Integrated Algebra Things to Remember Table of Contents Solving Multistep Equations
Page 1
Solving Inequalities
Page 2
Slope and the Equation of Lines
Page 3
Solving Systems of Linear Equations
Page 6
Solving Systems of Linear Inequalities
Page 9
Adding Polynomials
Page 10
Multiplying Polynomials
Page 11
Dividing Polynomials
Page 12
Factoring Polynomials
Page 14
Solving Quadratic Equations
Page 15
Rational Expressions
Page 16
Radicals
Page 18
Pythagorean Theorem
Page 19
Trigonometric Ratios
Page 20
Data and Statistics
Page 22
Practice Test
Page 45
Answer Key
Page 53 Solving MutiStep Equation
When solving equations our aim is to isolate the variable (get the variable by itself.) We use inverse operations to do this.
Example:
When you have parentheses you need to use the distributive property to get rid of the parentheses first.
Example:
2
To solve equations with fractional expressions, you need to multiply by the LCD first to get rid of the fraction.
Example: Solve for x: 1. Find LCD 16 4 2 16: 16, 32… 4, 8, 12, 16x 16 16 4: 16, 20, 24… = + 2 2: 2, 4, 6, 8, 10, 16 4 12, 14, 16, 18…
16 • 1 x + 16• 1 = 16• 1
x+4=8 -4 -4 x=4
2. Multiply each term by LCD
3. Simplify 4. Find the variable
Solving Inequalities Inequalities are sentences that compare two quantities that are not equal. The meaning of each of the inequality symbols is listed below.
3
You solve an inequality just like you solve an equation, but you reverse the direction of the inequality sign when you multiply or divide by a negative number. Example let’s say we begin with 8 > 4. If we divide both sides by 4 we get 2 > 1, which is false. We need to switch the sign because 2 < 1
Example:
The solution set includes all numbers that are less than negative 11.
Slope and the Equation of Lines Slope(m) is used to describe the steepness of a straight line. The higher the slope, the steeper the line. The slope of a line is a rate of change.
m=
( y2 − y1 ) rise vertical change = = ( x2 − x1 ) run horizontal change
Example: Find the slope of the line that passes through the points (-2, -2) and (4, 1).
(x1, y1) m=
( y2 − y1 ) ( x2 − x1 )
m=
(1 − (−2)) (4 − (−2))
m = ½ or 0.5 4
(x2, y2)
If you are given a line to find the slope, select any two points and use “rise over run”. Example:
Determine the slope of the line.
Slope (m) = “rise over run”. 6
rise 3 1 = = run 6 2
3
Vertical Lines Lines that are vertical have no slope (it does not exist). Vertical lines have "rise", but no "run". The rise/run formula for slope always has a zero denominator and is undefined. Example:
The equation of the line is x = 2. The equations for these lines describe what is happening to the xcoordinates. In this example, the x coordinates are always equal to 2.
5
Horizontal Lines Lines that are horizontal have a slope of zero. Horizontal lines have "run", but no "rise". The rise/run formula for slope always yields zero since the rise = 0. Example: The equation of the line is y = 3. This equation also describes what is happening to the ycoordinates on the line. In this case the y coordinates are always 3.
Remember the word “VUXHOY” Vertical lines Undefined slope X = number (equation of the line) Horizontal lines 0 - zero is the slope Y = number (equation of the line) To write the equation of a line, you need the slope (m) and the yintercept b (where the line crosses the yaxis). If given two points, use the slope formula to find m. Substitute (m) and any one of the points into the slope intercept formula to find the yintercept b. Write the equation of the line through the points (2, 1) and (3, 4) 1. Find Slope
2. Find yintercept
m=
y = mx + b
y2 − y1 x2 − x1
(4 − 1) (3 − 2) 3 m= =3 1 65
m=
4 = 3(3) + b 4= 9+b –9 –9 –5 = b 3. Substitute
y= 3x–5
6
Example:
We use the same method when writing the equation of parallel and perpendicular lines
W rite the equation of the line that is perpendicular to the line y = ½ x 1 and passes through the point (3, –3).
The slope of the line that is perpendicular will be the negative reciprocal (flip and change the sign) of ½, -2.
y = mx + b –3 = – 2(3) + b –3 = – 6 + b +6 +6 3 = b y = – 2x + 3
Solving Systems of Linear Equations A system of linear equations consists of two or more linear equations. In Integrated Algebra, we only study systems with two equations. The solution of the system is the value of the variables which make both equations true. We can solve systems using graphing. The solution of the system is where the graphs intersect. 7
Example:
Fin d t h e s o lut io n t o t h e fo llo win g s y s t e m :
2x + y = 4 y=x–2
+ 2x
1 .Graph bo t h equa ti o n s
y= 4
2 . Find w h ere t h e lin e s in t ers e c t (cro s s)
y
=x
–
2
(2, 0)
3 . Ch e c k: 2x + y = 4 2(2) + (0) = 4 y = x 2 (0) =(2) – 2
Both equations are true when x is 2 and y is zero.
We can also solve linear systems using substitution. This method is best when we have one variable in terms of the other. Example:
8
Solve the system using substitution: x+y=5 y=3+x x+y= 5 x + (3 + x) = 5 2x + 3 = 5 –3 –3 2x = 2 2 2 x=1
x+y=5 1+y=5 –1 –1 y=4
Sol ut ion: (1, 4) (1) + (4) = 5 (4) = 3 + (1)
We can also add or subtract to get rid of one of the variables. Before you add or subtract you may need to multiply so that the coefficients of one of the variables is the same or opposites of each other.
