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Math

Math Review of Binomial Distributions

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Overview

A binomial experiment  has a fixed number of independent trials, and each trial has only two possible outcomes.  Each of the trials has the same probability of success.  The probability distribution is called a binomial distribution.

What Types of Problems Are Binomial?

All the characteristics of a binomial experiment are present, then the distribution will be binomial.  A quiz has 8 questions, and each question has 4 alternatives.  If a student guesses randomly on every question, what is the probability of getting 5 or more correct?  It is a binomial experiment because it has a fixed number of trials (8), and the student is guessing randomly, so each question is independent of every other.  Each question can either be answered correctly or incorrectly.  In addition, each question has the same probability of success.

What Types of Problems Are Not Binomial?

In a problem that is not a binomial problem, all the characteristics are not present.  For example, suppose a standard deck of cards is used, and the number of aces in 5 trials are recorded, but the cards are not replaced after each trial.  It is not a binomial distribution, because every trial depends on one another.  During the first trial, the population of cards is 52, the second, 51, the third, 50, and the fourth 49, so the trials are not independent.  Similarly, if there are three possible outcomes, rather than two, the problem will not be binomial.

What Does a Binomial Distribution Look Like?

A binomial distribution is symmetrical, with the smallest values on either side of the mean, when it is graphed with the probability held constant, the number of trials on the x axis and the function on the y axis.  The more trials are done, the more the distribution spreads out, and the more the distribution flattens.

How Is the Binomial Distribution Used?

Many types of real-world situations have a fixed number of  independent trials with only two outcomes.  For example, treatments often have outcomes of success or failure, basketball players either hit or miss free throws, and potential voters either cast their votes or not in a particular election.  Binomial distributions can be used to determine the probabilities of those events and others like them.

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Properties of Probability

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Overview:

Probability is the measure of the likelihood of an event.  The basic mathematics of probability theory started with games of chance, but it can be applied to many situations, from weather forecasting to politics.  Probabilities range from 0 (no likelihood) to 1 (certainty), and are expressed as rational numbers.

What Is the Sample Space?

The sample space is the set of all possible outcomes of an event.  For a coin toss, the coin will either result in heads or tails.  For the roll of one die, the sample space is all the values on the faces of the die, or a set of {1, 2, 3, 4, 5, 6}.  For the roll of a pair of dice, the sums will be in a set from 2 (both dice give you a 1), the smallest sum possible, to 12 (both dice give you 6), the largest sum possible.

What Is A Fair Experiment?

In a fair experiment, all possible outcomes are equally likely.  The probability of any outcome is related to the total number of outcomes by a ratio of the number of outcomes in that event to the number of all possible outcomes of the event (the sample space).  Therefore, the probability that a coin will be heads is 1/2.  The probability that if one die is rolled, the number on top will be a 3 is 1/6.

What If Events Are Not Equally Likely?

Sometimes, possible outcomes can be combined in such a way so that not all outcomes are equally likely.  Suppose two fair coins are tossed: there are 4 possibilities in the sample space {HH, HT, TH, TT}.  The probability of each event when order is important equals 1/4 for each possibility. However, if the question is merely “How many heads come up when two coins are tossed?”, there are only 3 possibilities in the sample space, 0 heads, 1 head, or 2 heads.  The event 0 heads is defined as {TT}, and the event 2 heads is defined as {HH}.  However, there are two possibilities for 1 head, either {HT} or {TH}, so the outcomes are not equally likely.

What Are Mutually Exclusive Events?

If events are mutually exclusive, it means that neither sample space A or sample space B contain common elements.  Therefore, the probabilities can be added to form the probability of one event or the other occurring.  The sample space for tossing two fair dice and getting a sum of 7 consists of {(6,1), (1, 6), (2, 5), (5,2), (3,4), and (4,3)}.  The sample space for tossing two fair dice and getting a sum of 11 consists of {(6,5), (5,6)}.  The sample space for getting either a sum of 7 or a sum of 11 is the union of both sets.

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Math Review of Random Numbers

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Overview:  What Are Random Numbers?

