Statistics Exam 2

suppose two events are disjoint, what is the probabilithy of one event happening and the other event happening

0 if two events are disjoint then the probability of them occurring right after the other is 0

formal addition rule

P(A or B)=P(A)+p(B)-P(A+B) where P(A and B) rpresents the prob. that both occur at the same time

rules for complementary events

P(A) + P(Abar) = 1 P(Abar)= 1-P(A) P(A) = 1-P(Abar)

5% guidelines for cumbersome calculations

if a sample size is no more than 5% of the size of the population treat the selection as being independent event if replacement is used, this is because it can make calculations manageable since it is rare to select the same item twice

what does it mean to be unusual

if the outcome is far from what we typically expect. extreme events whic is consistent with the range rule of thumb

what is a continous random variables

infinitely many values and those values can be associated with measurements on a continous scale

population mean vs sample mean

population mean is all numbers in a sample added together, the sample mean is means multiplied by the probabilities added together

Capital P denotes....

probability

payoff odds

ration of net profit to the amount of bet

classical approach to probability

requires equally likely outcomes, assume that a given procedure has n different simple events and that each of those simple events has an equal chance of occuring. if event A can occur in s of these n ways then P(A)=s/n=number of ways A can occur/ total nuber of different simple events

relationship between mean and expected value

the mean is the expected value in the sense that if you continue indefinitely the average values would equal the expected values. an expected values does not need to be a whole number

Multiplication rule for several independent events

the probability for any sequence of independent events is simply the product of their corresponding probabilities

what does the second part of the binomial probability formula mean

the probability of x successes among n trials for any one particular order

what is P(A and B)

the probability that vent A occurs in the first trial and event B occurs in a second trial multiplied together

Conditional probability

the probabiliyt for thesecond event should take into account the fact that the first event A has already occured, notated as P(B|A)

if the variance decreased what would happen to the density curve

the spread would decrease and the graph will get horizontally shorter and vertically taller

requirements for probability distribution

there is a numerical random variable and its values are associated with corresponding probabilites. all probabilites equals 1, for every value of x 0 less than equal to P(X) which is less than equal to 1

table A-2 gives probabilites to __________ of the value

to the left

what are dependent events

two events are dependent if the occurences of one of them affects the probability of the occurences of the other,but this does not mean that one of the events causes the other one

Formal additive rule for disjoint events

P(A or B)= P(A) + p(B)

compound event

P(A or B)= P(event A occurs or event B occurs or they both occur)

Multiplication rule for independent events

P(A and B)= P(A) * P(B) only if a and b are independent

the probability that something equals exactly one value on a continuous distribution is

0

what are unusually low

: x successes among n trials is an unusually low number of successes if p (xor fewer) less than or equal to 0.05

TRUE OR FALSE: the binomial distribution utilizes statistics

FALSE: the binomial distribution describes a population therefore it uses parameters

TRUE OR FALSE: The values on Table A-2 are only for the standard normal distribution

True, if something is not on a standard normal distribution you are going to have to convert to use table A-2

uniform distribution

a continuous random variable has a uniform distribution if its values are spread evenly over a certain range, the graph of a uniform distribtion results in a rectangular shape

what are critical values

a z score separating unlikely values from those that are likely to occur

variance probbability distribution

it is x-mean squared multiplied by the probability

what does z alpha represent

it represents the z score separating the top alpha from the bottom 1- alpha

what are continuous random variables

it takes infinitely many values, those values can be associated with measurements on a continuous scale without gaps or interruptions

properties of sample mean

it targerts the value of the population mean (that is, the mean of the sample means is the population mean or the expected value) the distribution of sample means tends to be a normal distribution for large numbers

what values do unusal values lie below or above

maximum unusual value= mean + 2 *std dev minimum usual values= mean - 2 * std dev

counting

n! represents a way of ordering n numbers, it is espressed as n* n-1 * n-2.........

round off rule for mean std dev and variance

round results by carrying one more decimal place than the number of decimal places used for the random variable x, if the values of x are integers round all to one decimal place

letters A,B,and c denote.....