Solve the system using elimination: 2x y +2 =6 3x – y =5 6x y 3(2x y + 6 = 18 + 2 = 6) 6x – 2y 3x – y = 10 2( = 5) Keep 6x y + 6 = 18 ChangeChange – 6x + 2y = – 10 8y =8 8 8 y Solution: (2, 1) =1
Example: 9
3x – y =5 3x – (1 )= 5 +1 +1 3x 3 = 36 x =2
Linear systems can also be used to solve real world problems. Example:
Arielle has a collection of grasshoppers and crickets. She has 561 insects in all. The number of grasshoppers is tw ice the number of crickets. Find the nu mber of each type of insect that she has.
g + c = 561 L et g = gr assh opper s 2c + c = 561 L et c = cr ick ets 3c = 561 g + c = 561 g = 2c
10
3 3 c = 187 g = 2c g = 2(187) g = 374
Solving Systems of Linear Inequalities The solution set of a system of linear inequalities consists of all the points in the region where to two linear inequalities overlap. Example:
11
Solve the following system of linear inequalities x + y > 2 and x ≤ 2 graphically.
Graph the system of linear inequalities.
Solid lines
n tio
y≤2
lu
Dotted lines
So
x+ y >2 −x −x y > 2− x y > −x + 2
For > or ≥ we shade above or to the right of the line. For ≤ or < we shade below or to the left to the line.
Adding Polynomials Add like terms by adding the numbers in front of the variables (coefficients) of the terms, following the rules for adding signed numbers. Example: 12
Find the sum of 3x2 2 and 2x2 + 4x + 3 1. Arrange the like terms so that they are lined up under one another. 2. Add the like terms adding the coefficient of the terms
3x2 + 0x – 2 + 2x2 + 4x + 3 5x2 + 4x + 1 Subtracting Polynomials When subtracting polynomials, you must change signs of all terms being subtracted and follow rules for addition. Example: Keep – ChangeChange
2x² + 0x – 4 – (x²+ 3x 3)
2x² + 0x – 4 x² – 3x + 3 (signs changed) x² – 3x – 1
Multiplying Polynomials The Product of Powers rule states that for any number a and all integers m and n, am ∙ an = am + n. 13
Example 1: monomial • monomial
(4x 3 ) • (3x 2 ) = (4 • 3) • (x 3 • x 2 ) = 12 • x5 = 12x 5
Example 2:
Notice that the factors were regrouped and then multiplied. Also, multiply powers with the same base by adding the exponents.
This problem requires the distributive property. You need to multiply each term in the parentheses by the monomial (distribute the x across the parentheses).
monomial • binomial
= x 2 + 4x
Notice the distributive property at work again.
Example 3:
monomial • trinomial
= 2x 3 + 6x 2 + 8x Again, the distributive property is needed along with the rule for multiplying powers.
Example 4:
monomial • polynomial
= 3x 5 - 9x 4 + 18x 3 15x 2 To multiply two binomials, find the sum of the products of F the First terms, O the Outer terms, I the Inner terms, and L the Last terms. Example:
To find the power of a power, multiply the exponents. This is called the Power of a Power rule. 14
Dividing Polynomials The Quotient of Powers rule states that for any nonzero number a and all integers m and n, am = a m −n . n a
Example:
15
You can divide a polynomial by a monomial by separating the terms of the numerator. Example:
16
Factoring Polynomials Some polynomials can be factored using the Distributive Property. Example:
To factor polynomials of the form x2 + bx + c, find two integers m and n so that mn = c and m + n = b. Then write x2 + bx + c using the pattern (x + m)(x + n). Example:
17
Solving Quadratic Equations Quadratic equations generally have two solutions. We can solve quadratic equations by factoring and setting each factor equal to 0, then solving each equation. Example: Solve x2 + 3x – 18 = 0 x2 + 3x – 18 = 0 (x + 6)(x – 3) = 0 x + 6 = 0 OR x – 3 = 0 –6 –6 +3 +3 x=–6
Factor the left side Set each factor = 0 Solve each equation
x=3
The solution set of the equation is {–6, 3}. The solution set is also the roots of the equation. The roots of the equation are also the x intercepts (where the graph crosses the xaxis). At these points y = 0 Example: Based on this graph, what are the roots of the equation x 2 + 3x − 18 = 0 ?
Root Root
18
Rational Expressions A rational function is undefined when the denominator is zero. Example:
The function
y=
x −3 x2 − 4
is undefined
when the value of x is (1) 0 or 2 (3) 2, only (2) 2 or 2
(4) 2, only
x2 − 4 = 0
(x + 2) (x – 2 ) = 0 x = – 2 OR x = 2
we
2 2 x Simplify: 2 + 4 x x + 5x + 6
1. Factor the numerator
÷ 2x2 + 4x x 2x + 4
2 x ( x + 2) 2x 2 + 4x = x 2 + 5 x + 6 ( x + 2)( x + 3)
?