In a set of random numbers, the numbers do not follow any pattern.  Each number has an equal probability of occurring, and each number event is independent of any others .  Most of the time, numbers that are close to random are generated by computer programs or calculator programs designed to do just that.

What Are Some Examples of Random Events?

Many events are close to random. For example, individual molecules within a gas tend to move randomly, so that it cannot be predicted where an individual molecule will be.  Similarly,  the theory of radioactivity predicts that a percentage of atoms in a substance will decay into isotopes given an amount of time, but it does not predict precisely which particular atom will decay.

How Are Random Numbers Found?

Numbers that are close to random can be found by consulting random number tables, as the result of computer programs to generate random numbers, and by using a calculator to generate random numbers.  In addition, games that depend on giving all the players a fair chance are often determined by using dice or a spinner divided into equal parts.  That way each number has an equal probability of being chosen.

How Is Randomness Used In Statistics?

In scientific experiments, statistics are used as a tool to judge the results of an experimental treatment.  Subjects have an equal chance to be assigned to a treatment condition through random assignment.  Often a random number generator is used in order to assign subjects to treatment groups.  Not only does each subject have an equal chance of being assigned to any of the treatment groups, but the choice of any one subject is independent of all the others.  This minimizes errors that could occur if the assignment is not random, but confounded.

What Are Monte Carlo Methods?

Monte Carlo methods are mathematical simulations that use random numbers to generate solutions.  They are called Monte Carlo methods because the earliest studies used equipment to generate random numbers similar to the methods in gambling casinos, such as decks of cards and roulette wheels.  Some simulations include the common practice of airlines overbooking seats on flights, because there are probabilities generated for how many passengers will arrive to take seats.  The computer simulation can be run for the number of seats on the airplane as the number of trials, using random numbers to suggest a possible outcome.

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Measures of Central Tendency: Mean, Median, and Mode

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Overview:  What Are Measures of Central Tendency?

Measures of central tendency represent the most typical score or value in a group of scores on some measure in either a population or a sample of that population.  Along with measures of variability, they convey much information about a distribution.  The three most common measurements are the mean, median, and mode.

What is the Mean?

The mean is the average value of all the numbers in the data set, and is a representation of the population mean.  It is computed by summing all values in the data set, and dividing by the number of values.  Many statistical calculations use the mean value to determine if treatment conditions have had an effect by comparing the mean of a pretreatment sample with a post treatment sample.

What Is the Median?

The median is the value in the middle if all the numbers in a data set are arranged from largest to smallest.  It is often reported rather than the mean value , if the values between the largest value and the smallest value are widely separated, such as with salaries or the price of housing in a community,  This is because the median is less affected by extreme values.  When calculating the median, first arrange the values from largest to smallest, listing all instances of a value as many times as it occurs.   Suppose the president of a company makes $200,000 a year, the vice president makes $50,000 a year, 2 supervisors make $25,000 a year, 4 sales representatives make $21,000 a year,  2 custodians and 2 warehouse workers make $15,000 a year, and 3 clerical workers make $12,000 a year.  The median salary is $21,000.

What is the Mode?

The mode is the measure that occurs most frequently.  In the above example, there are 2 modes, the salary of the 4 sales representatives of $21,000, and the salary of the 2 warehouse workers and 2 custodians, at $15,000.  It is not as useful as a measure of central tendency because it can be affected by scores that are not the center of the data set, but are in other places in the data set.  If a third warehouse worker were hired at $15,000, the new mode would be below the mean or the median.

Which Measure Is the Most Useful?

If a distribution has only one mode and is symmetric, such as the normal curve that reflects scores on the SAT and ACT, the mean will be the most useful.  If there is a large difference between the mean and the median, as when a distribution has many scores that are higher than the mean or lower than the mean, the median may be the most useful.  It is less affected by outlying scores, and may give a better picture than the mean.  However, the mean is most often used in many other statistical measures and is the most often reported.

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Review of Solid Geometry on the SAT and ACT

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Overview:  What Is Solid Geometry?

