Applied Probability and Statistics C955

Changing Improper Fractions and Mixed Numbers

Improper fractions can be converted to mixed numbers by following these steps:

Write division problem with numerator divided by denominator.

Divide to determine quotient and remainder.

Write mixed number with the quotient as the whole number and the remainder as the numerator over the same denominator.

Changing Mixed Numbers Into Improper Fractions

Mixed numbers can also be converted to improper fractions by following these steps:

Multiply the whole number by the denominator of the fraction.

To the product given by step 1, add the number of the numerator.

Write the result of step 2 as the numerator of the improper fraction. The denominator of the improper fraction should be the denominator of the original fraction.

Simplify the improper fraction by diving the numerator and denominator by all common factors.

Discrete Data

Has distinct values, can be counted, has unconnected points (dots)

Continuous data

Has values within a range, measured (not counted) does not have gaps between data points. (connected lines or curves)

Sign rule for Multiplication and division

1. +# x +# = +#

2. -# x -# = +#

A product or division of two numbers of the same sign will result in a positive number

3. -# x +# = -#

4. +# x -# = -#

A product or division of two numbers of different signs will result in a negative number

Prime Number

A prime number is a number that has exactly two positive factors; 1 and itself.

Composite Number

A number that is not prime. It has 2 or more positive factor, including 1 and itself.

Prime Factorization

Writing the number as a product of only prime numbers.

Greatest Common Factor (GCF)

The larges number that divides all the given numbers evenly.

Multiples of a number

Numbers that can be obtained by multiplying the given number by 1, 2, 3, 4, etc.

Least Common Multiple (LCM)

the smallest positive number that can be divided by the given numbers

Centigrade/Fahrenheit Conversions

C = (F - 32) X 5/9

F = (C X 9/5) + 32

Unit Conversions for Household Measures of Volume

1 tablespoon = 3 teaspoons

1 oz = 2 tablespoons

1 cup = 8 oz

1 pint = 2 cups

1 quart = 1 pints

1 gallon = 4 quarts

Common Metric Conversions

1 L = 1000 mL

1kg = 1000 g

1 g = 1000 mg

1 mg = 1000 mcg

Conversions between Household and Metric Units

1 cc (Cubic Centimeter) = 1 mL

1 oz = 30 mL

1 L = 1.057 qt

1 tsp = 5 mL

1 kg = 2.2 lbs

1 oz = 28.35 g

Like Terms

Terms that have the same variable raised to the same power; they can be combined using addition and subtraction

Addition/Subtraction Principle

We can add or subtract the same number to both sides of an equation and the resulting expression remains equal.

Multiplication/Division Principle

We can multiply or divide the same number to both sides of an equation and the resulting expression remains equal. (Divide by 0 is not allowed)

Butterfly Method

Cross Multiply, if a/b = c/d then a x d = b x c

Slope-Intercept Equation

y = mx + b

Where "m" is the slope, "b" is the y-intercept, and "x" and "y" follow the coordinate formula (x,y)

Slope of a line

The slope of a line is the ratio of the vertical change between two points on the line to the horizontal change between those two points.

rise: (y2 - y1) / run: (x2 - x1)

Reducing Fractions Using Prime Factorization

The steps to reduce a fraction through prime factorization are as follows:

List the prime factors of both the numerator and denominator.

Cancel the factors that are common to both the numerator and denominator.

Multiply across the numerator and denominator.

6/8

6/8 = 2x3/2x2x2x

2x3/2x2x2x = 3/2x2

3/2x2 = 3/4

Reducing Fractions Using Common Factors

The steps for the common factors method are as follows:

Divide numerator and denominator by a common factor.

Continue to divide by common factors.

Write the reduced factor.

-28/42

-28/42 = (-28/7) / (42/7) = -4/6

Reduce -4/6 = (-4/2)/(6/2) = -2/3

Least Common Denominator

The least common multiple of the denominators of two or more fractions.

Let's determine the least common denominator of 13 and −27.

Ask "do 3 and 7 have a factor in common?" No.

So, this is situation 1. Multiply 3×7=21. 21 is the least common multiple of the numbers 3 and 7, therefore it is the least common denominator for 13 and −27.