2 x
3. Divide 2x out = x+ 3 common factors
+2
2x (x + 2) 2. Factor the denominator if possible x2 + 5x + 6
(x + 2 ) (x + 3 )
19
6 x 1 = 6, 6 + 1 = 7 2 x 3 = 6, 2 + 3 = 5
When we simplify rational expressions reduce them to their lowest terms Example:
In order to add or subtract (or "combine"), algebraic fractions, a common denominator is needed. If the denominators are the same, keep the denominator and add the numerators. If the denominator is different, first find the least common denominator (LCD) by multiplying each denominator by the factor(s) that it does not have in common with the other denominators. Example:
What is
4 1 − in simplest form? 6p 5p
4 •5 1 •6 − 6p • 5 5p• 6 20 6 = − 30 p 30 p
=
Radicals We can use We circle our sign once. The go under the
7 14 20 − 6 = = 30 p 30 p 15 p
Multiplication and division of rational expressions is actually an extension of reducing fractions, except that now there are two fractions in the problem instead of just one. Reducing may be done up down, NOT across (horizontally). Example:
Simplify 5 28 28 2
x
14
2
x
7
Example: 20
5 • 2 7 = 10 7
prime factorization to simply radicals. pairs and they go outside the square root product of the prime factors that remain radical sign.
To add and subtract radicals, the same number must be under the square root sign, i.e. they must be like radicals. If they are unlike radicals, you must simplify each tern first, and then combine them. Example:
28 + 63 = 2 7 +3 7=
28 7 x 4 2 x 2
(2 + 3) 7 =
28 = 2 7
5 7
7 x 9
63
3 x3
63 = 3 7
Pythagorean Theorem
21
In a right triangle, a2 + b2 = c2 where c represents the length of the hypotenuse (the longest side), and a and b represent the lengths of the other two sides (legs). We can use the Pythagorean Theorem to find the missing side of a right triangle when we know the length of the two other sides. Example:
If the length of the legs of a right triangle are 5 and 12, what is the length of the hypotenuse?
?
12 5
a2 + b 2 = c 2 52 + 122 = c 2 25 + 144 = c 2 169 = c 2
Take the square root of both sides.
169 = c 2
13= c
Example: A ramp was constructed to load a truck. If the ramp is 9 feet long and the horizontal distance from the bottom of the ramp to the truck is 7 feet, what is the vertical height of the ramp to the nearest tenth of a foot?
The height of the ramp is 5.7 feet.
22
Trigonometric Ratios S
O A O C T H H A
Soh Sine = opposite over hypotenuse Cah Cosine = adjacent over hypotenuse Toa Tangent = opposite over adjacent se u n o te
p hy any angle
opposite
adjacent
opposite
opposite
Sine =hypotenuse Cosine = adjacent Tangent =adjacent hypotenuse
Example:
23
We can use trig ratios to find a missing side of a right angle triangle when we are given the measure of an angle and the length of one of the sides. Example: A ladder is leaned up against a 16 m brick wall. If the ladder forms an angle of 360 with the wall, find the length of the ladder. We have the length of the adjacent side and we need to find the length of the hypotenuse. Therefore, we will use cosine Adjacent Cosine θ = Hypotenuse
US E
= 16
HY PO T
x(Cos 360 ) 16 = 0 (Cos 36 ) (Cos 360 ) 16 x = (Cos 360 ) x = 19.8m
16 m
x(Cos 360 )
360
EN
1
x
16 x
ADJ ACEN T
Cos 360 =
OPPOSITE
Trigonometric can also be used to find the measure of an angle, given the length of two sides of the right triangle. Example: In the diagram of ∆ABC shown below, AB = 8 and BC = 5. To the nearest hundredth of a degree, what is the measure of angle A in the triangle?
Tan A =
Opp= 5
hy p
opposite adjacent
5 8 5 x = tan −1 8
Tan x o =
xo adj = 8
We know: opposite adjacent We need: angle A
Use calculator
Use: Tan
x = 32.01o 24
Data and Statistics Qualitative vs Quantitative Data Examine the differences between qualitative and quantitative data.
Qualitative Data
Quantitative Data
Overview: Overview: • Deals with descriptions. • Deals with numbers. • Data can be observed but not measured. • Data which can be measured. • Colors, textures, smells, tastes, appearance, • Length, height, area, volume, weight, speed, beauty, etc. time, temperature, humidity, sound levels, cost, members, ages, etc. • Qualitative → Quality • Quantitative → Quantity •
•
Example 1: Oil Painting
Example 1: Oil Painting
• • • •
blue/green color, gold frame smells old and musty texture shows brush strokes of oil paint peaceful scene of the country masterful brush strokes
• • • •
picture is 10" by 14" with frame 14" by 18" weighs 8.5 pounds surface area of painting is 140 sq. in. cost $300
•
•
Example 2: Latte
Example 2: Latte
• • •
robust aroma frothy appearance strong taste burgundy cup
• • •
12 ounces of latte serving temperature 150º F. serving cup 7 inches in height cost $4.95
•
•
Example 3:
Example 3:
Freshman Class
Freshman Class
•
friendly demeanors civic minded
•
25
672 students 394 girls, 278 boys
•
environmentalists
•
68% on honor roll
•
positive school spirit
•
150 students accelerated in mathematics
Univariate vs Bivariate Data Examine the differences between univariate and bivariate data.
Univariate Data
Bivariate Data
involving a single variable
involving two variables
does not deal with causes or relationships
deals with causes or relationships
the major purpose of univariate analysis is to describe
the major purpose of bivariate analysis is to explain
central tendency mean, mode, median dispersion range, variance, max, min, quartiles, standard deviation. frequency distributions bar graph, histogram, pie chart, line graph, boxandwhisker plot
analysis of two variables simultaneously correlations comparisons, relationships, causes, explanations tables where one variable is contingent on the values of the other variable. independent and dependent variables
Sample question: How many of the students in the freshman class are female?