Solid geometry was developed after plane geometry as a way to describe the three-dimensional world and the objects in it.  In the ideal three-dimensional world, objects exist with faces and angles, depth and volume.  Objects are regular, or consist of a combination of regular objects, just as two-dimensional figures can be made from a combination of other two dimensional figures.  Solid geometry studies the properties, dimensions, and relationships of regular solids, as well as pyramids, cones, cylinders, prisms, spheres, and other solid objects.

What Are Regular Solids?

Regular solids have congruent faces, and the same number of faces meet at each vertex.  A solid with four regular triangular faces is called a tetrahedron, a cube has six square faces, an octahedron has eight triangular faces, a dodecahedron has 12 faces that are shaped like regular pentagons, and an icosahedron has 20 triangular faces.  They are symmetrical, and the only polygons that can fulfill all the requirements for sides, edges, and angles are triangles, squares, and pentagons.

 

Pyramids, Cones, Cylinders, Prisms

Some other solid figures do not fit the image of strict regularity.  For example, polygons other than triangles form the base of pyramids, yet the other faces are triangular.  Cones usually have a circular or elliptical base and an infinite number of faces tapering to a single point.  Cylinders are formed by the solid enclosed when two circles in parallel planes are joined. Prisms are a special class of solids with a polygonal base, a translation of that base in a parallel plane (similar to a cylinder) and sides that join them.

Spheres and Spheroids

Spheres are perfectly round in three dimensions.   They have many distinct properties, such as every point on the surface of the sphere is the same distance from the center, and that the width and the girth do not vary in a perfect sphere.  Objects that are close to spheres are spheroids and approximate many of the properties of spheres.  They may be elliptical rather than perfectly round, or polyhedrons that have a large number of sides.  Spherical geometry is the branch of trigonometry dealing with spheres and spheroids.

Applications of Solid Geometry

Many geometric solids have applications in nature and technology. Common crystal shapes include the tetrahedron, cube, and octahedron.  In addition, most viruses have the shape of a regular polyhedron because a basic unit protein can be repeated and packed into the smallest space possible.  The most common sphere is a ball or globe, and the most common spheroid is the Earth itself.

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Review of Math Functions on the SAT and ACT

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Overview:  What Are Functions?

Functions in mathematics describe two different types of relationships between numbers.  The simplest definitions are that a function can be either the set of ordered pairs in which the first element is paired with a second element or the relationship between two sets of elements in which every element in Set A can be paired with an unique element in Set B.  The definitions are different from each other to reflect the many ways that functions can be represented — as pairs in tables or lists, in rules made up of words, or in graphs.  The first elements in the ordered pair or in Set A are the domain or independent variable, represented on the x-axis; and the second elements in the ordered pair or in Set B are the range or dependent variable, represented on the y-axis.

Linear Functions

Linear functions follow the general form y= mx +b, where m is a constant representing the slope and b is a constant representing the y-intercept (the point at which the line crosses the y-axis).  In the real world, most data sets do not follow an exact line, but a line of best fit can be approximated using regression techniques.  One common type of linear function used frequently in the social sciences, the correlation coefficient, measures how strongly one variable is related to another.

Quadratic Functions

Quadratic functions are one type of nonlinear model in which the general form of the equation of best fit follows the quadratic formula ax2 + bx +c = 0, where a is not equal to 0, and a, b, and c are all constants.  The resulting graph is a parabola.  There are many real-world functions that are quadratic, including the Newtonian measurement of the relationship between gravity, velocity, and time.  In the equation the variable h represents height the object is initially thrown from, g is the acceleration constant due to gravity, v is velocity, and t is time, so that h=-1/2gt2-v0t +h0.

Exponential and Logarithmic Functions

An exponential function occurs when the form of the function f(x) = abx, when a is not equal to 0, b is greater than 0 and not equal to 1.  (If b is equal to 1, then the function is linear.)  Many different types of exponential functions, such as population growth, occur in the natural world.  Logarithmic functions are solved for y rather than x, so they are the inverse.  They also form a family of curves.

