Survey of Mathematics
Survey of Mathematics MTH 150
Monroe Community College
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This 84 page Class Notes was uploaded by Marion Morissette on Thursday October 15, 2015. The Class Notes belongs to MTH 150 at Monroe Community College taught by Brigitte Martineau in Fall. Since its upload, it has received 8 views. For similar materials see /class/223548/mth-150-monroe-community-college in Mathematics (M) at Monroe Community College.
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Your name Your section MTH 150 SURVEY OF MATHEMATICS Chapter 9 GEOMETRY 93 Perimeter and Area 94 Volume and Surface Area MTH 150 AAR Brigitte Martineau Survey of Mathematics Chapter 9 Geometry 93 Perimeter and Area Perimeter Area Summary of Perimeter and Area Formulas W Perimeter Rectangle Area 1 s Perimeter Square Area s Circumference Circle Area Diameter 51 52 Perimeter Triangle Area 4 b b1 Perimeter TrapeZOId SI 52 Area 4 b2 Page 2 of 12 MTH 150 AAR Brigitte Martineau Survey of Mathematics Chapter 9 Geometry Perimeter Parallelogram W Area b C a Pythagorean Theorem a2 12 c2 4 b 1 l The area of a rectangle is 40 square meters and the length is 8 meters What is the width You may have heard the terms monolith and obelisk An obelisk is a tall narrow four sided tapering monument which ends in a pyramidal top Think Washington Monument If an obelisk is made from a single piece of stone it is a monolith Think 2001 A Space Odyssey You can learn a great deal about ancient construction by reading about the huge obelisk at St Peter s Square in Rome It was built in Egypt and transported by barge to Rome It was then placed on a stone base that is a square with 7foot sides What is the area of the base rememberto label your answer Mary drove around Lake Hollingsworth and saw that the distance was 3 miles How far would it be for her to swim across the lake on a path through the exact center of the lake Use 314 as an approximation for 1T Page 3 of 12 MTH 150 AAR Survey of Mathemaucs Bngme Mamneau Chapter 9 7 Geometry 4 Fwd the area of the shaded reg on f the bases of he paraHe ogram are 5 5 cm the swdes 1 1 3 Cm as We 5 Fwd the area of the shaded reg on knowmg that he dwameter and he smaH base are 0m 2 5 andthe ong base measures4 Cm 0 You a parce H taugmav V Based on me 0 d mumcwpa mans the area of the property 5 approwmate y 100000 ydz and me mm of me p ot 5 250 yds What 5 me pemmeter of your mom Page40f 12 MTH 150 AAR Survey or Mathemaucs Brrgrne Mamneau Chapter 9 7 Geometry 7 The Mammaer rs consrderrng a awn servrce A pram ormerr property rs shown berow How muchwm tcostm e Mamnut s to have her awn cut rr me servrce charges 0 002 per square room 400 n TheMa39hnut s yard m n 300 n lt gt 20 hi Pagesor 12 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 9 Geometry 8 Kathy and Sally want to meet at a pavilion across from a park in New York City but Kathy has rollerblades on and can t go through the grass She will skate on the sidewalk around the perimeter of the park Sally however is walking and will cut across the grass in a diagonal line toward the pavilion How much further will Kathy travel than Sally Kathy 132R 1056 9 The picture below is an overhead view of the pentagon Each side is 921 feet What is the perimeter of the pentagon Rememberto label your answer Page 6 of 12 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 9 Geometry 10Judy and her brother Sid bought their grandmother a new garage door to replace the one Judy broke when she backed the car out of the garage after forgetting to pull up the door rst Judy wants to decorate the garage door with some special ribbon She wants to buy enough ribbon to make a big cross as indicated below lfthe garage door has dimensions 9 feet by 6 feet what is the smallest number of feet of ribbon she must purchase to decorate in this manner If ribbon costs 25 cents per foot what is the minimum she must pay forthe ribbon 6 feet high 9 feet wide Page 7 of 12 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 9 Geometry 94 Volume and Surface Area What is volume What is surface area Volume 3D and Surface Area Volume Rectangular Solid Surface Area 4 h w l 5 Volume 5 Surface Area s r a h Cube Q Volume Cylinder Surface Area Page 8 of12 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 9 Geometry 1 Volume m2h 3 Cone h Surface Area m2 7rmr2 h2 Volume 7273 Sphere Surface Area 4an found using calculus tools A Volume lBh Pyramid 3 v where B area ofthe base Volume Bh Any Regular Polyhedron where B area ofthe base 33 Units of Measure Area units2 Examples cm2 ft2 Volume units3 Examples cm3 ft3 Examples 1 How much water does it take to ll a circularjacuzzi with a diameter of 5 feet They plan to ll the jacuzzi to a height of2 feet Use 314 as an approximation for 7239 Page 9 of 12 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 9 Geometry 2 Aimee s ower bed is rectangular in shape and is 6 ft long by 3 ft wide She wants to spread mulch and have it be 2 inches thick a Find the amount of mulch she needs in cubic feet b If mulch costs 2 per cubic foot how much will she spend 3 Jim wants to paint the walls ofa 12 ft X 14 ft room The ceiling is 8 ft high He won t be painting the ceiling or 3 windows in the room which are 4 ft X 2 ft a How many square feet need to be covered in paint b lf1 gallon of paint covers 300 square feet for one coat how much does he need for 2 coats of paint You can t buy a part of a can Page 10 of12 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 9 Geometry 4 Fifteen candy pebbles can t into a space oftwo cubic centimeters A candy maker is considering a holiday packaging of her multicolored candy pebbles Which