21 is the LCD for 13 and −27

Transforming Fractions

Divide the least common denominator by the current denominator.

Multiply both the numerator and the denominator by this integer.

Let's now transform 1/3 and −2/7 each into their equivalent fractions that share a common denominator of 21 , which we found in the example above.

First, convert 1/3 to an equivalent fraction with a denominator of 21 .

Divide the LCD (the new denominator) by the current denominator.

21÷3=7

Multiply the numerator and denominators by 7 .

(1×7)/(3×7)=721

Adding Fractions with the Same Denominator

Add or subtract the numerators of all the fractions in the expression

Keep the same denominator! (The temptation to add the two denominators is very strong—resist.)

If necessary, reduce the answer.

Adding Fractions with Different Denominators

Find the least common denominator (LCD).

Use the LCD to find the equivalent fractions and rewrite the expression.

Add or subtract the numerators of all the fractions in the expression.

Keep the denominator the same—do not add them.

If necessary, reduce the answer.

Adding and Subtracting Mixed Number Fractions

Change the mixed numbers to improper fractions.

Find the least common denominator (LCD) if the fractions have different denominators and convert to equivalent fractions with the LCD.

Add or subtract the numerators of all the fractions in the expression.

Keep the denominator the same.

Change improper fractions to a mixed number (if needed).

If necessary, reduce the fraction to lowest form.

Subtract the following mixed numbers:

8 5/6 − 5 1/2

Step 1: Change the mixed number to an improper fraction.

8 5/6 = 53/6

and

5 1/2=11/2

Step 2: Find equivalent fractions with the least common multiple.

Convert the fractions to equivalent fractions with the LCM. The least common multiple is 6 , therefore:

53/6 does not need to change

Multiply the numerator and denominator of 11/2 by 3

53/6 − 11/2 = 53/6 − 33/6

Step 3: Subtract like fractions.

Next, subtract the fractions.

53 / 6−336=206

Step 4: To complete the problem, convert any improper fractions to lowest terms.

20/6 = 3 2/6

Finally, reduce the fraction to its lowest form.

3 2/6 = 3 1/3

Multiplying Fractions

Multiply the numerators to obtain a new numerator.

Multiply the denominators to obtain a new denominator.

Write the answer in fraction form and reduce it to the lowest terms, if necessary.

Change any improper fractions to mixed numbers.

Mixed Number Fractions:

Change any mixed numbers to improper fractions.

Multiply the numerators to obtain a new numerator.

Multiply the denominators to obtain a new denominator and write the answer in fraction form.

Change the improper fraction back to a mixed number.

Reduce the mixed number to the lowest terms, if necessary.

Dividing Fractions

Change the division sign ( ÷ ) to a multiplication sign ( × ).

Write the reciprocal of the second fraction.

Multiply the numerators.

Multiply the denominators.

Write the answer in the form of a fraction.

Reduce the fraction to the lowest terms, if necessary.

Mixed Number Fractions:

Change any mixed numbers to improper fractions.

Change the division sign ( ÷ ) to a multiplication sign ( × ).

Write the reciprocal of the second fraction.

Multiply the numerators and denominators as usual.

Change the improper fraction to a mixed number.

Reduce the fraction to the lowest terms, if necessary.

Coefficient

In algebra, constants often take the form of coefficients. A coefficient is a number by which a variable is being multiplied. Coefficients are written in front of variables. So, in 16x , 16 is the coefficient and x is the variable.

If a variable is without a number in front of it, the coefficient is 1 . Though it is not written, there is essentially an invisible 1 in front of any variable without a numerical coefficient.

Steps for Combining Terms

Identify Like Terms

Move Terms Next to Each other

Add or Subtract

Distributive Property

Distribute the term outside the parentheses to each of the terms inside the parentheses

a(b + c) = ab + ac

Distribution of Negative Numbers

When in doubt, change all subtraction operations to the addition of negative numbers.

Principle of Equality

If you perform equivalent operations to both sides of an equation, the result will always be an equivalent equation.