Sample question: Is there a relationship between the number of females in Computer Programming and their scores in Mathematics?
Possibly Biased Data Statistics are influenced by a multitude of factors. It is even possible that statistics can be manipulated so that they tell the story that the person using them wants to tell. Because of these influencing factors, it is important to understand how to evaluate statistical information. When viewing statistics, you should consider:
Who collected the data? Does the group collecting the data have an interest in how the results turn out?
Example: A study on the hazards of cigarette smoking being done by a tobacco company. (may not be reliable findings conflict of interest)
What is the sample size of the study? How Example: A study is done on the favorite color of 14 year olds. The sample group for the study is Mrs. Smith's third many people/items were studied? period class containing 20 students. 26
(too few participants to generalize finding to all 14 year olds) •
Do the statistics show any bias?
Read more about BIAS below ↓
Example: The study of how many people can walk a balance beam is conducted with students from a gymnastics class. (the results are biased due to the very specific selection of the participants)
Sources of bias: Selection bias: In a statistical study, it is important that the smaller group used for the study (the sample) be truly representative of the larger group to whom the findings will be directed (the population). Preferably the sample group should be chosen at random. Dexterity Study: 500 people are invited to a research center for an experiment in dexterity and flexibility. 100 of the people show up. The researchers document the number of people who can clasp their right foot with their right hand behind their backs, by reaching over their right shoulders (as seen at the right). They conclude that an amazing 62% of people can perform this act of dexterity and flexibility. Are these finding reliable? What is wrong with this study? Since only 100 of the 500 invited participants showed up for this study, this is not a representative sample. It may also be the case that the people who were confident about their dexterity and flexibility showed up and the results are biased in their favor. This situation presents two problems for the study. First, the weights are unreliable for participants weighing over 150 pounds. Second, the question should be asked if the participants' heights were also measured. If not, what was the definition of an "overweight" participant?
27
Practice with
Categorizing Data
Try the following questions: 1. Jed's new horse: • • •
15.2 hands high 1250 pounds costs $2,200
age: 3 years, 4 months Choose: quantitative data qualitative data •
2. The tree: • • •
rough brown bark red berries wandering branching
small, sharply edged leaves Choose: quantitative data qualitative data •
28
Choose:
Pennsylvania coal:
•
florescent black black residue earthy smell
•
rough texture
• •
quantitative data qualitative data of Form
4. The students in the senior class at LHS High School: • • •
578 students 236 honor students 150 scholarship winners
51% male Choose: quantitative data qualitative ddata •
For questions 5 9, determine if the problem deals with univariate or bivariate data.
5. Determine if the number of hours a student studies will improve his/her final examination scores. Choose: univariate data bivariate data
6.
Determine the mean score on the last math test. Choose: univariate data
bivariate data
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7.
Determine whether lowering cholesterol lowers the risk of heart disease.
Choose: univariate data bivariate data
8. Prepare a frequency table for the number of students scoring A, B, C, and D on the Chemistry test.
Choose: univariate data bivariate data
9. Which situation should be analyzed using bivariate data? Choose: Ms. Saleem keeps a list of the amount of time her daughter spends on her social studies homework. Mr. Benjamin tries to see if his students' shoe sizes are directly related to their heights. Mr. DeStefan records his customers' best video game scores during the summer. Mr. Chan keeps track of his daughter's algebra grades for the quarter.
For questions 10 13, determine if the situation is biased.
30
10. A study is conducted to determine whether office workers
have high blood pressure. The participants in the study were friends of the researcher who shared the same doctor.
Choose: biased unbiased
11. A study is conducted to determine the number of students who wore a free promotional Tshirt given to all students at a local university rock concert. Five hundred students were chosen at random from the 5000 students attending the concert and asked if they wore the Tshirt during the concert.
Choose: biased unbiased
12. A study is conducted to estimate the average speed of drivers using the fast lane of the
motorway. To determine the drivers' speeds, a police car will follow the drivers on the motorway and record their speeds using the police car's speedometer.
Choose: biased unbiased
13. A movie theater is conducting a survey on their customer service. Customers willing to complete the survey are entered in a drawing for a free iPod.
Choose: biased unbiased
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Box-and-Whisker Plots In a set of data, quartiles are values that divide the data into four equal parts. Th e 5 Nu m b er Su m m ar y
Box and W h isk er Plot s and t h e 5 nu m ber su m m ar y
•
The five number summary is another name for the visual representation of the box and whisker plot.
•
The five number summary consist of : The median ( 2nd quartile) The 1st quartile The 3rd quartile The maximum value in a data set The minimum value in a data set
Mr. J.D. Miles Turner Middle School Atlanta Georgia milesmath@gmail.com www.JovanMiles.com
Co n st r uct i n g a b o x an d w h i sk er p lo t
Co n st r uc t i n g a b o x an d w h i sk er p lo t
• Step 1 - Find the median.
• Step 2 – Find the lower quartile. • The lower quartile is the median of the data set to the left of 68.
• Remember, the median is the middle value in a data set.
(18, 27, 34, 52, 54, 59, 61,) 68, 78, 82, 85, 87, 91, 93, 100
18, 27, 34, 52, 54, 59, 61, 68, 78, 82, 85, 87, 91, 93, 100
52 i s the lower qu artil e
68 is the medi an o f th is d ata set.