will hold more candy pebbles a triangular prism with a base area equal to 12 squared centimeters and a height of 15 centimeters or a cylinder with a circular base of approximately 16 squared centimeters with a height of 11 centimeters How many more candy pebbles does the larger container hold Susan wants to cover the front two sides and bottom ofthe following cornice with fabric The back will not be covered because it is against the wall The top will not be covered because lights will be installed to shine on the ceiling 30 inches 01 6 inches 5 11 indies 114 inches l 24 inc es a How many square inches of fabric will she need b If 144 square inches 1 square foot how many square feet are needed 0 If 9 square feet 1 square yard how many square yards are needed Page 11 of12 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 9 Geometry 6 What is the cost of concrete for a driveway that is 50 feet long 14 feet wide and 12 inches deep ifthe concrete sells for 70 per cubic yard Express your answer to the nearest dollar Brain Teaser What happen to the perimeter ofa square is we double the measurements Triple the measurements What happen to the area ofa square ifwe double the measurements Triple the measurements What happen to the volume of a square if we double the measurements Triple the measurements lfthe radius of a sphere is doubled then what is the corresponding change in its volume Page 12 of12 Your name Your section MTH 150 SURVEY OF MATHEMATICS Chapter 6 ALGEBRA GRAPHS AND 61 62 63 64 FUNCTIONS Order of Operations Linear Equations in One Variable Formulas Applications of Linear Equations in One Variable Page 1 of 16 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 6 Algebra Graphs and Functions 61 Order of OperationsSome vocabulary first Algebra Variable Constant Coefficient Algebraic expression Page 2 of 16 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 6 Algebra Graphs and Functions Evaluating an expression Equation Solution to an equation Solving an equation Exponent and Base Page 3 of 16 MTH 150 AAR Brigitte Martineau Survey of Mathematics Chapter 6 Algebra Graphs and Functions ORDER of OPERATIONSAGAN and AGAIN 1 P 2 E 3 MampD Examples Evaluate the expression for the given value of the variable x x9 x x9 x26x 4 x5 x3y2 x 1 8x3 4x27 x3 Determine whether the value is a solution to the equation 4x 715 x 2 4 AampS 5x2 7x 11 x 1 4 y3 yx23x 5 x1 y 1 Page 4 of 16 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 6 Algebra Graphs and Functions Worksheet Is 4 a solution of 2x 8 2 Is 1 a solution of2b 1 3 A ls1 a solution of 4 2m3 4 Is 2 a solution of2x2 1 4x 1 5 Is 2 a solution of m2 4m3 6 Is 6 a solution of rt 22 n 4n4 l Is 3 a solution of 2aa 1 3a3 G Is a solution of5m110m 3 9 Is a solution of4y13 10 Is a solution of8x 112x3 Page 5 of 16 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 6 Algebra Graphs and Functions 62 Linear Equations in One Variable Vocabulary Terms Like terms Simplify an expression Review of the Properties of Real Numbers Distributive property ab c ab ac Commutative property of addition ab ba Commutative property of multiplication ab ba Associative property of addition a b c a b c Associative property of multiplication abc abc Examples 0 9x 6x 0 5x 4y 3y8x3 o x 4x3 o 6r 3 2r510 o 15x9 2x Page 6 of 16 MTH 150 AAR Brigitte Martineau Solving a Goal Rule How Usi 1 A A monsoon Survey of Mathematics Chapter 6 Algebra Graphs and Functions n Equation ng the Four Properties of Equations Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Procedure to Solve Linear Equations f fractions multiply both sides by the lowest common denominator to clear the fractions Use distributive property to eliminate the parentheses Combine like terms on one side Write as ax b using addition and subtraction property Solve X c using multiplication and division property Check Examples Solve the following 3y 411 23xpn Page 7 of 16 MTH 150 AAR Brigitte Martineau 2 5 12x 12 3x15 3x2x1x What is a proportion What is a Survey of Mathematics Chapter 6 Algebra Graphs and Functions 3t 42t 1 10 53p 4 3x212x1x Proportions Page 8 of 16 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 6 Algebra Graphs and Functions How do we solve What are the steps 1 De ne a variable X 2 Set the proportion Be careful about the units 3 Crossmultiply 4 Answer the question Examples ln Northampton County the property tax rate is 8025 per 1000 of assessed value lfa house and lot have been assessed at 132600 determine the amount of tax the owner will have to pay A machine manufactures 700 toys every 3 hours How many toys does it manufacture in 60 hours How many hours does it take to manufacture 2800 toys Page 9 of 16 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 6 Algebra Graphs and Functions 63 Formulas What is a formula How to evaluate a formula Examples 0 P abc determine Pwhen a 25 b 53 and c 32 Does this formula look familiar o pi2r determine rwhen p 62500 and i 5 electronics formula 03w 2 A person s body mass index BMI is found by the formula B 7 where w is the person s weight in pounds and h is the persons height in inches Lance Bass is 6 ft tall and weighs 200 lb Determine his BMI Page 10 of16 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 6 Algebra Graphs and Functions 0 R OV Dr determine 0 when R 670 V 100 D 10 and r 4 economics formula SIMPLE INTEREST FORMULA I PRT R T Example Jeff Hubbard borrowed 800 from his brother for 2 years At the end of 2 year he repaid the 800 plus 128 in interest What simple interest rate did he pay EXPONENTIAL FORMULA Exponential Growth and Decay Radium226 is a radioactive isotope that decays exponentially at a rate of 00428 per year The amount ofradium226 