Steps for Solving an Equation With Complex Expressions

Substitute any variable's known value for the variable itself

Simplify expressions on either side of the equation following order of operations:

Distribute

Combine Like terms

Add and subtract constants

Complete any other process that serves to simplify the expression

Move terms across the equation, using the Addition and Subtraction Principles of Equality:

Move all constants to one side of the equation

Get all terms with the variable to be solved on the opposite side of the equation

Simplify the expressions on either side of the equation:

Combine like terms on one side, if necessary

Add and subtract constants on the other side, if necessary

Isolate the lone variable on one side of the equation, using the Multiplication and Division Principles of Equality:

The variable will be across from its value

Check your answer:

Plug in your solution to the original equation. Perform the arithmetic on both sides of the equation. If the two sides of the equation are equal, you have successfully solved the equation!

Butterfly Method in Algebraic Equations

Example

Solve the following algebraic equation: 23/21 x j = 25/13 .

Write the algebraic equation in the form ax/b=c/d

Remember that a/b(x)=ax/b since x can be converted into x/1 .

Now draw "butterfly wings" around the opposite terms.

Multiply the numbers in each of the butterfly wings:

Getting and equation into Slope-Intercept Form

First, make sure the y term is on the left side of the equation.

Put all x 's and constants on the right side of the equation.

Multiply or divide to make the coefficient of y be 1 .

Point-Slope Form

y−y1=m(x−x1) ,

where m is the slope of the line and (x1,y1) are the coordinates of a known point.

A line has a slope of 5 and passes through the point (3,7) . What is the equation of the line in slope-intercept form?

To find the equation, we start with the point-slope formula:

y−y1=m(x−x1)

Next, we fill in the information that is known. Remember, m is the slope of the line and (x1,y1) are the coordinates of a known point (3,7)

y−y1=m(x−x1)y−7=5(x−3)

Finally, we can use algebra to manipulate the equation into slope-intercept form:

y−7=5(x−3)y−7=5x−15y=5x−8

As you can see, starting with just a point and the slope of a line, we can express its linear equation in slope-intercept form.

Direction of Inequalities

Multiplying or dividing both sides by a negative number will cause the direction of an inequality to reverse.

Quantitative Data

Numerical data, consists of data values that are numerical, representing quantities that can be counted or measured.

Standard Deviation Rule

A standard proportion or percentage of data points that lie within each standard deviation away from the mean for a normal distribution.

68% percent of the data will fall within 1 standard deviation of the mean, 95% of the data will fall within 2 standard deviations of the mean, and 99.7% of the data will fall within 3 standard deviations of the mean.

Bimodal

Data set with more than one mode

Skewed Right

A skewed distribution with a tail that stretches right, toward the larger values.

The long tail of the curve is on the positive side of the peak.

Measure of Central Tendency

A summary measure that is used to describe an entire set of data with one value that represents the middle or center of the distribution.

There are 3 main measures: mean, median, or mode.

Skewed Left

A skewed distribution with a tail that stretches left, toward the larger values.

The long tail of the curve is on the negative side of the peak.

Outlier

An observation point that is distant from other observations

Unimodal

One mode

Five-Number Summary (FNS)

Minimum, 1st quartile, median, 3rd quartile, and maximum

Statistics

the science that deals with the collection, classification, analysis, and interpretation of numerical facts or data

Interquartile Range (IQR)

The difference, in value, between the bottom and top 25% of the sample or population. Q3-Q1

Frequency Distribution

A record of the number of times data occurs withing a certain category

Measures of Spread

Measures used to describe the distance of data from the center of the dataset, such as range and standard deviation

Check Sheet

A structured form or table that allows data to be collected by marking how often an event has occurred in a certain interval.

Multimodial

Bimodial, A data set that has more than two modes

Qualitative Data

Non-numeric information based on some quality or characteristic

Categorical Data

Also called qualitative data, consists of values that can be sorted in to groups or categories.

Consists of data that are groups, such as names or labels, and are not necessarily numerical.

Stem Plot (Stem and Leaf Plot)

A visual representation of data in which individual data points are plotted to the right of a vertical line, or chart, and the left (stem) shows the interval categories.

Box Plot

A graph that displays the highest and lowest quarters of data as whiskers, the middle two quarters of the data as a box, and the median

1.5 IQR Criterion Rule

Outliers are defined to be any points that are more than 1.5 × IQR above Q3 or below Q1 .

