Praxis Core Math 5733

Box Plot

(also known as a box & whisker plot) plots a data set on the number line by segmenting the distribution into four quantities that are divided equally in half by the median.

Indirect Proportion

(also referred to as indirect variations, inverse proportions, or inverse variations) as one quantity increases, the other decreases.

Proportion

Population mean alternative to finding μ. Part of the population with the given characteristic. The proportion can be expressed as a decimal, fraction, or a percentage, and can be given as a single value or range of values.

Fractions come in 3 different varieties:

Proper, improper, and mixed numbers

Equivalent fractions

a term used to describe these fractions that are made up of different numbers but represent the same quantity

Scatter plot

a way to visually represent the relationship between two variables. Each variable has its own axis. The independent variable is plotted on the horizontal axis & the dependent on the vertical axis.

Perfect Square

a whole number that is the square root of another whole number. Ex: 16 and 64 are perfect squares because 16 is the square of 4 and 64 is the square of 8.

Composing & Decomposing Multi-Digit Numbers

aids in conceptualizing what each digit of a multi-digit number represents. The standard form in which numbers are written consists of series of digits representing a given value based on their place value. Ex: 592.7. This number is composed of 5 hundreds, 9 tens, 2 ones and 7 tenths.

fourth quartile

all of the data above the third quartile

Associate property

allows any terms to be "associated" by parenthesis and retain the same final answer. Order does matter for subtraction and division.

Commutative property

allows the order of the terms in an expression to change while keeping the same final answer. Both addition and multiplication can be completed in any order and still obtain the same result.

Joint Probabilities

also middle entries in the two-way frequency tables.

Marginal Probabilities

also the totals in the two-way frequency table

Area Model

another method of multiplication. A rectangle that is divided into rows and columns that match up to the number of place values within each number

Standardizing

any data set that resembles a normal distribution to a standard normal distribution using the z-score formula.

Integers

any number that does not have a fractional part. This inludes all positive, negative, whole numbers and zero. Fractions & decimals (which aren't) whole numbers, aren't integers.

Rational numbers

any numbers that can be written as a fraction, such as 1/3, 7/4 and -25.

Direct Proportion

as one quantity increases, the other also increases.

Margin of Error

because population mean and proportion are estimates, there a margin of error, which is the difference between the actual value and the expected value.

Unimodal

graph with a single peak

Bimodal

graph with multiple peaks

Geometric shapes can be used to

help visualize fractions

Mean/Median/Mode

how the center of a set of data (statistical value) is represented

a Terminating decimal number

if it stops at some point. Every rational number can be written as a terminating decimal.

Constant variables

remain unchanged by the scientist across all trials. They are held constant for all groups in an experiment, they aren't being measured in the experiment, and they are usually ignored. Can either be controlled directly (like nutrition, water, and sunlight given to plants) or they can be selected specifically (like certain animal species or people of a certain age group)

Histogram

represents a data set as a collection of rectangles, which illustrate the frequency distribution of the data.

Decomposing Numbers

requires taking a number written in standard form and breaking it apart int o the sum of each place value. Ex: the number 83.17 is decomposed by breaking it in to the sum of 4 values (for each of the 4 digits): 8 tens, 3 ones, 1 tenth, and 7 hundredths. The decomposed or "expanded" form of 83.17 is 80+3+0.1+0.07.

Mean

same as Average. To find the mean, add up all the data then divide by the total number of data points. Ex: A class of 10 students, the test scores were, 50, 60, 65, 65, 75m 80, 85, 85, 90, & 100. Add all of the scores together than divide that total by 10. The mean would be 75.5.

Observational Study

similar to an experiment, but without a designed control and test population because of circumstances (ex: lack of funding, unrealistic expectations). Instead, existing control, and test populations must be used, so this method has a lack of randomization.

Percentiles

similar to quartiles. If a data point lies in the n-th percentile, this means it lies in the range of the first n% of the data. If a data point lies AT the n-th percentile, it mean that n% of the data lies below this data point.

Data Gathering Techniques

the 3 most common types of data gathering techniques are sample surveys, experiments, and observational studies.

Area under the curve

the area under the curve of a standard normal distribution is equal to 1. Areas under the curve can be estimated using the z-score and a table. The area is equal to the probability that a data point lies in that region in decimal form.

Mode

the data point that appears most frequently. If two or more data points all tie for the most frequent appearance, then each of them is considered the mode. Ex: the test scores, 50, 60, 65, 65, 75, 80, 85, 85, 90, & 100. There are two modes: 65 & 85.