Trigonometric Functions

Trigonometric functions include circular functions such as sines, cosines, and tangents, which are derived from the relationships between angles and sides within triangles.  As with the graphs of quadratic, exponential, and logarithmic functions, the graphs of trigonometric functions show consistent patterns.  The wave patterns illustrate periodic events such as sound and light waves.

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Math Review of Combinations and Permutations

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Overview
Mathematical combinations occur in problems that involve the number of ways that sets can be combined, where order does not matter.  Permutations are similar arrangements, but order does matter.  They are related through principles of multiplication and arrangements involving factorials.

How Many Choices Are There?
If the problem is merely a combination of a choice from one set and a second choice from another set, there are only the number of choices in the first set multiplied by the number of choices in the second set.  For example, if there are two different colors of chairs (black or purple), and three different desk colors (light, medium, and dark), the possibilities are 2 X 3 or 6.  This can be generalized to more than two sets as N(A1) ∙ N(A2) ∙ … N(Ak), where k is a constant that represents the number of sets.

What Are Factorials?
Suppose the problem involves choices without replacement.  That is, supposing there are five possibilities for the first choice (A, B, C, D, E).  No matter which one is chosen for the first choice, only 4 remain for the second choice.  The first and second have already been chosen, so there are only 3 possibilities for the third choice.  There are only two for the fourth choice, and only 1 for the fifth choice.  The number can be written as 5 ∙ 4 ∙  3 ∙ 2 ∙ 1 =120 possible choices.  Another way to represent that is 5!, where the number 5 and the exclamation point afterwards are read as five factorial.

What Are Permutations?
Suppose the problem involves a situation where order matters, such as the probability in a number n! that particular alternatives will be either first, second , or third.  For example, suppose there are 8 teams in a conference.  The total number of rankings of all 8 teams are choices without replacement, or 8!, or 40320 possibilities.  If only the first, second, and third place teams are chosen for the next round, the number of possible permutations can be represented by 8!/3! or 8∙7∙6∙5∙4∙3∙2∙1 /3∙2∙1 or 6720.

What Are Combinations?
Suppose the problem involves a situation where order does not matter.  Suppose there are 8 teams in a conference, as in the previous example.   How many different combinations of teams could be selected, in order that teams would be either first, second, or third?  This could be represented by 8C3, where C represents the number of combinations, or 8!/(8-3)!3!, or 40320/5!3! , 56 different combinations.

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How to Read Math Questions on the SAT and ACT

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Overview: A Different Style of Reading
Reading math questions on the SAT and ACT takes special attention to exact detail, following directions precisely, and watching for key words.  It is a different type of reading than students usually do in other sections of the test, such as reading comprehension or vocabulary sections.  The “plot” of a math question has a beginning, middle, and the solution provides the end.

Attention to Detail
Other sections of the SAT or ACT may ask for the main ideas, interpretations, or points of view in a selection.  Usually, comprehension of the entire selection doesn’t hinge on a single word or detail.  However, successful solutions to math problems depend upon attention to detail.  Questions like “1/3 is 1/4 of what number?” and “What is 1/3 of 1/4?” do not have the same solution.

Following Directions
Filling in the answer sheet correctly, as well as working the correct section in the time limit given are more obvious examples of following directions, but there are also some special symbols on some math problems. The goals of those problems are to test skill in substitution and logical reasoning.  Those questions require suspension of disbelief, as well as following directions.

Watching for Key Words
Words such as exactly, at most, at least, fewer, and between refer to quantities.  If an answer to a problem is “nonnegative”  it can be zero,  but is the answer is positive it cannot. Make sure all quantities in the question stem are accounted for when solving the problem.

Watch for Unnecessary Information
Math problems on tests, as well as math problems in homework, sometimes contain information that is not necessary to solve the problem.  Sometimes this information is within the problem itself, and sometimes it is in the alternative answers.   For example, suppose the statement given on a math question is “Union Station has exactly 5 gates”.  One of the alternatives is “South Station has 7 gates.”  Since nothing else in the question was about South Station, it is unnecessary information.

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Using Math Proportions on the SAT and ACT

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Overview:  Why Proportions?
Solving proportions is an important math skill, both on tests and in everyday life.  In addition, understanding proportions is critical in the sciences, as elements exist in varying quantities in compounds and mixtures.  Understanding proportions within populations and how they work can also be essential as part of statistical analysis in social sciences.