Co n st r uc t i n g a b o x an d w h i sk er p lo t
Thd e 5 m be Su m m ar y Con st r uct in g a b ox an wN hui sk err pl ot
• Organize Const r uct ing a box and w hisker plot the 5 number summary Median – 68 • Step 4 – Find the maximum and minimum Lower Quartile – 52 • The upper quartile is the median of the data set values in the set. quartile. • Step 3 – Find the upper Upper Quartile – 87
•
Step 3 – Find the upper quartile. to the right of 68.
18, 27, 34, 52, 54, 59, 61, 68, (78, 82, 85, 87, 91, 93, 100)
Max – 10 0
upper quartile is is thethe median of the data •• The The maximum greatest value Min –set 1 8 in the data to the right of 68. set.
• 87 i s th e up per qu ar til e
The minimum is the least value in the data set. 18, 27, 34, 52, 54, 59, 61, 68, (78, 82, 85, 87, 91, 93, 100)
18, 27, 34, 52, 54, 59, 61, 68, 78, 82, 85, 87, 91, 93, 100 87 is the upper quartile
18 is the minimum and 100 is the maximum.
Co n st r uc t i n g a b o x an d w h i sk er p lo t •
Step 5 – Find the inter-quartile range (IQR).
•
The inter-quartile (IQR) range is the difference between the upper and lower quartiles.
Up pe r Quartile = 8 7 Lowe r Quartile = 5 2 87 – 5 2 = 3 5 35 = IQR
32
Analyzing The Graph
Gr aphing The Dat a
• The data values found inside the box represent the middle half ( 50%) of the data.
• Notice, the Box includes the lower quartile, median, and upper quartile.
• The line segment inside the box represents the median
• The Whiskers extend from the Box to the max and min.
Practice •
Use the following set of data to create the 5 number summary. 3, 7, 11, 11, 15, 21, 23, 39, 41, 45, 50, 61, 87, 99, 220
The 5 Num ber Sum m ar y • Median - 39 • Lower Quartile - 11 • Upper Quartile - 61 • Max - 220 • Min - 3
33
Scatter Plots and Correlation Have you ever been curious to know if one event affects another event? For example, if I study longer, will I get a better grade on my Regents examination? Let's decide if studying longer will affect Regents grades based upon a specific set of data. Given the data below, a scatter plot has been prepared to represent the data. Remember when making a scatter plot, do NOT connect the dots. Study Hours 3 5 2 6 7 1 2 7 1 7
Regents Score 80 90 75 80 90 50 65 85 40 100 Notice: Certain values may have more than one result, such as (7,90) and (7,85) and (7,100).
The data displayed on the graph resembles a line rising from left to right. Since the slope of the line is positive, there is a positive correlation between the two sets of data. This means that according to this set of data, the longer I study, the better grade I will get on my Regents examination. If the slope of the line had been negative (falling from left to right), a negative correlation would exist since the slope of the line would have been negative. Under a negative correlation, the longer I study, the worse grade I would get on my Regents examination. YEEK!! If the plot on the graph is scattered in such a way that it does not approximate a line (it does not appear to rise or fall), there is no correlation between the sets of data. Check out these graphs for visual interpretations of types of correlations:
34
While the points "tend" to be rising, it is not a clearly positive relationship since points are not clustered as to show a clear straight line.
The points are clustered as to resemble a rising straight line with a positive slope.
The points are clustered as to resemble a falling straight line with a negative slope.
While the points "tend" to be falling, it is not a clearly negative relationship since points are not clustered as to show a clear straight line.
See how to use your TI83+/TI84+ graphing calculator with scatter plots. Click calculator.
There is no way of determining from these points, if the pattern is rising or falling. There is no evidence of a straight line.
Warning!!
Correlation does not necessarily mean Causation. Just because there is a strong correlation between data, does not necessarily mean that one set of data is causing the affect 35
that is occurring in the other set of data. During the months of February and March, the weekly number of jars of strawberry jam sold at a local market in New York was Weekly Data Collection recorded. For the same time frame, the number of copies The jars of strawberry jam The number of CDs sold of a popular classical music sold in New York in Florida CD sold in Florida was 5 jars 25 CDs recorded. The data was 7 30 examined and was plotted From looking at the graph, it can be seen that there is a high positive correlation between these two sets of data.
9
35
10
42
11
48
11
52
12
56
So, this must mean that the number of jars of strawberry jam sold in New York was causing an increase in the number of classical music CDs sold in Florida. Of course this is not true! Always be careful what you infer from your statistical analyses. Be sure the relationship makes sense. Also keep in mind that other factors may be involved in a causeeffect relationship.
36
Line of Best Fit When data is displayed with a scatter plot, it is often useful to attempt to represent that data with the equation of a straight line for purposes of predicting values that may not be displayed on the plot. Such a straight line is called the "line of best fit." It may also be called a "trend" line. A line of best fit is a straight line that best represents the data on a scatter plot. This line may pass through some of the points, none of the points, or all of the points.
Materials for examining line of best fit: graph paper and a strand of spaghetti
Is there a relationship between the fat grams and the total calories in fast food?
Sandwich
Total Fat Total (g) Calories
Hamburger
9
260
Cheeseburger
13
320
Quarter Pounder
21
420
Quarter Pounder with Cheese
30
530
Big Mac
31
560
Arch Sandwich Special
31
550
Arch Special with Bacon
34
590
Crispy Chicken
25
500
Fish Fillet
28
560
Grilled Chicken
20
440
Grilled Chicken Light
5
300
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Can we predict the number of total calories based upon the total fat grams?