R remaining after tyears can be found by the formula R R0e where R0 is the original amount present lfthere are originally 10 grams of radium226 determine the amount of radium226 remaining after 1000 years Page 11 of16 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 6 Algebra Graphs and Functions 64 Applications of Linear Equations in One Variable Write the following sentences as mathematical expressions 1 The sum of8andy l a less than 16 A tincreased by 10 A The product of 8 and y 01 p decreased by 7 O q multiplied by 13 l 6 times the difference between m and 7 G The sum ofthreefourths ofn and 12 O b decreased by the product of2 and b 108 increased by the quotient of n and 4 11The product oft and the sum of tand 16 Page 12 of16 MTH 150 AAR Brigitte Martineau Survey of Mathematics Chapter 6 Algebra Graphs and Functions Write the following sentences as linear equations and solve 1 A 01 A number decreased by 9 is 5 A number multiplied by 7 is 42 Four times a number decreased by 10 is 42 Six more than ve times a number is 7 times the number decreased by 18 A number divided by 3 is 4 less than the number The product of3 and a number decreased by 45 is 4times the number Miguel purchases two new pairs of pants at The Gap for 60 If one pair was 10 more than the other how much was the more expensive pair Page 13 of16 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 6 Algebra Graphs and Functions 8 Budget Warehouse has a plan whereby for a yearly fee of 80 you save 8 ofthe price of all items purchased in the store What is the total Vito will need to spend during the year for his savings to equal the yearly fee 9 Jim is building a rectangular deck and wants the length to be 3 ft greater than the width What will be the dimensions of the deck if the perimeter is to be 54 ft 10A health club is offering two new membership plans Plan A costs 56 per month for unlimited use Plan B costs 20 per month plus 3 for every visit How many visits to the health club must Doug make for Plan A to result in the same cost as Plan B Page 14 of16 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 6 Algebra Graphs and Functions 11The Gilberts purchased a car lfthe total cost including a 5 sales tax was 14 512 nd the cost ofthe car before tax 12 Rachel has been told that with her halfoff airfare coupon her airfare from New York to San Diego will be 22700 The 22700 includes a 7 tax on the regular fare On the way to the airport Rachel realizes that she has lost her coupon What will her regular fare be including tax 13The cost of renting a small truck at the UHaul rental agency is 35 per day plus 020 a mile The cost of renting the same truck at the Ryder rental agency is 25 per day plus 032 a mile How far would you have to drive in one day for the cost of renting from UHaul to equal the cost of renting from Ryder Page 15 of16 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 6 Algebra Graphs and Functions 14 Chuck Salvador has 140 ft of fencing in which he wants to build two connecting adjacent square pens What will be the dimensions if the length ofthe entire enclosed region is to be twice the width Page 16 of16 MTH 150 Chapter 5 Brigitte Martineau Section 53 53 The Rational Numbers Definition Set of all numbers of the form where p and q are integers and q at 0 Examples Is 2 a rational numbers Why Reducing fractions Ex Mixed Numbers and Im proper Fractions What is a mixed numbers What is an improper fraction How to convert a positive mixed number into an improper fraction Integer denominator numerator denominator Examples 2 1 Convert 4 into an improperfraction Same with 52 Page 1 of 3 MTH 150 Chapter 5 Brigitte Martineau Section 53 How to convert a positive improper fraction to a mixed number 1 Divide the numerator by the denominator 39 d 2 The mixed number Quotient remain er denommator Examples 4 5 Convert g Into a mixed number Convert 3 Into a mixed number Terminating versus repeating decimals Rational numbers can be expressed as decimal numbers either terminating or repeating Examples Adding and subtracting fractions NEED COMMON DENOMINATORllll 2 5 6 4 9 9 9 3 2 7 31 12 10 3 7 14 ii 12 10 Page 2 of 3 MTH 150 Chapter 5 Brigitte Martineau Section 53 a C ac Multi in fractions bd 0 py g b d bd 21 2gtxlti 5 8 5 9 Dividing fractions Use the reciprocal What is the reciprocal of a number A number its reciprocal always equal Find the reciprocal of 13 Find the reciprocal of 4 a General Rule for lelSlon of fractions Page 3 of 3 Your name Your section 13 13 13 13 13 13 13 2 00 A 5 6 l MTH 150 SURVEY OF MATHEMATICS Chapter 13 STATISTICS Sampling Techniques The Misuses of Statistics Frequency Distributions Statistical Graphs Measures of Central Tendency Measures of Dispersion The Normal Curve MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics Sampling Techniques and Misuses of Statistics Highlights Statistics the art of gathering data analyzing data and making inferences about data Two types of Statistics 0 concerned with gathering and analyzing data concerned with making generalizations or predictions from the collected data Probability versus Statistics 0 Statistics 0 Probability Population versus Sample A is a subset of the population The is the entire contents 3c the mean of the u the mean of the Standard deviation 3 the standard deviation ofthe o the standard deviation ofthe Why using samples and not the population Page 2 of 26 MTH 150 AAR Brigitte Martineau Survey of Mathematics Chapter 13 Statistics Sampling Techniques are very important Why What is an unbiased sample Sampling Techniques Random Sampling and how to do it Systematic Sampling and how to do it Cluster Sampling and how to do it Strati ed Sampling