specific events

sampling distribution of the proportion

the distribution of sample proportions, with all samples habing the same sample size n taken from the same population p tilde is equal to sample proportion and p is equal to population proportions

sampling distribution of the variance

the distribution of sample variances with all samples having the same sample size n taken from the same population

what does independent mean

two events are independedn if the occurence of one does not affect the probability of the occurences of the others, ex rolling a die, flipping a coin

probability histogram

very similar to a relative frequency istogram but vertical scale shows probabilities the probabilities is equivalent to the areas under the curve

what is subjective probability

when the probability of a particular event is estimated by using knowledge of the relevant circumstances

what are unusually high

x success among n trials are unusually high if P(x or more) < or equal to 0.05

what is distance along the horizonal scale on table A-2

z-scores

disjoint

events are disjoint or mutally exclusive if they cannot occur at the same time ex rolling a die on an even number

TRUE OR FALSE: selections without replacement are always independent and selections with replacement are always dependent

False, it is the other way around

actual odds against

expressed a:b and is a ration of P(Abar)/P(A)

Formal multiplication rule for dependent events

P(A and B)= P(A) * P(B|A)

Rare Event Rule

if under a given assumption the probability of a particular observed event is exceptionally small we conclude that the assumption is probably not correct.

probability distribution

a description that gives the probability of each value of the random variable often expressed in the format of a table, graph, or formula

Principle of redundancy

a design feature contributing to realiability, cricial components are duplicated so that if one fails the other will work

what is a simulation

a process that behaves in the same way as the procedure itself so that the same results are produced.

What is a random variable

a variable typicaly represented by X that takes a numerical value determined by change, for each outcome of a procedure x takes a certain value but for different outcomes that value may be different

what is a sample space

all possible siple events; that is the sample space consists of all outcomes that cannot be broken down further

probability limits

always express as a fraction or a decimal number between 0 and 1, the probabililty of an impossible event is 0, the probability of an event that is certain is 1, for any event A the proabability of A is between 0 and 1 inclusively (o<equalto P(A),equalto 1

what does it mean to be unlikely

an event is unlikely when the probability is very small 0.05 or less

what is a simple event

an outcome or an event that cannot be further broekn down into simpler components

what is an event

any collection of results or outcomes of a procedure or an experiment, or a game

wat is the region under the curve in the body of table A-2

areas

Law of large numbers

as the procedure is repeated again and again the relative freqency of an event tends to approach its actual probability

actual odds in favor

b:a indicates odds in favor to A and is calculated by dividing P(A) by P(Abar)

expected value decision theory

calulating the expected value you can determine how much you would lose at a casino

completmentary events

denoted by some letter bar, consists of all outcomes in which that certain event does not occur, to find subtract 1 from P(A)

P(z<a)

denotes the probability taht the z-score is less than a

P(a<z<b)

denotes the probability that the z score is between a and b

P(z>a)

denotes the probability that the z score is greater than a

finding z scores when given a probability using table A-2

draw a bell shaped curve and identify the region that it wants you to find use the cumulative area from the left to locate the closes probability in the body of table A-2 and identify the corresponding z score

what is a discrete random variable

either a finte number of values or countable number of values

Fundamental counting rule (multiplication rule)

for a sequence of two events in which the first event can occur in m ways and the second event can occur in n ways the events togethre can occur a total of m*n ways

practical rules when sing central limit theorem

for sample sizes larger than 30 the distribution of the sample mean can be approximated by a normal distribution, if the original population is normally distributed hen any sample size n, the sample means will be normally distributed we can apply the central limit theorem if either n>30 or the original population is normal

standard normal distribution

has 3 properties: 1. the graph is bell shaped 2. it is symetric about its center 3. its mean is equal to 0 4. its standard deviation is equal to 1

continous random variable example

heights of individuals, time it takes to finish a test, in practice we don't measure accurately enough to truly se all possible values of a continuous random varianle.