IQR * 1.5 =

Q3 + (IQR * 1.5) = Upper Outlier

Q1 - (IQR * 1.5) = Lower Outlier

When can two way tables be used?

When both the explanatory and response variables are categorical

Symmetric

Left is roughly the same as the right

Right Skewed

Positively skewed, tail stretches to the right of the peak

Left Skewed

Negatively skewed, tail stretches to the left of the peak

U-Shaped

Contains a valley rather than a peak

Uniform

Straight, all data appears to be equal

Mode

The value that occurs most frequently in a given data set.

Median

the middle score in a distribution; half the scores are above it and half are below it

Mean

Average, extreme values greatly influence the mean in the direction of the skew

Range

the difference between the highest and lowest scores in a distribution

InterQuartile Range (IQR)

the difference between the first and third quartiles

Quartiles

Values that divide a data set into four equal parts

Second Quartile

The median of the data set

First Quartile

The median of the lower half of the data set

Third Quartile

The median of the upper half of the data set

Standard Deviation

The average distance each data point is from the mean

Empirical Rule

For the normal distribution (bell-shaped distribution), approximately 68% of the measurements are within one standard deviation of the mean, approximately 95% of the measurements are within two standard deviations of the mean, and approximately 99.7% of the measurements are within three standard deviations of the mean.

Which measures of center and spread should you use?

Normal, symmetric data: Use the mean and standard deviation.

Skewed: Use the median or IQR.

Categorical: Use the mode and no measure of spread.

Misrepresenting Data with Visualizations

Scale of Axis: The vertical scale should start at 0.

Omitting Labels or Units: Leaves the size and categories unspecified.

Using a 2-dimensional graph to represent a 1 dimensional measurement: When visualizing data that represents size (big circle vs small circle), our eyes see area. This distorts the true differences we are trying to illustrate.

Explanatory Variable (Relationship between two variables)

(x) - presumed to possibly cause changes in the response variable; also known as the independent variable.

A variable that helps explain or influences changes in a response variable. Otherwise known as an independent variable.

Response Variable (Relationship between two variables)

(y) - presumed to be affected by the explanatory variable; also known as the dependent variable.

Variable affected by an explanatory variable. Otherwise knows as a dependent variable.

Graphical Displays of Two Variable Data

Categorical: Categorical, (C -> C), Two-way table

Categorical: Quantitative, (C -> Q), Side-by-side Box Plot

Quantitative: Quantitative, (Q -> Q), Scatterplot

Two-way Frequency Table (Contingency Table)

C -> Q, rows show one variables categories, columns, columns show the other variable's categories

Joint Frequencies

Values at the middle of the table, the amount of data falling in to both the corresponding row and column

Marginal Frequencies

Values on the right and bottom side of tables. Totals of the corresponding row or column

Grand Total

Bottom Right corner of table. The total size of the dataset

Conditional Percentages

Computed by dividing each joint frequency by the corresponding explanatory variable marginal frequency

Overall Percentages

Computed by dividing each frequency by the grand total

Positive Correlation

As the explanatory variable increases, the response variable increases

Negative Correlation

As the explanatory variable increases, the response variable decreases

No Correlation

Scatterplot reveals no trend between variables

Non-Linear Relationship

Scatterplot reveals a trend is not a straight line

Correlation Coefficient (r)

Measures the strength of the linear relationship between variables.

r is always between -1 and 1

The closer r is to 1, the stronger the positive linear correlation

The closer r is to -1, the stronger the negative linear correlation.

R=0 indicates no linear correlation, but that does not rule out non-linear relationships.

Effect of Outliers

When far off the regression line, outliers weaken r.

On a scatterplot, the closer the points are laid out in a line, the stronger the correlation

Population (Sampling Method)

The group you want to study

Sampling Frame (Sampling Method)

The list of people or things you pull the sample from

Sample (Sampling Method)

The subset of the population that is being studied

Bias in Sampling

Occurs when the sample frame does not accurately represent the population

Simple Random Sample

Participants are randomly chosen from the entire population

Voluntary Sample

Researchers invite everyone from the sampling frame to participate, those who respond are the sample.