and how to do it Convenience sample Page 3 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics Examples 0 Stacey wants to nd the average height of students at MCC Since it s too hard to determine the height of every MCC student she collects a sample She decides to use the heights of members of the mens and womens basketball teams for her sample a Is this an unbiased sample Explain b Would the actual average height ofMCC students be higher or lowerthan the average height of her sample Why 0 Joe wants to nd the average weight of newborns and decides to choose his sample from babies in the neonatal intensive care unit a Would this be an unbiased sample Explain b Would the actual average weight of newborns be higher or lower than the average of Joe s sample Why 0 Should the United Nations continue to have its headquarters in the United States A television program asked its viewers to call in with their opinions on that question There were 186000 callers 67 ofwhom said No A nationwide random sample of 500 adults found that 72 answered Yes to the same question Explain to someone who knows no statistics why the opinions ofonly 500 randomly chosen respondents are a better guide to what all Americans think than the opinions of 186000 callers Page 4 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics 0 Some subscribers of Consumer Reports respond to an annual questionnaire regarding their satisfaction with new appliances cars and other items The information obtained from these questionnaires is then used as a sample from which frequency of repairs and other ratings are made by the magazine Are the data obtained from these returned questionnaires representative ofthe entire population or are they biased Explain your answer A group of hotel owners in a large city decide to conduct a survey among citizens of the city to discover their opinions about casino gambling a Describe the population b One ofthe hotel owners suggests obtaining a sample by surveying all the people at six ofthe largest nightclubs in the city on a Saturday night Each person will be asked to express his or her opinion on casino gambling Does this seem like a goodidea Misuses of Statistics Most accidents occur on Saturday night This means that people do not drive carefully on Saturday night 0 Consider the following two graphs Which car would you buy Repair Costs per year Repair Costs per year 1000 1000 750 l 1 l 39U 39U 3913 g g g 81 81 81 395 395 850 F8 81 81 s s s o o o Page 5 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics 133 Frequency Distribution What is a frequency distribution Two types of frequency distribution Why using frequency distribution Ungrouped Frequency Distribution Look at these data 0 2 1 1 1 O 00 N00 What could they mean Page 6 of 26 MTH 150 AAR Brigitte Martineau Survey of Mathematics Chapter 13 Statistics Grouped Frequency Distribution How to create a frequency distribution table Each class should be the same width The classes should not overlap Each piece of data should only belong in one group The is also called the midpoint of the class Class mark lower 11m1t 4 Upper 11m1t The ofa frequency distribution is the class with greatest frequency Does the following set of numbers make any sense Interval Frequency Class Mark Page 7 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics EXAMPLES 1 Construct a frequency distribution on the data you see below representing the number of phones in a household 1 2 O 2 4 6 4 2 3 1 3 3 4 2 1 1 4 4 2 2 2 2 2 2 O 4 3 2 1 3 4 3 4 o What is the sample size 0 What is the most popular value mode 0 What is the frequency of the most popular value mode 2 The town of Brighton is planning to improve the local park The response of 32 families who were asked how many times per year they visit the park are shown below Construct a frequency distribution letting the rst class be 20 22 2O 21 24 25 26 27 29 32 2O 23 24 25 26 27 3O 32 2O 23 24 26 26 28 31 33 21 23 24 26 26 28 31 34 Midpoint What is the class width How many classes What is the third class What is the lower class limit of the third class What is the upper class limit ofthe third class What is the modal class Page 8 of 26 MTH 150 AAR Brigitte Martineau Survey of Mathematics Chapter 13 Statistics Why doing graphs 1 Bar Graph Purpose What does it look like 134 Statistical Graphs Delecls 1000Cals 2 Comparative Bar Graph What does it look like Characteristics Monthly Unline vs Catalog Sales For 2001 93 9 s 45 H as Revenue 510005 mil m Mar 4M PM Pia5v Jaw Jul M Rug mm Month Ionian cumog Page 9 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics 3 Picture Graph aka as Pictogram ii What does it look like Characteristics Number Enigma WIEIIW II Punch Lemonade Orange Soda Beverage Circle Graph Pie Chart 7 n125 What does It look like M5 nn75 Characteristics n5 I n15 How to Interpret a Circle Graph If pounds or 39 number of pounds for each category Page 10 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics 4 Histogram Hislugram What does it look like WED 1EE Characteristics Mu a 12EI 5 1EE o E39 EU 39 EU 0 AU ZEI El El 3 E 9 1 15 21 24 27 SD INDLIS How to construct an histogram Create a frequency distribution from the data given Find the class mar or Draw the graph Make sure the bars are Label Axis Examples Page 11 of 26 MTH 150 AAR Survey or Matnernatrcs Bngrtte Mamneau cnapter 13 e Statrstrcs 5 Frequency Polygon What does 1 took We Charactertsucs sax m Frequency Purygun mx 225w 275w 32m 375w um n5u 52m 575u How to construct a frequency potygon Create a rrequency drstnbutton rrorn tne data gryen Ftndme dass mark atso known as Draw tne grapn Make sure to connect tne Labe ANS Examples Page 12 or 26 MTH 