPONENTS AND RADICAL FUNCTIONS 3-1 Evaluate n th Roots and Use Rational Exponents Real nth Roots of a n th Root of a

n m CHAPTER 3 RATIONAL EXPONENTS AND RADICAL FUNCTIONS 3-1 Evaluate n th Roots and Use Rational Exponents Real nth Roots of a n th Root of a CHAPTER RATIONAL EXPONENTS AND RADICAL FUNCTIONS Big IDEAS: 1) Usig ratioal expoets ) Performig fuctio operatios ad fidig iverse fuctios ) Graphig radical fuctios ad solvig radical equatios Sectio: Essetial