what does n, x, p, q, and P(X) denote

n=fixed number of trials x=denotes a specific number of successes in n trials, so x can be any whole number between 0 and n inclusively p=denotes the probability of success in one of the n trials q=denotes the probability of failure in one of the n trials P(X) denotes the probability of getting exactly x succeses among n trials

combinations

nCk=n!/(n-k)!k! is the number of ways to choose k objects from n (without replacement) and rearrangements of the same item are counted as being the same.

permutations

nPk=n!/(n-k)! is the number of ways to choose k items from n without replacement where rearrangements of the same item are counted as being different

examples of discrete random variables

nmber of heads in 4 coin flips, number of classes missed last week number rolled on a 6 sided die

does disjoint equal independent

no there can be two events with differing probabilities that are independent but are not disjoint since the probabilities are not compliments of each other

examples of random variables

number of boys in a random family with three children, x-0,1,2,3 random weight: x>0

examples of a binomial distribtion

number of correct guesses at 30 true or false questions when randomly guessing number of left handers in a randomly selected sample of 100 unrelated people

what does the first part of the binomal probability formula mean

number of outcomes with exactly x successes among n trials

relative frequency probability

number of times A occured/ number of imes the procedure was repeated

what does p and q symbolize

p=sucess=P(S) q=1-p=P(F)

sampling distribution of the sample mean

the distribution of all possible sample means with all the samples having the same sample size n taken from the same population

binomial probability distribution

results from a procedure that meets all of the following requirements : 1. the procedure has a fixed number of trials 2. the trials must be independent. (the outcome of any individual trial doesnt affect the probabilities in the other trials) 3. Each trial must have all outcomes classified into two categories (sucess or failure) 4. the probability of success remains the same in all trials

properties of sample proprotions

sample proprtions target the value of the population proportions (that is the mean of the sample proprotions or the expected value of the sample proportions is equal to the population proportions distribution of the sample proportion tends to be a normal distribution

properties of sample variances

sample variances target the value of the population variance, that is the sample variances is the population variances or the expected value of the sample varaince) the distributionof the sample variances tends to be a distribution skewed to the right

why sample with replacement?

sampling without replacement would have very practical advantage of avoidin wasteful duplications when something was selected more than once, however we are interested in sampling with replacement because when selecting a relativelysmall sample from a larg pop. it makes no significant difference whether we sample with replacement or witout and sampling with replacement results in independent events that are unaffected by previous outcomes which are easier to analyze and calculate

when drawing a density curve how do you force the area to be 1

set the vertical axis to 1/ n so that area=1

procedure for nonstandard normal distribution

sketch a normal curve, label the mean and any specific x values, ten shade the region representing the desired probability for each relevant value x that is a boundary for the shaded region, convert that value to the equivalent z score find the area of the shaded region as before

if the variance is increased what happens to the density curve

spread increases and it gets longer and the graph height gets shorter

sampling distribution of a statistic

such as sample mean or sample proportions is the distribution of all values of the statisic when all possible sample size n are taken from the same population

tree diagram

summarizes the possible outcomes for a true/false question followed by a multiplechoice question

central limit theorem

the distribution of the sample mean for a sample size n appraches a normal distribution as the sample size n increases. given the random variable x has a distribution which may or may not be normal with a mean of mew and a standard deviation sigma and he sample siz n is selected from the population Conclusion: the distribution of the sample mean will as the sample size increases appraoach a normal distribution, te mean of that normal distribution is the same as the pop, mean and the standard deviation of that normal distribtion is sigma over root n

what is the expected value

the expected value for a discrete random variable is denoted by E and is the mean value of outcomes

density curve

the graph of a continuosu probability distribitoin, must satisfy: 1. the total area under the curve must equal 1 2. every point on the curve must have a vertical height that is 0 or greater (cannot fall below x-axis)

confusion of inverse

the incorrect ideolgy that the probability of b occurring assuming a occurred is the same as a occurring assuming that b occurred

P(A) denotes.........

the probability of event A occurring