150 AAR Brigitte Martineau Number of People in C39 O 0 Survey of Mathematics Chapter 13 Statistics The frequency polygon shows a distribution of IQ scores Which one of the following is true based on the graph Distribution of IQ Scores 70 75 80 85 90 95 100 105 110 115 120 125 130 IQ Scores The graph is based on a sample of approximately 50 people More people had an IQ score of 100 than any other IQ score and as the deviation from 100 increases or decreases the scores fall off in a symmetrical manner More people had an IQ score of 1 10 than a score of 90 The percentage of scores above any IQ score is equal to the percentage of scores below that score Page 13 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics 135 Measures of Central Tendency Some examples of measures of Central tendency are What do they measure The many meanings ofthe word Average 1 Y is the symbol for i2 7 2 The is the value in the middle ofa set or ranked data 3 The is the piece of data that occurs most frequently 4 The is the value halfway between the lowest and the highest value in the data lowest value highest value midran e g 2 Page 14 of 26 MTH 150 AAR Brigitte Martineau Examples Survey of Mathematics Chapter 13 Statistics Consider the following 2 sets of data Calculate the Averages for each lfthere is no mode say so Data Set Mean Median Mode Midrange BeSt Measure 21 19 20 1 2 3 4 90 5 10 15 20 50 10 20 30 40 Page 15 of 26 MTH 150 AAR Brigitte Martineau Examples Survey of Mathematics Chapter 13 Statistics a Assume that the mean ofa set of5 numbers is 50 What is the sum of the 5 numbers b You scored 70 80 65 and 90 on your 4 rst tests What score do you need on your fth test in orderto have a mean of 75 c The results of3 tests of MTH 150 are shown in the following table Explain what is going on Mean Median Exglanation Test 1 74 73 Test 2 73 80 Test 3 65 6O Page 16 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics 136 Measures of Dispersion Measures of Dispersion are used to measure the What are the common measures ofdispersion Why do we need measures of dispersion Range verses Standard Deviation O O O O O O O d5 d6 1 1 Range d3 d4 I mt dz po Average Standard Deviation HOW to nd the range range highest Value lowest Value How to nd the Standard Deviation s Page 17 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics Examples Find the range and standard deviation ofthe following set of numbers Range Standard Deviation Step 1 Find the mean Y Step 2 Fill in the table below Data Set Sum XXX ff Page 18 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics 0 Find the range and the standard deviation ofthe following data 8 8 12 14 6 6 Data Set x X Sum XXX ff o The results of2 tests of MTH 150 are shown in the following table Explain Mean Sta39ld Explanation DeVIatIon Test 1 74 4 Test 2 73 22 Page 19 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics 137 Normal Distribution Different shapes of the histogram o Rectangular o JShaped BiModal Skewed Normal Page 20 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics Examples Many products come with owner registration or warranty cards Usually the consumer is asked a few questions about his or her family and household income Random samples of warranty or registration cards for 5 different products revealed the household income distributions shown below Name the distributions 0 microwave ovens b electric fans 0 bigscreen televisions d snow blowers Characteristics of the Normal Curve Page 21 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics Examples Assume that the number of hours college students spend working per week is normally distributed with a mean of 18 hours and a standard deviation of 4 hours gt Suppose a college student works 14 hours a week Describe in a complete sentence how the of hours this college student works compares to the number of hours all college students work w Suppose a college student works 26 hours a week Describe in a complete sentence how the of hours this college student works compares to the number of hours all college students work 0 Suppose a college student works 30 hours a week Describe in a complete sentence how the of hours this college student works compares to the number of hours all college students work U Would a standard deviation of 5 make sense in this problem quot39 What part ofthe problem tells us that the zscore table or empirical rule can be used to answer the question Page 22 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics ZScore What is a zscore What is the formula for zscore Notation Be careful 1 The calculated value ofz is rounded to the nearest 2 The zscore measures the number of above or below the 3 zscores range from to Examples A certain data set has mean 76 and standard deviation 10 Find the zscores for 90 and 60 Bill and Joe both got 79 on their statistics test Bill is in section 1 where the mean was 75 and the standard deviation was 10 Joe is in section 5 where the mean was 77 and standard deviation was 21 Who has the best relative score Page 23 of 26 MTH 150 AAR Brigitte Martineau How can we use zscore to find probabilities Survey of Mathematics Chapter 13 Statistics Find the area underthe standard normal curve to the left ofz 152 Find the area underthe standard normal curve to the left ofz 152 Find the area underthe standard normal curve to the right ofz 152 Find the area underthe standard normal curve between 2 O and z 152 Find the area underthe standard normal curve between 2 152 and z 0 Find the area to the left ofz 081 Page 24 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics 0 Find the area to the right ofz 231 c Find the area between 2 215 and z 132 c Find the area