psych stat unit 2

normal distribution symmetrical

exactly fit together overlap

+d

shifted above population mean

-d

shifted below population mean

sample design

specific plan or protocol for how individuals will be selected or sampled from a population of interest address 2 questions 1. does the order of selecting participants matter? 2. Do we replace each selection before the next draw?

standard error of the mean

standard deviation of a sampling distribution of sample means it is standard error or distance that sample mean values deviate from the value of population mean

cohen's effect size conventions

standard rules for identifying small medium and large effects based on typical finding behavioral research

identify the hypothesis to test

state the hypothesis null hypothesis alternative hypothesis

alternative hypothesis

statement directly contradicts null hypothesis by stating actual value of a population parameter is less than greater than or not equal to the value in null hypothesis

normal distribution tails

tails always approach x-axis but never touch it

p value

probability of obtaining a sample outcome given value stated in null hypothesis is true compared to level of significance

type 1 error

probability of rejecting a null hypothesis that is actually true researchers directly control the probability of committing this type of error

type 2 error

probability of retaining a null hypothesis is actually false (beta error)

rules about probability

probability varies between 0&1 written as a fraction decimal percent or proportion probability can NEVER be negative either probable or impossible

bivariate random vairable

any random variable with only 2 possible outcomes

unbiased estimator

any sample statistic obtained from a randomly selected sample that equals the value of its respective population parameter on average sample mean is an unbiased estimator because it equals the population mean on average

normal distribution total area under the curve

between 0-1 NEVER negative

normal distribution theoretical

can be normally distributed in theory

normal distribution standard deviation

can equal any positive value

normal distribution list

check for normality compute mean and standard deviation find real limits locate z score for each real limit find proportion located within the real limits

make a decision

compare population mean to sample mean probability greater than or equal to 5% null hypothesis is true

dependent

conditional probability of 1 outcome is dependent on the occurrence of the other outcome an outcome is dependent when the probability of its occurrence is changed by the occurrence of the other outcome

decisions researchers can make

reject null hypothesis retain null hypothesis

independent

2 outcomes in depend when the probability of 1 outcome doesn't affect the probability of the second outcome

additive rule

additive law when 2 outcomes for a given event are mutually exclusive the probability that any one of these outcomes occurs is equal to the sum of their individual probabilities

random event

any event in which outcome observed can vary

fixed event

any event in which the outcome observed is always the same

level of significance

criterion of judgement upon which a decision is made regarding clause stem in null hypothesis criterion based on probability of obtaining a statistic measured in a sample if value stated in a null hypothesis were true

critical values

cutoff value defines boundaries beyond which less than 5% of sample means can be obtained if null hypothesis is true sample means obtained beyond a critical value will result in a decision to reject null hypothesis

power

hypothesis testing is probability

significance

decision made concerning a value stated in null hypothesis when null hypothesis is rejected we reach significance null hypothesis is retained we fail to reach significance

effect

difference between a sample mean and population mean stated in null hypothesis in hypothesis testing an effect is not significant when we retain the null hypothesis an effect is significant when we reject the null hypothesis

distribute the frequencies

distribute relative frequencies probabilities for 2 outcomes can be mutually exclusive independent complementary and conditional

sampling distribution

distribution of all sample means of sample variances that could be obtained in samples of a given size from same population

binomial probability distribution

distribution of probabilities for each outcome of a bivariate random variable

probability distribution

distribution of probabilities for each outcome of a random variable sum of probabilities in a probability distribution is equal to 1 ranges from 0 to 1 NEVER negative

binomial distribution

distribution of probabilities for each outcome of a variable with only 2 possible outcomes larger the sample size more closely binomial distribution will approximate a normal distribution

mathematical exception

expected value mean or average expected outcome of a given random variable the expected outcome of a random variable is the sum of the products for each random outcome multiplied by the probability of its occurrence

central limit theorem

explains that regardless of the distribution of scores in a population the sampling distribution of sample means selected at random from that population will approach the shape of a normal distribution as the number of samples in the sampling distribution increases probability distribution for obtaining a sample mean from a population is normal

sampling error

extent to which sample means selected from the same population differ from one another this difference which occurs by chance is measured by standard error of the mean

probability

frequency of times an outcome occurs divided by the total number of possible outcomes

nondirectional tests

hypothesis test in which alternative hypothesis is stated as not equal to a value stated in the hypothesis

directional test

hypothesis test in which the alternative hypothesis is stated as greater than or less than a value stated in null hypothesis

patterns

increase sample size, power by reducing standard error increase effect size sample size and alpha level increases power decrease beta population standard deviation and standard error increases power

law of large numbers

increasing the number of observations or samples in a study will decrease the standard error. larger samples are associated with closer estimates of the population mean on average

z statistic

inferential statistic used to determine the number of standard deviations in a standard normal distribution that a sample mean deviates from the population mean stated in the null hypothesis

set criteria for a decision

level of significance

alpha level

level of significance or criterion for a hypothesis test largest probability of committing a type 1 error that we will allow and still decide to reject the null hypothesis

column A unit normal table

lists Z scores only positive Z scores equal to negative ones above or at mean

Column B unit normal table

lists area between z score and mean 0-.5 never more increases

column C unit normal table

lists area from Z score toward tail .5-0 decrease closer to tail

cutoff score for a proportion

locate Z score associated with given proportion in the unit normal table transform z score into raw score (x)

test statistic

math formula that identifies how far or how many standard deviations a sample outcome is from the value stated in a null hypothesis allows researchers to determine likelihood of obtaining sample outcomes if null hypothesis were true used to make a decision regarding null hypothesis

