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A measure of the shape of a data distribution. Data skewed to the left result in negative skewness; a symmetric data distribution results in zero skewness; and data skewness to the right result in positive skewness.

An exploratory data analysis technique that uses five numbers to summarize the data; smallest value, first quartile, median, third quartile, and largest value.

A method of assigning probabilities that is appropriate when all the experimental outcomes are equally likely.

A method of assigning probabilities that is appropriate when data are available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated a large number of times.

A random variable that may assume either a finite number of values or an infinite sequence of values.

A function, denoted by f(x), that provides the probability that x assumes a particular value for a discrete random variable.

A probability distribution for which each possible value of the random variable has the same probability.

A probability distribution showing the probability of x occurrences of an event over a specified interval of time or space.

For data skewed to the left, the skewness is negative; for data skewed to the right, the skewness is positive. If the data are symmetric, the skewness is zero.

A symmetric distribution, the mean and the median are equal. When the data are positively skewed, the mean will usually be greater than the median; when the data are negatively skewed, the mean will usually be less than the median.

3. By using the interquartile range, IQR = Q3 - Q1, limits are located. The limits for the box plot are 1.5 (IQR) below Q1 and 1.5 (IQR) above Q3. Data outside these limits are considered outliers.

An advantage of the exploratory data analysis procedure is that they are easy to use; few numerical calculations are necessary. We simply sort the data values into ascending order and identify the five-number summary.

Always assigned on a scale from 0 to 1. A probability near zero indicates an event is unlikely to occur; a probability near 1 indicates an event is almost certain to occur.

Give some examples of experiments and their associated outcomes to follow. (A = experiment, B = Experimental outcome)

The sample space for an experiment is the set of all experimental outcomes. Also known as sample point.

How can you describe the sample space for a coin-tossing experiment, and figure out how many experimental outcomes are possible?

If an experiment can be described as a sequence of k steps with n1 possible outcomes on the first step, n2 possible outcomes on the second step, and so on, then the total number of experimental outcomes is given by (n1) (n2) ... (nk)

There are two possible outcomes for the coin toss meaning that (n1=2), (n2=2), etc... Therefore for six coin tosses we will (2)(2)(2)(2)(2)(2) = 64.

When data are available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated a large number of times.

When one cannot realistically assume that the experimental outcomes are equally likely and when little relevant data are available. When the subjective method is used to assign probabilities to the experimental outcomes,

we may use any information available, such as our experience or intuition. After considering all available information, a probability value that expresses our degree of belief (on a scale from 0 to 1) that the experimental outcome will occur is specified.

In the physical sciences, researches usually conduct an experiment in a laboratory or a controlled environment in order to learn about cause and effect. In statistical experiments, probability determines outcomes.

Even though the experiment is repeated in exactly he same way, an entirely different outcome may occur. Because of this influence of probability on the outcome, the experiments of statistics are sometimes called random experiments.

The probability of any event is equal to the sum of the probabilities of the sample points in the event.

That the experimental outcomes are equally likely. In such cases, the probability of an event can be computed by counting the number of experimental outcomes in the event and dividing the result by the total number of experimental outcomes.

Examples of Discrete Random Variables (A = Experiment, B = Random variable (x), C = Possible Values for the Random Variable)

Experimental outcomes based on measurement scales such as time, weight, distance, and temperature can be described by continuous random variables.

Examples of Continuous Random Variables (A = Experiment, B = Random variable (x), C = Possible Values for the Random Variable)

Think of the values of the random variable as points on a line segment. Choose two points representing values of the random variable.

Once the probability distribution is known, it is relatively easy to determine the probability of a variety of events that may be of interest to a decision maker.

Evaluating f(x) for a given value of the random variable will provide the associated probability. For example, using the preceding probability function, we see that f(2) = 2/10 provides the probability that the random variable assumes a value of 2.