between 2 085 and z 206 Assume that the number of hours college students spend working per week is normally distributed with a mean of 18 and standard deviation of 4 Draw and label the normal curve with appropriate values ofthe Empirical Rule a Find the percent of college students who work at least 18 hours Page 25 of 26 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 13 Statistics b What percent work between 14 and 26 hours What percent work more than 22 hours a week 0 Q What percent work less than 14 hours a week 390 What percent work less than 26 hours a week n What percent work less than 16 hours a week What percent work more than 21 hours a week 9 339 What percent work between 13 and 23 hours a week Page 26 of 26 Your name Your section MTH 150 SURVEY OF MATHEMATICS Chapter 12 PROBABILITY 121 The Nature of Probability 122 Theoretical Probability 123 Odds 124 Expected Value Expectation 125 Tree Diagrams 126 Or and And Problems MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability 121 The Nature of Probability In which areas ofyour life have you heard or used probabilities lately Vocabulary Experiment Outcomes Event Theoretical Probability versus Empirical Probability Empirical Probability ofan event Page 20f19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability Let s practice Experiment Event Results Combined Name H T Pmb T Probability This experiment is an example ofthe LAW OF LARGE NUMBERS Examples 0 The theoretical probability of rolling a 4 on a die is 16 Does this probability mean that ifa die is rolled six times one 4 will appear If not what does it mean 0 A multiplechoice test has four possible answers for each question lfyou guess at an answer what is the probability that you select the correct answer for one particular question Explain the meaning ofthis probability in a complete sentence Page 3of19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability 122 The Theoretical Probability Experiment Throwing a die Theoretical Probability ofan Event Ex A die is rolled P3 Px gt 6 Peven Px 5 6 Podd P4 P2ltx55 Pnota4 Ex Selecting one card from a deck of card What is the probability that o P9 o P2 lt card lt 8 o Pspade o P7and red 0 Pnotanace o Pblack o Pheart and club P H d o no ace car 0 PJ or Q or K o Pred and spade o P8 lt card lt 9 o Pblack face card Page 4of19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability Characteristics of Probabilities o P impossible event 0 P event that is sure to happen 0 A probability will always have a value between 0 If you add all probabilities of all outcomes o P event A P not event A o P not event A Examples 0 Mr Clark gave his class a question to work on at the beginning of math class Sarah answered that the probability something would happen is Jeffwalked in class late and didn t hear the question Explain in a complete sentence how Jeff knew this answer was wrong without even hearing the question 0 Tim and Denise are studying for a math test Tim says that the probability something would happen is Explain in a complete sentence how Denise could explain to Tim that his answer is wrong Page 50f19 MTH 150 AAR Brigitte Martineau Survey of Mathematics Chapter 12 Probability Mrs Green told her math class that the theoretical probability of her letting the class out early the day before spring break was equal to zero Explain in a complete sentence what Mrs Green was telling her math class Mrs Green told her math class that the empirical probability of her letting the class out early the day before spring break was equal to zero Explain in a complete sentence what Mrs Green was telling her math class Mr Howlett told his math class that the theoretical probability of having a test at the end of each chapter is equal to one Explain in a complete sentence what Mr Howlett was telling his math class Mr Howlett told his math class that the empirical probability of having a test at the end of each chapter is equal to one Explain in a complete sentence what Mr Howlett was telling his math class John randomly chooses a marble from a bag There are blue yellow pink and green marbles in the bag 0 A What information do you need to know in order to determine the probability that John picks a green marble from the bag 0 What information would have to be given in order for you to determine that the probability that John chooses a green marble is Page 6of19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability 123 Odds What does it mean Notation Odds against Odds in favor Widespread Misunderstanding Examples Roll a die What are the odds against rolling an odd number What are the odds against rolling a number less than 3 Page 7of19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability Ifthe odds against are a b then the odds in favorare Examples In a deck of card 0 What are the odds in favor ofa heart 0 What are the odds against heart Ifthe odds against being promoted are 4 11 what is the probability of being promoted Ifthe odds against winning a contest are 2 7 what is the probability ofwinning the contest What is the probability of loosing the contest P 747 38 The odds in favor of Boris Penzed winning the chicken wing eating contest are 3 8 Determine the probability that Boris will a win the contest b not win the contest Page 80f19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability 124 Expected Value Expectation What is it Ifthe expected value is negative 9 Ifthe expected value is O 9 Also called fair price Ifthe expected value is positive 9 Expected Value where Pn is and An is Example You ip a coin You win 1 if you ip a Head and you lose 1 ifyou ip a Tail What is the expected value in this game Examples At an investment