Let's find out! 1. Prepare a scatter plot of the data. 2. Using a strand of spaghetti, position the spaghetti so that the plotted points are as close to the strand as possible.
Our assistant, Bibs, helps position the strand of spaghetti.
3. Find two points that you think will be on the "best fit" line. 4. We are choosing the points (9, 260) and (30, 530). You may choose different points. 5. Calculate the slope of the line through your two points. Choose two points that you think will form the line of best fit.
rounded to three decimal places.
6. Write the equation of the line.
Predicting: 7. This equation can now be used to predict information that was not plotted in the scatter plot. Question: Predict the total calories based upon 22 grams of fat.
ANS: 427.141 calories
If you are looking for values that fall within the plotted values, you are interpolating. If you are looking for values that fall outside the plotted values, you are extrapolating. Be careful when extrapolating. The further away from the plotted values you go, the less reliable is your prediction.
So who has the REAL "lineofbestfit"? In step 4 above, we chose two points to form our lineofbestfit. It is possible, however, that someone else will choose a different set of points, and their equation will be slightly different. 38
Your answer will be considered CORRECT, as long as your calculations are correct for the two points that you chose. So, if each answer may be slightly different, which answer is the REAL "lineofbestfit? To answer this question, we need the assistance of a graphing calculator. See the next lesson.
Line of Best Fit Using a Graphing Calculator
So which line is the REAL "lineofbestfit"?
Simply stated, the graphing calculator has the capability of determining which line will actually represent the REAL lineofbestfit. The directions on this page were prepared using the TI83+/TI84+ graphing calculator. Use of the graphing calculator will help you to better understand the concept of line of best fit.
Let's examine our fast food problem again with the use of the graphing calculator. (This example and chart are from MathBits.com and were used with permission.)
Is there a relationship between the fat grams and the total calories in fast food?
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Sandwich
Total Fat Total (g) Calories
Hamburger
9
260
Cheeseburger
13
320
Quarter Pounder
21
420
Quarter Pounder with Cheese
30
530
Big Mac
31
560
Arch Sandwich Special
31
550
Arch Special with Bacon
34
590
Crispy Chicken
25
500
Fish Fillet
28
560
Grilled Chicken
20
440
Grilled Chicken Light
5
300
Can we predict the number of total calories based upon the total fat grams? 1. Enter the data in the calculator lists. Place the data in L1 and L2. STAT, #1Edit, type values into the lists
2. Prepare a scatter plot of the data. Set up for the scatterplot. 2nd StatPlot choices shown at right. Choose ZOOM #9 ZoomStat. Graph shown below.
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3. Have the calculator determine the line of best fit. STAT → CALC #4 LinReg(ax+b) Include the parameters L1, L2, Y1. (Y1 comes from VARS → YVARS, #Function, Y1)
You now have the values of a and b needed to write the equation of the line of best fit. See values at the right. y = 11.73128088x + 193.8521475 4. Graph the line of best fit. Simply hit GRAPH. To get a predicted value within the window, hit TRACE, up arrow, and type the desired value.
Question: Predict the total calories based upon 22 grams of fat. ANS: 451.940 calories
Practice with Scatter Plots and More Try the following questions:
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1. The correlation seen in the graph at the right would be best described as:
Choose: high positive correlation low positive correlation high negative correlation low negative correlation
2.
You are asked to write an equation for the line of best fit for the scatter plot shown at the right without the use of a graphing calculator. What should you do first? Choose: Connect the points together. Find the slope using (1,2) and (9,12). Find the slope using (4,6) and (5,8). Decide which two points give the most representative straight line.
3. True or False: When data is graphed and a positive correlation is observed, the first set of data is always causing the affect seen in the second set of data.
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4. The scatter plot at the right shows the number
of chapters in a book in relation to the number of typos found in the book. If you predict the number of typos that would occur in a book containing 12 chapters, you would be:
Choose: extrapolating interpolating
5. When making a scatter plot, you should never: Choose: label the axes connect the dots plot more than one y value for any x value use a graphing calculator
6. If you were asked to interpolate information from this graph, you would have to be careful to limit the number of chapters to: Choose: between 2 and 8 chapters between 1 and 9 chapters chapters 2, 4, 6, and 8 only there is no need to limit chapters
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7. Another term for line of best fit is Choose: scatter plot trend line tangent line slope
8.
Prepare a scatter plot for the following data. Determine from your graph whether there is a correlation between the number of hours spent in the mall and the number of dollars spent. Hours in Mall
10
8
9
3
1
2
5
6
7
8
2
3
Dollars spent
40
15
24
20
10
35
50
70
18
25 100
60
9. The following chart shows the prices of gasoline and
milk at a local convenience store for a period of three weeks.
Exam this data for correlations between the price of gasoline and the price of milk. Comment on what a correlation between this data is saying.
10.
Week:
Gasoline
Milk
1st week
$2.89
$2.35
2nd week
$2.95
$2.40
3rd week
$3.04
$2.43
Which situation describes a situation that is not a causal relationship?
Choose: The rooster crows and the sun rises. The more miles driven the more gasoline needed. The more powerful the microwave the faster the food cooks. The faster the pace of the runner the quicker the runner finishes.