In computing the variance of a random variable, the deviations squared and then weighted by the corresponding value of the probability function. The sum of these weighted squared deviations for all values of the random variable is called variance.

When computing a Poisson probability for a different time interval, we must first convert the mean arrival rate to the time period of interest and then computer the probability.

Skewness

Five-number summary

Box Plot

A graphical summary of data based on a five-number summary.

Probability

A numerical measure of the likelihood that an event will occur.

Experiment

A process that generates well-defined outcomes.

Sample Space

The set of all experimental outcomes

Sample Point

An element of the sample space. A sample point represents an experimental outcome

Tree Diagram

A graphical representation that helps in visualizing a multiple-step experiment

Classical method

Relative Frequency Method

Subjective Method

A method of assigning probabilities on the basis of judgement.

Random Variable:

A numerical description of the outcome of an experiment.

Discrete random variable:

Continuous Random Variable:

A random variable that may assume any numerical value in an interval or collection of intervals.

Probability Distribution

A description of how the probabilities are distributed over the values of the random variable.

Probability Function

Discrete Uniform Probability Distribution

Expected Value

A measure of the central location of random variable.

Variance

A measure of the variability, or dispersion, of a random variable.

Standard Deviation

The positive square root of the variance.

Poisson Probability Distribution

Poisson Probability Function

The function used to compute Poisson probabilities.

What does the negative and positive integer for skewness indicate?

How do the median and mean relate to the skewness of a data?

What average should be preferred when the data are highly skewed?

The median provides the preferred measure of location when the data are highly skewed.

What are the four components of the five-number summary?

- Smallest value
- First quartile
- Median
- Third quartile
- Largest Value

What are the steps in creating a box plot? (PART I)

- A box is drawn with the ends of the box located at the first and third quartiles. This box contains the middle 50% of the data.
- A vertical line is drawn in the box at the location of the median

What are the steps in creating a box plot? (PART II)

What are the steps in creating a box plot? (PART III)

4. The dashed lines are called whiskers. The whiskers are drawn from the ends of the box to the smallest and largest values inside the limits computed in step 3.

5. Finally, the location of each outlier is shown with the symbol *.

What is an advantage of the exploratory data analysis procedure? (PART I)

What is an advantage of the exploratory data analysis procedure? (PART 2)

The box plot can then be constructed. It is not necessary to compute the mean and the standard deviation for the data.

Probability values

- (A) Toss a Coin (B) Head, Tail
- (A) Select a part for inspection (B) Defective, non-defective
- (A) Conduct a sales call (B) Purchase, no purchase
- (A) Roll a die (B) 1, 2, 3, 4, 5, 6
- (A) Play a football game (B) Win, lose, tie

Sample Space

How would use S to denote the sample space?

- S = {Head, Tail}
- S = {Defective, Non-Defective}
- S = {1, 2, 3, 4, 5, 6}

S = {(H,H), (H,T), (T,H), (T,T)} Thus, we can see that four experimental outcomes are possible.

Counting Rule for Multiple-Step Experiments

Example of Counting Rule for Multiple-Step Experiments using tossing coins six times.

What are the two basic requirements for assigning probabilities? (PART I)

- The probability assigned to each experimental outcome must be between 0 and 1, inclusively. If we let
*Ei*denote the*i*th experimental outcome and*P*(Ei) its probability, then this requirement can be written as ---- 0 <= P(*Ei*) <= 1 for all*i ---*

What are the two basic requirements for assigning probabilities? (PART II)

2. The sum of the probabilities for all the experimental outcomes must equal 1.0. For *n* experimental outcomes, this requirement can be written as ---- P(*E*1) + P(*E*2) + ... + P(*E*n) = 1 ----

When is the classical method of assigning probabilities appropriate? (PART I)

*n* experimental outcomes are possible, a probability of 1/*n* is assigned to each experimental outcome.

When is the classical method of assigning probabilities appropriate? (PART II)

When using this approach, the two basic requirements for assigning probabilities are automatically satisfied.