tax seminar Judy Johnson estimates that 20 people will attend if it does not rain and 12 people will attend if it rains The weather forecast indicates there is a 40 chance it will not rain and a 60 chance it will rain on the day ofthe seminar Determine the expected number of people who will attend the seminar Page 90f19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability The Palm Coast investment club is considering purchasing a certain stock After considerable research the club members determine that there is a 60 change of making 8000 a 10 chance of breaking even and a 30 chance of losing 6200 Find the expectation ofthis purchase Mike and Dave play the following game Mike picks a card from a deck of cards If he selects a heart Dave gives him 5 If not he gives Dave 2 Find Mike s expectation and Dave s expectation A multiplechoice exam has four possible answers for each question For each correct answer you are awarded 5 points For each incorrect answer 2 points are subtracted from your score For answers left blank no points are added or subtracted o If you do not know the correct answer to a particular question is it to our advantage to guess Explain Page100f19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability o If you do not know the correct answer but can eliminate one possible choice is it to your advantage to guess Explain This week the expected number ofviewers for the show Lost is 13 million people Does this guarantee that next week there will be 13 million viewers for Lost Explain your answer in complete sentences A poker player s expectation is 10 for a hand of poker Is it possible for this poker player to lose money on her next hand of poker Explain your answer in complete sentences The concept of Fair Price The fair price is the Fair Price Cost to Play Expectation Page11of19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability Example Ten thousand raf e tickets are sold for 5 Four prizes will be awarded one for 10000 one for 5000 and one for 1000 Eric purchases one ofthese tickets Find his expected value and nd the fair price of a ticket Page120f19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability 125 Tree Diagram When computing probabilities we need to know the total number of outcomes Sometimes it will be tedious to nd all possible outcomes ofan experiment We will use tree diagram as well as the counting principle to help us nd out how many possible outcomes exist A tree diagram is o How many outcomes when you ip a coin o How many outcomes when you roll a die 0 How many outcomes when you flip a coin then roll a die Counting Principle lfone experiment can be done in M distinct ways and a 2nd experiment can be done in N distinct ways then the 2 experiments in that order can be done in MN ways Let s draw a tree diagram to nd out how many outcomes we have when flipping a coin then rolling a die Page13of19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability Sample Space Sample Point Without replacement versus With replacement Example lftwo states are selected at random from the 50 states determine the number of possible outcomes ifthe states are selected with replacement V thout replacement Example Four students Bob Jim Mike and Lucy volunteered to become president and vice president ofa committee Draw a tree diagram to illustrate all the possible outcomes Page14of19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability Example Here is a diagram of all the different routes you can take from Buffalo to Albany Buffalo Rochester Syracuse Albany 0 Construct a tree diagram to illustrate this experiment 0 What is the total number of possible routes 0 P taking route 2 o P taking route 4 o P taking route 6 o Proute 1 and 7 o Proute1 or 3 or 7 o Proute 2 and route 5 in the same trip 0 Pnot route 2 or not route 5 because of traf c jams Page150f19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability 126 OR and AND Problems In the previous section we learned how to use tree diagrams to help us nd the total number ofoutcomes When the total number of outcomes is really large drawing tree diagram is not really useful In this section we will try to solve OR and AND problems without using the tree diagram method OR Problems Let s look at an experiment we are familiar withrolling a die Find Podd or a number less than 5 PA or B PA PB PA and B Examples If PA or B 06 PB 03 and PA and B 01 nd PA o In a deck of card Find Pred or 8 Pred or spades Two events are mutually exclusive if Page16of19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability AND Problems In AND Problems two selections will occur one after another You will select 2 cards you will choose 2 students IT WILL ALWAYS INVOLVED 2 EVENTS FOLLOWED BY EACH OTHER PA and B PA PB Assuming that event A has occurred Examples In a deck of card 2 cards are selected with replacement Find the probability that you selected 2 face cards Is the probability of selecting a face on the second card affected by the fact that you selected a face card on the rst selection What if the previous experiment was done without replacement Is the probability of selecting a face on the second card affected by the fact that you selected a face card on the rst selection Page17of19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability SUMMARY V th replacement I I V thout replacement I I Example A cooler contains 6 cans of soda 3 colas 2 lemonades and 1 orange You and your friend are suddenly thirsty and you select two cans at random o Is this an experiment with or without replacement 0 What isthe probability that the rst can is a cola and the second an orange soda 0 What is the probability that you do not