Module 2 Flashcards
what is probability
random process generates outcome A equals the proportion of times the outcome would occur across a very larger number of independent repetitions of the process
PROBABILITY IS LONG-TERM RELATIVE FREQUENCY
probability is estimated by observing many many occurrences.
Frequentist definition of probability
Probability Rules
- The probability of an outcome must be a number between 1 and 0
- All possible outcomes together must have a total probability equal to 1
- Probability of A + Probability of B must = 1 - Complement Rule - probability that an even does NOT occur equals 1 minus the probability that the event does occur.
- Addition Rule
- if events A and B are mutually exclusive if they have no outcomes in common and can never occur simultaneously
If not mutually excludive
PrAandB = PrA + PrB - Pr(AandB) - Multiplication Rule
- if A and B are indepdendent
PrAandB = PrA x PrB
Independent if knowing that one occurs does not give information about the probability of the other - not mutually exclusive because its possible for both to occur simultaneously - separate events two dice
Complement Rule -
probability that an even does NOT occur equals 1 minus the probability that the event does occur.
Addition Rule
- if events A and B are mutually exclusive if they have no outcomes in common and can never occur simultaneously
If not mutually excludive
PrAandB = PrA + PrB - Pr(AandB)
Multiplication Rule
- if A and B are indepdendent
PrAandB = PrA x PrB
Independent if knowing that one occurs does not give information about the probability of the other - not mutually exclusive because its possible for both to occur simultaneously
- separate events two dice would be independent
Tversky and Kahneman
Likelihood of feminist movement, bank teller, bank teller and feminist movement
Multiplication rule makes it so bank teller and feminist movement is less likely because of multiplication rule
relative frequency is __________ to probability
eqivalent
The probability of randomly selecting a score that is exactly equal to 3 is ________ because on a continuous scale of infinite values ___________
0 bc the value of 3 is a single point and points have a width of 0 because on a continuous scale of an infinite number of values, the value of 3 is a signle point and points have a width of 0
thus with continuous distributions, non-zero probabilities exist only for an interval of possible observations.
Probability density function
describes the population probability distribution of a random variable
the probability of any outcome or event is ___________
the area under the curve bounded by the values that define the event
The total area under the curve defined by a probably density function always equals
1
Probability rule 2
For a continuous variable X, the probability of observing a value between a and b equals the area under the curve defined by the probability density function of X that falls between point a and b which is found via _______
integration
Normal distribution or curve is also called
normal probability density function
u =
population mean
population mean symbol
u
o =
population standard deviation
population standard deviation
o
Z score
Standardized value
take an observation from a normal distribution with any mean and standard deviation and standardize it or transform it, to put it on the standard normal distribution.
follow a normal distribution
positive z-scores are greater than the mean, negative are less than the mean
What units are z-scores in?
Z scores are in standard deviation units
e.g. 1 z score is one standard deviation unit greater than the mean
what is special about about z scores
any set of scores rescaled to have a a mean of 0 and a sd of 1 are standard scores. Howefer z scores are a special type of standard scores that come from the standard normal population
z score formula
z = X-u/o
is based on known or assumed population mean and SD
X-Xbar/S
standardized scores based on sample data
= Z IF X is known to come from a normal population.
SD is the standard deviation of the sample data
sampling distribution
Imagine collecting an infinite number of samples leading to an infinite number of means and sds
if we assume that each sample came from the same population, then each sample mean is an estimate of the same population mean u and population standard deviation o
because of sampling error very few if any of these mean and sd estimates will exactly equal the true population mean and sd
Create frequency distribution table for the collection of sample means = this collection would form the sampling distribution of the mean
the sampling distribution is the distribution of a
statistic
Sampling distribution alternate
infinite samples are drawn from the same population and a statistic is calculated for each of the many different samples, then the distribution of that statistic is the sampling distribution
sampling distributions are _______________ population distributions
theoretical population distributions
Central limit theorem
Sampling distribution of the mean
For means calculated from samples drawn from any parent population with mean u and standard deviation o, the sampling distribution of the mean (each based on the same number N of independent observations) will converge to a normal distribution with mean u and SD o/sqrtN as N approaches infinity
Standard error
standard deviation of that statistic’s sampling distribution
SD of the sampling distribution of sample means
o/sqrtN is the standard error of the mean and is often represented as oxbar
what does the standard deviation of the sampling distribution of the mean represent?
average amount that a sample mean xbar is expected to be different from the true population mean u
zscore for an individual observation X -formula
Z = X-u/o
Z-score for a sample mean Xbar formula
Z = Xbar-u/osqrtN