Using the classical method, denote the likelihood of each number on the die being rolled

P(1) = 1/6, P(2) = 1/6, P(3) = 1/6, P(4) = 1/6, P(5) = 1/6 and P(6) = 1/6

When is the relative frequency method of assigning probabilities appropriate?

When is the subjective method of assigning probabilities most appropriate? (PART I)

When is the subjective method of assigning probabilities most appropriate? (PART II)

When is the subjective method of assigning probabilities most appropriate? (PART III)

that the experimental outcome will occur is specified. Because subjective probability expresses a person’s degree of belief, it is personal.

How are the notion of an experiment differ from the statistics and statistics? (PART I)

How are the notion of an experiment differ from the statistics and statistics? (PART II)

What do you do when drawing a random sample without replacement from a population of size *N*?

*n* that can be selected.

Event

An event is a collection of sample points.

The probability of an event

Is the sample space, S, considered an event?

*P(S)* = 1.

What is the assumption made when the classical method is assigned?

Random Variable

A random variable is a numerical description of the outcome of an experiment.

- (A) Contact five customers (B) Number of customers who place an order (C) 0, 1, 2, 3, 4, 5
- (A) Sell an automobile (B) Gender of the customer (C) 0 if male; 1 if female.

What are some example experimental outcomes applications?

- (A) Operate a bank (B) Time between customer arrivals in minutes (C) x => 0
- (A) Construct a new library (B) Percentage of project complete after six months (C) 0 <= x <= 100

What is a way to determine whether a random variable is discrete or continuos? (PART I)

What is a way to determine whether a random variable is discrete or continuos? (PART II)

If the entire line segment between the two points also represents possible values for the random variable, then the random variable is continuous.

What is a primary advantage of defining a random variable and its probability distribution?

Required conditions for a discrete probability function

f(x) => 0

∑f(x) = 1

Discrete Uniform Probability Function

f(x) = 1/n

where...

n = the number of values the random variable may have

What is the probability function for rolling dices discrete uniform random variable?

- f(x) = 1/6
- x = 1, 2, 3, 4, 5, 6

What can be accomplished by evaluating f(x) for a given value of the random variable?

Expected value of a discrete random variable

E(x) = *μ = *∑xf(x)

To compute the expected value of a discrete random variable, we must multiply each value of the random variable by the corresponding probability f(x) and then add the resulting products.

Variance of a discrete random variable (PART I)

Var(x) = σ* ^{2}*= ∑(x-

An essential part of the variance formula is the deviation, x-μ, which measures how far a particular value of the random variable is from the expected value, or mean, μ.

Variance of a discrete random variable (PART II)

Variance of a discrete random variable (PART III)

The notations Var(x) and σ^{2} are both used to denote the variance of a random variable.

How is the standard deviation measured?

* ^{2}* is measured in squared units and is thus more difficult to interpret.

Properties of a Poisson Experiment

- The probability of an occurrence is the same for any two intervals of equal length.
- The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval.

Poisson Probability Function

f(x) = (μ^{2} e^{-^}μ) - x!

Where...

f(x) = the probability of x occurrences in an interval

μ = Expected value or mean number of occurrences in an interval

e = 2.71828

What do you have to do when computing a Poisson probability for a different time interval?

About this deck

Author: Alexandre I.

Created: 2011-01-24

Updated: 2011-07-07

Size: 73 flashcards

Keywords: flash card flashcards digital flashcards note sharing notes textbook wiki college dorm class classroom exam homework test quiz university college education learn student teachers tutors share, study blue studyblue studyblu

Views: 12

Created: 2011-01-24

Updated: 2011-07-07

Size: 73 flashcards

Keywords: flash card flashcards digital flashcards note sharing notes textbook wiki college dorm class classroom exam homework test quiz university college education learn student teachers tutors share, study blue studyblue studyblu

Views: 12

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