Proportions as Ratios
Proportions are used most simply in problems like ” If c cans of soup can make x cups, how many cans will make y cups?”  This can be represented simply as c/x = n/y.  It can be solved by cross-multiplication, or xn =cy, so n =cy/x.  If the x, y, c, n don’t make much sense, try substitution on the practice problem.  If 2 cans of soup can make 6 cups, how many cans will make 9 cups?  Then, 2/6 = n/9, or 6n = 2(9) or 6n/6 = 18/6, or 3.  Granted, the problems on the SAT or ACT won’t be that easy, but those that use proportions are extensions of problems like that.

Proportions as Fractions
Most proportion problems are fractions in plain sight.  (This includes percentage problems, which are fractions of 100 and can be converted easily to decimals).  The numerator can be one expression and the denominator can be another expression.  As long as the numerators and denominators match on both sides, they can be cross multiplied and solved like the cans and cups in the ratio problem.

Proportions in Science Reading Passages
Math proportions can also be tested during reading comprehension and on science questions.  For example, a reading passage may be presented about the process of photosynthesis.  According to the chemical equations for photosynthesis, free oxygen molecules are produced from water and chlorophyll after the action of sunlight.  Understanding that the amount of oxygen that is produced is in proportion to the elements before photosynthesis will help with comprehension questions.

Proportions in Statistics
Statistical analysis is used as a tool in most social science research.  The essential question in any experiment or collection of data is “Does the data deviate significantly from the expected proportions on some measure?”  For example, suppose teachers in high school Algebra II classes in a particular school want to know how their classes this year perform as compared with the previous year.  They can record data on students taking a test last year, and compare the proportion of student scores above a certain cutoff on the same test given this year.

Do you need to know more about how test questions on the SAT or ACT are scored? Learn more about how we are assisting thousands of students each academic year.

Test Prep Academy is the premier test prep and private tutoring company for college-bound students. Our highly qualified test prep tutors deliver one-on-one personalized instruction that fit our student’s busy schedule.

Do You Have Enough Information to Solve the Math Problem?

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Overview:  Common Strategies

Some math problems on the SAT and ACT  present an equation or series, and then give a number of alternative solutions.  Like all problems of that type, only one answer is correct.  However, besides numerical solutions, there’s an additional choice: usually labeled (E)  Not enough information is available to solve this problem.

Enough Info for Geometry Problems

Often, there is a figure included, not necessarily drawn to scale.  Adjacent, complementary, or corresponding angles may be labeled.  Also, check to make sure similar triangles are not buried in other figures.  Practice solving geometry problems long before the test is given, in order to detect the information hidden in such figures.  Test questions will ask for applications rather than complicated proofs.

CSI for Arithmetic

Investigate the alternative answers that contain numbers.  That way, a cumbersome algebra problem can be turned into arithmetic.  If the alternative is right there, there is enough information, especially if the problem is a difficult one.  More practice on such problems long before the test will help the investigator spot the hidden clues to the answer.

Algebra Alternatives

Alternative answers to algebra problems on the SAT and ACT test thinking and reasoning skills as much as student knowledge of algorithms.  Students solve algebra problems during math homework by repeating strategies and solving the same types of problems.  However, the math questions on standardized tests focus on reasoning the correct answer, no matter what way it is solved.

Is That Your Final Answer?

In reality, there are very few problems on the SAT or ACT with not enough information available.  When the math problem is difficult, it is tempting but often incorrect to choose that choice.  Most of the time, there is more information to go on than the student actually needs.  The question stem sometimes will contain irrelevant information, especially if there is a figure included.  Read the problem carefully for all instructions, key words, and hidden clues.

Do you need to know more about how test questions on the SAT or ACT are scored? Learn more about how we are assisting thousands of students each academic year.

Test Prep Academy is the premier test prep and private tutoring company for college-bound students. Our highly qualified test prep tutors deliver one-on-one personalized instruction that fit our student’s busy schedule.

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