Randomized experiment

used to compare two treatments by using statistics involving a T-test, which tests whether two data sets are significantly different from one another. To use a T-test, test statistics must follow a normal distribution.

second quartile

when a data point lies between the first quartile AND the median

first quartile

when a data set lies in the 1st quartile, it means it is less than or equal to the median of the first half of the data points. If it lies in the AT the first quartile then it is equal to the first quartile.

Complex Numbers

Composed of both real and imaginary numbers. Made up of the sum of real & imaginary numbers.

Categorical Data

Data that isn't described using numbers. Ex: age is numerical data but hair color is categorical. Two way frequency tables are used to summarize categorical data.

joint frequency

Entries in the middle of a two-way frequency table

__________ numbers are all divisible by the number 2.

Even

Imaginary Numbers

Imaginary numbers are the result of taking the square root of -1, and sr of -1=i.

base 10 system

Numbers used in everyday life are constituted in a base 10 system. Each digit in a number, depending of location, represents some multiple of 10, quotient of 10 when dealing with decimals. Each digit to the left of the decimal point represents a higher multiple of 10. Each digit to the right of the decimal point represents a quotient of a higher multiple of 10 for the divisor.

PEMDAS - Order of Operations

Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Addition/Subtraction & Multiplication/Division should be completed in order from left to right.

Conversion Factor

Ratios & rates can be used together to convert rates into different units. Ex: if someone is driving 50 kilometers per hour, that rate can be converted into mph by using a ratio known as the Conversion Factor.

Adding/Subtracting Complex Numbers

Similar to collecting like terms. Real numbers are added together & imaginary numbers are added together. Ex: 6=2i-3=7i, 6 and (-3) are combined to make 3, and the 2i and 7i are combined to make 9i.

Radical symbol

The check mark with a flat top looking symbol.

Expanded form

a number written in expanded form expresses it as a sum of the value of each of its digits. Ex: 7,631.42 written in expanded form is 7,000+600+30+1+0.4+0.02

Every fraction can be built from

a combination of unit fractions

Conjugate

a complex number is a technique used to change the complex number into a real number. Ex: the conjugate of 4-3i is 4+3i. Multiplying (4-3i)(4+3i)=16+12i-12i+9, which has a final answer of 16+9=25.

root of a number

a different way to write an exponent when the exponent is the reciprocal of a whole number.

Common factors

a factor shared by two numbers.

Unit fraction

a fraction in which the numerator is 1, and the denominator is a positive whole number

In a division problem:

a is the dividend, b is the divisor, and c is the quotient. a/b=c

Experiment

a method in which a hypothesis is tested using trial & error process. A cause and effect of that cause are measured, and the hypothesis is accepted or rejected. Completed in a controlled environment where results of a control population are compared to the results of a test population. Randomization is involved.

T-value

a ratio of the difference between the two sample means and the standard error

Ratios

can also be compared if the units in each ratio are the same. Ex: The ratio of boys to girls in math class can be compared to the ratio of boys to girls in a science class by stating which ratio is higher and which is lower.

Composite numbers

can be divided by other numbers. All numbers greater than 1 that aren't prime numbers are composite numbers.

Converting improper fractions to mixed numbers

can be helpful in comparing fractions because the whole number portion of the number is more visible

Proportional reasoning

can be used to solve problems involving ratios, percentages, and averages. Ratios can be used in setting up proportions and solving them to find the unknown.

Prime numbers

cannot be divided except by 1 and itself. Has no other factors, which means that no other combination of whole numbers can be multiplied to reach that number. Ex: the set of prime numbers between 1 & 7 is (2,3,5,7,11,13,17,19,23)

Two-Way Frequency Table

counts the relationship between two sets of categorical data. There are rows & columns for each category, and each cell represents frequency information that shows the actual data count between each combination.

Describing a Set of Data

data can be described in terms of its center, spread, shape, and any unusual features. Can be measured by mean, median & mode. There can be small or large spreads. When displayed as a graph, shapes can be indicated normal, symmetrical, or skewed. There is also right-skewed and left-skewed. There can be gaps or outliers.

normal distribution

data follows the shape of a bell curve and the data set's median, mean, and mode are equal. Therfore, 50% of its values are less than the mean and 50% are greater than the mean. Data sets that follow this shape can be generalized by using normal distributions.

Outliers

data points that are far away from the rest of the data.

third quartile

data that lies between 1/2 and 3/4 of the way through the data set. Between the median and the median of the top half of the data.

