Probability And Statistics

Front 1

For given set U being everything and A and B beung subsets of U What is A u B in statistics? What's is A n B?

Back 1

A u B is everything in A and B. A n B is only the numbers that exist in A and B.

Front 2

What is a combination vs a permutation? What are formulas for each?

Back 2

Combination is represented by C (n,r) and is not order specific. n is total # of items and r is # that is allowed at a time. Ex: number of combinations for out of 6 people to get 4 promotions.

C(n,r) = P(n,r)/r! = n!/(r!(n-r)!)

Permutation is P(n,r) and is order specific. Ex: ID code begins with three letters can only be used once. 5 possible letters.

P(n,r) = n!/(n-r)!

Front 3

What formula do I use when I have n objects and they are split into different categories?

Back 3

So basically it is factorial of total number of objects divided by factorial of number of objects in each category (type of color, shape, etc) multiplied together.

P(n1,n2,n3,...nk) = n!/(n1!n2!...nk!)

Front 4

What are the two ways events can be combined in probability? What are these denoted by?

Back 4

Can either be connected by "and" and "or". What is probability of A and B denoted by P(A,B), A n B. Probability of A or B denoted by P(A + B), A u B.

Front 5

What is law of total probability? What is law of compound probability? How does compound eq differ if A and B are independent events? What is P(B/A) denote?

Back 5

Gives probability of even A OR B occurring.

P(A+B) = P(A) + P(B) - P(A,B)

Gives probability that A AND B will occur.

P(A,B) = P(A)*P(B/A) = P(B)*P(A/B)

P(A,B) = P(A)P(B)

Denotes the probability of B given A has occurred.

Front 6

A person exhibits a symptom that occurs p(s) in population. This symptom is exhibited by all people with a disease. That disease occurs p(d) in all people. What is probability this person has the disease? What is this problem an example of and what is general formula?

Back 6

P(D/S) = [ P(D) P(S/D) ]/[ P(S/D) P(D) + P(S/Not D) P(Not D) ]

Bayes theorem

P(Bj/A) = P(Bj)*P(A/Bj) / sum(P(A/Bi)*P(Bi))

Front 7

What are mode, median, and mean?

Back 7

Mode is # occurs most frequently, median is number in the middle of data, mean is sum of all numbers in sample divided by total numbers in sample

Front 8

What is a weighted mean?

Back 8

It is the mean but have to take into account the weighted part. So four calculating final grade it is score of test times weight of that test plus. Or for GPA. It is class grade times credit hours of that class divided by total credit hours. Do this for each class and add together.

Front 9

When is it appropriate to use geometric mean? What is formula?

Back 9

When you are trying to find average of values that are used as multipliers. Ex: interest on an account over n years.

= nth root of (X1*X2*X3....Xn)

Front 10

What is Stan dev of population? What is stand dev when combining two or more data sets with known stand devs? What if a series of data sets are taken from same population and Stan devs are known what is stand dev of combined data? What is the standard dev of the mean values of these samples? Where do I find formula for stand dev of product of two random variables?

Back 10

Stan dev pop = sq root [ 1/ N sum(Xi - mean)^2)].

Stand dev sum = sq root [stan dev 1^2 + Stan dev 2^2 + ....].

Stan dev = stand dev * sq root(n).

Stand dev mean = Stan dev / sq root (n).

Stan dev product = sq root [ A^2(stan dev b)^2 + B^2(stand) dev a)^2 ]

Front 11

How does sample Stan dev differ from pop Stan dev? What is variance? What is sample coefficient of variance?

Back 11

1/N term is 1 / (n-1).

It is Stan dev squared.

It is CV = s / mean.

s is sample standard deviation

CV is relative dispersion of sample data

Front 12

What is a probability density function?

Back 12

It is a graph that shows the probability (y axis) for random values of x. Area under the graph is always equal to 1. Because the integral across a range 0 to a gives probability of x being less than a.

Front 13

When a probability mass function is provided how do you find expected value?

Back 13

Need to plug each value in data set into mass function to get its value. Then you sum all products of value by its value in mass function. Essentially a weighted average ( I think).

Front 14

When dealing with continuous variables where do I find eqs? How do you find expected value? Variance? Stand dev? Coefficient of variance?

Back 14

Search expected value. This involves functions for each variable.

u = E[x] = integral (x*f(x)) bounds -infinity to infinity

f(x) is probability density function

(Stand dev)^2 = V[x] = E[(x-u)^2] = integral((x-u)^2*f(x)dx) bounds -infinity to infinity

= √(V[x])

= stand dev / u

Front 15

What is Binomial probability function? What is it used for?

Back 15

= n!/(x! (n-x)!) * P^x * q^(n-x).

Is used when all outcomes are discrete and there are only two outcomes (success or failure). P is prob of success. q is prob of failure. x is # of times you want it to be successful. N is total number of tries.

Front 16

For anything regarding Bell curve or large real world experiments what do I search to find tables? What is conversion formula to find Z value for tables?

Back 16

For tables search unit normal distribution.

Then Z =( X - u )/ stan dev.

Front 17

With Bell curve what percent is within + or - one stab dev? Two Stan dev? 3 Stan dev?

Back 17

68.2%. 95.4% 99.7%

Front 18

What is central limit theorem?

Back 18

Basically States that mean and stan dev of really large number of samples or multiple samples is equal to that of pop.

Front 19

What is t- distribution? What is exceedance? What is chi-squared distribution? What is confidence level?

Back 19

The probability distribution function with v degrees of freedom with V being 1 less than sample size. It is tabulated with t as a function of V and exceedance.

Exceedance is the probability of being incorrect (unfinished sideways 8 symbol or alpha). Essentially total area under upper tail of probably density function f(x).

Chi Is used with hypothesis testing of variances known as critical values of F squared table.

Is 1 - the probability of being incorrect signifies by C.

All of these have tables you can look up for values.

Front 20

What are confidence intervals? If Stan devs known what tables do I use? Where is Z? if they aren't known what tables do I use? What is t? What is degrees of freedom?

Back 20

They are the limits given a certain %confidence. Can be between two means of different samples or a normal distribution or its mean.

If the are known use unit normal distribution table.

Z is the value on left side where confidence level is area under graph.

Is Stan dev not known use t distribution table.

Degrees of freedom in NCEES handbook under confidence interval formulas.

Front 21

What eqs do I use if the question involves testing a hypothesis and finding what the confidence level is of the test being accurate?

Back 21

Search test statistics to find eqs. If Stan dev known use Z and if not use t. Simple relation of Z vs x (value from unit normal distribution table) to see if confidence level is for or against hypothesis.

Front 22

What is least squares method used for? How do I do it?

Back 22

To find equation of line from series of data points.

Find these values

Sum (xi)

Sum (yi)

Sum (xi^2)

Sum(yi^2)

[Sum(xi)]^2

[Sum(yi)]^2

Sum(xi*yi)

Then use formulas given I'm NCEES to find slope and y intercept.

Front 23

What is the standard error of estimate? What is sample correlation coefficient? What is goodness of fit?

Back 23

Is also known as mean squared error. Can look up equation under least squares method.

It is R.

If R is greater than .85 fit is good.