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11. For 10 days, Romero kept a record of the number of hours he spent listening to music. The information is shown in the table below. Day
1
2
3
4
5
6
7
8
9
10
Hours
9
3
2
6
8
6
10
4
5
2
Which scatter plot shows Romero’s data graphically? Choose:
(1) (2) (3) (4)
Percentiles and More Quartiles
Percentiles are like quartiles, except that percentiles divide the set of data into 100 equal parts while quartiles divide the set of data into 4 equal parts. Percentiles measure position from the bottom. Percentiles are most often used for determining the relative standing of an individual in a population or the rank position of the individual. Some of the most popular uses for percentiles are connected with test scores and graduation standings. Percentile ranks are an easy way to convey an individual's standing at graduation relative to other graduates. Unfortunately, there is no universally accepted definition of "percentile". Consider the following two slightly different definitions:
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Definition 1: A percentile is a measure that tells Definition 2: A percentile is a measure that tells us what percent of the total frequency scored at or us what percent of the total frequency scored below that measure. A percentile rank is the below that measure. A percentile rank is the percentage of scores that fall at or below a given percentage of scores that fall below a given score. score. Formula: To find the percentile rank of a score, x, out of a set of n scores, where x is included:
Formula: To find the percentile rank of a score, x, out of a set of n scores, where x is not included:
Where B = number of scores below x E = number of scores equal to x n = number of scores See this formula in more detail in the Examples section.
Example: If Jason graduated 25th out of a class of Example: If Jason graduated 25th out of a class of 150 students, then 125 students were ranked 150 students, then 125 students were ranked below below Jason. Jason's percentile rank would be: Jason. Jason's percentile rank would be:
Jason's standing in the class at the 84th percentile is as higher or higher than 84% of the graduates. Good job, Jason!
Jason's standing in the class at the 83rd percentile is higher than 83% of the graduates. Good job, Jason!
The slight difference in these two definitions can lead to significantly different answers when dealing with small amounts of data. Note: We will be using Definition 1 for the rest of this page. (other interpretations are also possible check with your teacher)
About Percentile Ranks:
• percentile rank is a number between 0 and 100 indicating the percent of cases falling at or below that score. • percentile ranks are usually written to the nearest whole percent: 74.5% = 75% = 75th percentile • scores are divided into 100 equally sized groups • scores are arranged in rank order from lowest to highest • there is no 0 percentile rank the lowest score is at the first percentile • there is no 100th percentile the highest score is at the 99th percentile. • you cannot perform the same mathematical operations on percentiles that you can on raw scores. You cannot, for example, compute the mean of percentile scores, as the results may be misleading.
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Consider: 1. Karl takes the big Earth Science test and his teacher tells him that he scored at the 92nd percentile. Is Karl pleased with his performance on the test? He should be. He scored as high or higher than 92% of the people taking the test.
2. Sue takes the Chapter 4 math test. If Sue's score is the same as "the mean" score for the math test, she scored at the 50th percentile and she did "as well or better than" 50% of the students taking the test. 3. If Ty scores at the 75th percentile on the Social Studies test, he did "as well or better than" 75% of the students taking the test.
Examples: Finding Percentiles 1. The math test scores were: 50, 65, 70, 72, 72, 78, 80, 82, 84, 84, 85, 86, 88, 88, 90, 94, 96, 98, 98, 99. Find the percentile rank for a score of 84 on this test. Be sure the scores are ordered from smallest to largest. Locate the 84. Solution Using Formula:
Solution Using Visualization: Since there are 2 values equal to 84, assign one to the group "above 84" and the other to the group "below 84". 50, 65, 70, 72, 72, 78, 80, 82, 84, | 84, 85, 86, 88, 88, 90, 94, 96, 98, 98, 99
The score of 84 is at the 45th percentile for this test.
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2. The math test scores were: 50, 65, 70, 72, 72, 78, 80, 82, 84, 84, 85, 86, 88, 88, 90, 94, 96, 98, 98, 99. Find the percentile rank for a score of 86 on this test. Be sure the scores are ordered from smallest to largest. Locate the 86. Solution Using Formula:
Solution Using Visualization: Since there is only one value equal to 86, it will be counted as "half" of a data value for the group "above 86" as well as the group "below 86". 50, 65, 70, 72, 72, 78, 80, 82, 84, 84, 85, 8|6, 88, 88, 90, 94, 96, 98, 98, 99
The score of 86 is at the 58th percentile for this test.
3. Quartiles can be thought of as percentile measure. Remember that quartiles break the data set
into 4 equal parts. If 100% is broken into four equal parts, we have subdivisions at 25%, 50%, and 75% creating the: First quartile (lower quartile) to be at the 25th percentile. Median (or second quartile) to be at the 50th percentile. Third quartile (upper quartile) to be a the 75th percentile.
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Test Scores
Frequency
Cumulative Frequency
For the table at the left, find the intervals in which the first, second and third quartiles lie.
7680
3
3
8185
7
10
8690
6
16
9195
4
20
If there are a total of 20 scores, the first quartile will be located (25% ∙ 20 = 5) five values up from the bottom. This puts the first quartile in the interval 8185.
In a similar fashion, the second quartile will be located (50% ∙ 20 = 10) ten values up from the bottom in the interval 8185. The third quartile will be located (75% ∙ 20 = 15) fifteen values up from the bottom in the interval 8690. Practice Questions 1. For 10 days, Romero kept a record of the number of hours he spent listening to music. The information is shown in the table below. Which scatter plot shows this data graphically?
[1]
[2]
[3]
[4]
2. Throughout history, many people have contributed to the development of mathematics. These mathematicians include Pythagoras, Euclid, Hypatia, Euler, Einstein, Agnesi, Fibonacci, and Pascal. What is the probability that a mathematician's name selected at random from those listed will start with either the letter E or the letter A? [1] 2/8
[2] 3/8
[3] 4/8
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[4] 6/8
3. Which expression represents [1]
in simplest form?