Decimal place

denotes how far to the right of the decimal point a numeral is. First digit to the right is 'tenths', next is 'hundredths', third is 'thousandths'.

The bar in a fraction represents:

division

Exponents

express a number or variable multiplied by itself a certain number of times. Ex: x^3 means x is multiplied by itself 3 times. X is called the base and 3 is the exponent.

T-test Test statistic

falls between the p values of 0.05 and 0.025

degrees of freedom

for two samples are calculated as the following (rounded to the lowest whole number).

Unlike fractions

fractions that are different and do not represent equal quantities

Covariance

general term referring to how two variables move in relation to each other. Ex: an employee gets paid by the hour. For them, hours worked and total pay have a positive covariance. As hours worked increase, so does pay.

Median

in a data set, the median is in the point in the middle. The middle refers to the point where half the data come before and half comes after, when recorded in numerical order. Ex: Speeds of a fastball of a pitcher during the recorded last inning (least to greatest), 90, 92, 93, 93, 95, 96, 97, 97, 97. There are 9 numbers so the median is in the middle so it would 95 in this case. Ex2: In cases where the data points is an even number, the average of the two middle points is taken. Add the two medians, divid by 2 = your answer.

Sample surveys

involve collecting data from a random sample of people from a desired population. To have accurate data, the sampling should be unbiased and random.

Prime factorization

involves another step after breaking down a number's factors: breaking down the factors until they are all prime numbers. Ex: 129. First the factors are 3 & 43. Both are prime numbers. But if 43 was not a prime number then it would require being factorized until all of the factors are expressed as prime numbers.

Place Value System

make infinite number of values possible. The position in which a digit is written corresponds to a given value. Starting from the decimal point (which is implied, if not physically present), each subsequent place value to the left represents a value greater than the one before it. Each subsequent value to the right represents a value less than the one before it.

frequency distribution

normal distributions in which the data set is plotted as percentages rather than the true data points.

Factors

numbers multiplied to achieve a product. Every product in a multiplication equation has at minimum, 2 factors. Some products will have more but for the most part, factors are positive integers.

relative frequency distribution

one where the y-axis is between zero and 1, which is the same as 0% to 100%.

Least common denominator

or the product of two denominators can be used as the common denominator

PEMDAS stands for?

parenthesis, exponents, multiplication/division, addition/subtraction. Multiplication & division and Addition & subtraction are performed in the same step (left to right), whichever comes first.

variables

the key to analyzing data, especially when data is in a graph or table. Can represent anything, including objects, conditions, events, and amounts of time.

Greatest common factor

the largest number among the shred, common factors of 2 numbers.

Z-score

the number of standard deviations that a data point falls from the mean.

Addends

the numbers being added

dimensional analysis

the process of converting between different units using equivalent measurement statements. Ex: running 5 kilometers is approximately the same as running 3.1 miles. This conversion can be found by knowing that 1 kilometer is equal to approximately 0.62 miles.

Scale Factor

the ratio between two similar geometric figures

Unit Rates

the simplest form of rate. The denominator in the comparison of two units is one. Ex: If someone can type at a rate of 1000 in 5 minutes, then his/her unit rate is 1000/5=200 words in per minute. ANY rate can be converted into a unit rate by dividing to make the denominator one.

Least common factors

the smallest number that's a multiple of two numbers.

standard deviation

the square root of the variance

Marginal frequency

the totals in a two-way frequency table

One of the most fundamental concepts of fractions is:

their ability to be manipulated by multiplication or division

Irrational numbers

those that cannot be written as a fraction, such as numbers with never-ending, non repeating decimal values. Many are from taking roots of numbers.

one-tailed test

used if an assumed difference between two treatments. When the p value lies between 0.05 and 0.025

two-tailed test

used if no difference is assumed. When the p values need to be doubled so the corresponding p value falls between 0.1 and 0.05.

Condition relative frequencies

used to avoid errors in two-way frequency tables

Two-way relative frequency table

when the frequencies are changed into percentages based on totals. Percentages can be calculated using the table total, the row totals, or the column totals.

significance level (alpha)

where a typical value is alpha=0.05. Once all info is compiled, the decision is made to use a one-tailed test or two-tailed test.

z score formula

z = (x - μ)/σ : z=(x-mean)/standard deviation, x is the data point.

Population Mean

μ: the average value or the parameter for the entire population. Due to size constraints, finding the exact value of μ is impossible, so the mean of the sample population is used as an estimate instead. The larger the sample size, the closer the sample mean gets to the population mean.