select any cola o What is the probability of selecting at least one cola Page180f19 MTH 150 AAR Survey of Mathematics Brigitte Martineau Chapter 12 Probability Example Here s a chart which shows how often American women exercise Ifthree women are selected at random determine the probability that 0 They all exercise daily 0 The rst two exercise about once a week and the third exercises daily 0 The rst exercises at least three times a week the second exercises daily and the third never exercises 0 They all exercise about once a week or more Page190f19 Your name Your section MTH 150 SURVEY OF MATHEMATICS Chapter 1 Critical Thinking Skills 12 Estimation 13 Problem Solving MTH 150 Survey of Mathematics Chapter 1 Brigitte Martineau Estimation An important step in solving mathematical problems or in fact any problem is to make sure that the answer you ve arrived at makes sense One technique for determining whether an answer is reasonable is to estimate 12 Estimation What is estimation Symbol used Examples 1 A Estimate the cost of 52 thirtysevencent stamps B Is 370 a reasonable estimate Why or why not Is the actual answer higher or lower C Is 52 a reasonable estimate Why or why not Is the actual answer higher or lower 2 A Estimate the income earned for 32 hours at 795 per hour B Is 795 a reasonable estimate Why or why not Is the actual answer higher or lower Page 2 of5 MTH 150 Survey of Mathematics Chapter 1 Brigitte Martineau Estimation 3 A Estimate the total frequent ier mileage awarded for four trips of 1521 1897 2324 and 2817 miles B Would an estimate of less than 6000 miles be reasonable Why or why not 4 A Estimate the weight ofone hamburger in a package of six hamburgers if the weight of the package is 312 lb B Is 15 lbs a reasonable estimate Why or why not 5 A lfthe weight limit of Mathers Bridge is 10 tons approximately how many cars can be on the bridge at the same time if each car has an average weight of 2300 pounds Note 1 ton 2000 pounds B Is 10 cars a reasonable estimate Would the answer be more or less than this Page 3 of3 MTH 150 Survey of Mathematics Chapter 1 Brigitte Martineau Estimation 13 Problem Solving Guidelines for solving word problems 1 Understand the Problem What do we know 2 Variables Equations Patterns Tables 3 Solve 4 Check Examples 1 Chalon Bridges an architect is designing a shopping mall The scale of her plan is 1 in 12 ft lfone store in the mall is to have a frontage of 82 ft how long will the line representing that store s frontage be on the blueprint 2 A2bag of fertilizer covers 6000 ft2 How many bags are needed to cover an area of 32000 ft Page 4 of0 MTH 150 Survey of Mathematics Chapter 1 Brigitte Martineau Estimation 3 Sam Stevens and four other friends want to schedule a 6hour fishing trip on Lake Ontario with Reel Easy Sport Fishing The cost for five people is 445 The cost for six peoples is 510 How much money per person will Sam and each of his friends save if they can find one more friend to go on their fishing trip Assume that they will split the total cost equally The price ofa u shot at Maxim Health Systems increased by 25 from 2005 to 2006 In 2005 Maxim Health Systems charged 20 for a flu shot If Maxim Health sold u shots to 2 million people in 2005 and to 2 million people in 2006 how much more money did they earn from selling u shots in 2006 than in 2005 A 5 On four exams Wallace Memmer s grades were 79 93 91 and 68 What grade must he obtain on his fifth exam to have an 80 average Page 5 of6 MTH 150 Survey of Mathematics Chapter 1 Brigitte Martineau Estimation 6 Wendy Weisner fills her gas tank completely and makes a note that the odometer reads 384514 miles The next time she put gas in her car lling the tank took 126 gal and the odometer read 386870 miles Determine the number of miles per gallon that Wendy s car got on this tank of gas 7 Assume that the rate ofin ation is 6 per year for the next 2 years What will the price of a dishwasher be 2 years from now if the dishwasher costs 799 today 8 V th a certain medical insurance policy the customer must first pay an annual 100 deductible then the policy covers 80 ofthe cost ofXrays The rst insurance claims for a speci c year submitted by Yungchen Cheng are for two Xrays The rst Xray cost 640 and the second Xray cost 920 How much in total V ll Yungchen need to pay for theses Xrays Page 6 of0 MTH 150 AAR Brigitte Martineau Survey of Mathematics Chapter 11 Consumer Mathematics How to Find the Number of Days Between Two Specific Dates Table 1 11 31 Day of Month Jan Day 1 I Day 2 2 Du 3 3 Day 4 4 Day 5 5 Day 6 6 Day 397 7 Day 8 8 Day 9 9 Day 10 10 Day 1 1 11 Day 12 12 Day 13 13 Day 14 14 Day 15 15 Day 16 16 Day l7 17 Day 18 18 Day I 19 Day 211 20 Day 21 21 Day 22 22 Day 23 23 Day 24 24 Day 35 25 Day 26 26 Day 27 27 Day 28 28 Day 29 29 Day 30 30 Day 31 31 28 Feb 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 31 Mar 60 61 62 63 64 65 66 67 68 69 70 71 72 73 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 30 Apr 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 Days in Each Month 31 30 31 31 May June July Aug 121 152 182 213 122 153 183 214 123 154 184 215 124 155 185 216 125 156 186 217 126 157 187 218 127 158 188 219 128 159 189 220 129 160 190 221 130 161 191 222 I31 162 192 223 132 163 193 224 133 I64 68 225 I34 165 195 226 135 166 196 227 I36 167 197 228 137 168 I98 229 138 I69 199 230 139 170 200 231 140 171 201 232 141 172 202 233 142 173 203 234 143 174 204 235 144 175 205 236 145 176 206 237 146 177 207 238 147 178 208 239 148 179 209 240 149 180 210 241 150 181 211 242 151 212 243 30 Sept 244 245 246 247 248 249 250 251 252 253 254 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