[2]
[3]
[4]
4. Which interval notation represents the set of all numbers from 2 through 7, inclusive? [1] (2,7]
[2] (2,7)
[3] [2,7)
[4] [2,7]
5. Which property is illustrated by the equation ax + ay = a(x + y)? [1] associative
[2] commutative
[3] distributive
[4] identity
[3] (x + 4)(x 4)
[4] (x + 8)(x 8)
6. The expression x² 16 is equivalent to [1] (x + 2)(x 8)
[2] (x 2)(x + 8)
7. Which situation describes a correlation that is not a causal relationship? [1] The rooster crows, and the sun rises.
[3] The more powerful the microwave, the faster the food cooks. [2] The more miles driven, the more gasoline needed. [4] The faster the pace of a runner, the quicker the runner finishes.
8. The equations 5x + 2y = 48 and 3x + 2y = 32 represent the money collected from school concert ticket sales during two class periods. If x represents the cost for each adult ticket and y represents the cost for each student ticket, what is the cost for each adult ticket? [1] $20
[2] $10
[3] $8
[4] $4
9. The data set 5, 6, 7, 8, 9, 9, 9, 10, 12, 14, 17, 18, 19, 19 represents the number of hours spent on the Internet in a week by students in a mathematics class. Which boxandwhisker plot represents the data?
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[1]
[3]
[2]
[4]
10. Given: Set A = {(2,1), (1,0), (1,8)} and Set B = {(3,4), (2,1), (1,2), (1,8)} What is the intersection of sets A and B? [1] {(1,8)} [2] {(2,1)}
[3] {(2,1), (1,8)} [4] {(3,3), (2,1), (1,2), (1,0), (1,8)}
11. Tanya runs diagonally across a rectangular field that has a length of 40 yards and a width of 30 yards, as shown in the diagram. What is the length of the diagonal, in yards, that Tanya runs? [1] 50
[2] 60 [3] 70 [4] 80
12. A cylindrical container has a diameter of 12 inches and a height of 15 inches, as illustrated in the diagram. What is the volume of this container to the nearest tenth of a cubic inch? [1] 6,785.8 51
[2] 4,241.2 [3] 2,160.0 [4] 1,696.5
13. What is an equation for the line that passes through the coordinates (2,0) and (0,3)? [1]
[3]
[2]
[4]
14. Which situation should be analyzed using bivariate data? [1] Ms. Saleem keeps a list of the amount of time her daughter spends on her social studies homework. [2] Mr. Benjamin tries to see if his students' shoe sizes are directly related to their heights. [3] Mr. DeStefan records his customers' best video game scores during the summer. [4] Mr. Chan keeps track of his daughter's algebra grades for the quarter.
15. An electronics store sells DVD players and cordless telephones. The store makes a $75 profit on the sale of each DVD player (d) and a $30 profit on the sale of each cordless telephone (c). The store wants to make a profit of at least $255.00 from its sale of DVD players and cordless phones. Which inequality describes this situation? [1] 75d + 30c < 255 [2] 75d + 30c < 255
[3] 75d + 30c > 255 [4] 75d + 30c > 255
16. What is the slope of the line containing the points (3,4) and (6,10)? [1] 1/2
[2] 2
[3] -2/3
[4] -3/2 52
17. Which type of graph is shown in the diagram?
[1] absolute value [2] exponential [3] linear [4] quadratic
18. The expression
is equivalent to
[1]
[3]
[2]
[4]
19. Daniel's Print Shop purchased a new printer for $35,000. Each year it depreciates (loses value) at a rate of 5%. What will its approximate value be at the end of the fourth year? [1] $33,250.00 [2] $30,008.13
[3] $28,507.72 [4] $27,082.33
20. Which inequality is represented by this graph? [1]
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[2]
[3]
[4]
21. In triangle MCT, the measure of angle T = 90 degrees, MC = 85 cm, CT = 84 cm, and TM = 13 cm. Which ratio represents the sine of angle C? [1] 13/85
[2] 84/85
[3] 13/84
[4] 84/13
22. The diagram in the box shows the graph of y = | x 3 |. Which diagram shows the graph of y = | x 3 | ?
54
[1]
[2]
[3]
[4]
23. The groundskeeper is replacing the turf on a football field. His measurements of the field are 130 yard by 60 yards. The actual measurements are 120 yards by 54 yards. Which expression represents the relative error in the measurement?
[1]
[3]
[2]
[4]
24. Which value of x is in the solution set of the inequality 2x + 5 > 17 ? [1] 8
[2] -6
[3] 4
25. What is the quotient of [1]
[2]
[4] 12
? [3]
[4] 55
26. The length of a rectangular window is 5 feet more than its width, w. The area of the window is 36 square feet. Which equation could be used to find the dimensions of the window? [1] w² + 5w + 36 = 0 [2] w² - 5w - 36 = 0
[3] w² - 5w + 36 = 0 [4] w² + 5w - 36 = 0
27. What is the sum of
expressed in simplest form?
[1]
[3]
[2]
[ 4]
28. For which value of x is [1] -2
undefined?
[2] 0
[3] 3
[4] 4
29. Which verbal expression represents 2(n 6) ? [1] two times n minus six [2] two times six minus n [3] two times the quantity n less than six [4] two times the quantity six less than n
30. Which graph represents a function?
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[1]
[3]
[2]
[4]
Answer Key
57
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