Bayes's theorem

mathematical formula that relates the conditional and marginal (unconditional) probabilities of 2 conditional outcomes that occur at random

normal distribution is

mathematically defined theoretical symmetrical mean median and mode all located in 50th percentile standard deviation can equal any positive value total area under the curve or a normal distribution is equal to 1 tails of a normal distribution are asymptotic mean can be positive or negative and standard deviation positive

normal distribution mean

mean can equal any value

cohen's d

measure of effect size in terms of the number of standard deviations that means scores shifted above or below the population mean stated by non hypothesis larger the value of d the larger the effect in the population

standard deviation of a probability distribution

measure of variability for the average distance that outcomes for a given random variable deviate from the expected value or mean of a probability distribution it is calculated by taking the square root of the variance of a probability distribuition

variance of a probability distribution

measure of variability for the average squared distance for that outcomes for a given random variable deviate from the expected value or mean of a probability distribution

hypothesis testing

method for testing a claim or hypothesis about a parameter in a population using data measured in sample. In this method we test a hypothesis by determining the likelihood that a sample statistic would be selected if the hypothesis regarding the population parameter were true

sampling without replacement

method of sampling in which each participant or item selected is not replaced before next selection this method of sampling is most common method used in behavioral research

sampling with replacement

method of sampling in which each participant or item selected is replaced before the next selection this method of sampling is used in development of statistical theory

multiplicative rule

multiplicative law when 2 outcomes for a given event are independent the probability that both outcomes occur is equal to the product of their individual probabilities

standard normal distribution

normal distribution with a mean equal to 0 and a standard deviation equal to 1 standard normal distribution is distributed in z score units along the x axis

null hypothesis

null statement about population parameter like population mean that is assumed to be true

low effect size

one common solution to overcome is increase sample size

experimental sampling

order does Not matter we sample without replacement total number N!/n!(N-n)!

theoretical sampling

order matters sample with replacement total number of samples possible N^n

sample space

outcome space total number of possible outcomes that can occur in a given random event

rejection regions

region beyond a critical value in a hypothesis test when the value of a test statistic is in the rejection region we decide to reject the null hypothesis otherwise we retain null hypothesis

normal distribution mean median and mode

same value all located in 50th percentile

retain null hypothesis

sample mean associated with high probability of occurrence when null hypothesis is true

reject null hypothesis

sample mean associated with low probability of occurrence when null hypothesis is true

sample mean related to population mean

sample mean is an unbiased estimator follows central limit theorem has minimum vairance

mean

sample mean unbiased estimator of population mean sampling distribution approximately normal locate probability -population mean -standard error of mean

sample variance

sample variance unbiased estimator distribution of sample variances follows skewed distribution rule approach positively skewed distribution as sample # increases a distribution of sample variances has no minimum variance

hypothesis

statement or proposed explanation for an observation a phenomenon or scientific problem that can be tested using the research method often a statement about the value for a parameter in a population

effect size

statistical measure of size of an effect in a population allows researchers to describe how far scores shifted in a population which allows researchers to describe how far scores shifted in the population or the percent of variance that can be explained by a given variable how far scores shifted in the population percent of variance that can be explained by a given variable

one-sample z test

statistical procedure used to test hypotheses concerning the mean in a single population with a known variance

complementary

sum of their probabilities is equal to 1 when 2 outcomes are complementary they are exhaustive of all possible outcomes so the two outcomes constitute 100% of the sample space

compute test statistics

test statistic

normal distribution

theoretical distribution in which scores are symmetrically distributed above and below the mean median and mode at center of the distribution

Z transformation locate proportion

transform a sample mean into a z score locate corresponding proportion for z score in unit normal table

locate proportion

transform raw score (x) into Z score locate corresponding proportion for the Z score in the unit normal table

type 3 error

type of error possible with one tailed test in which a decision would have been to reject the null hypothesis but researcher decides to retain null hypothesis bc the rejection region was located in wrong tail

unit normal table

type of probability distribution table displaying a list of scores and the corresponding probabilities associated with each z score listed

sample mean characteristics

unbiased estimator distribution of sample means follow central limit theorem distribution of sample means has minimum variance regardless of the distribution of scores in a population the sampling distribution of sample variances selected at random from that population will approach shape of a positively skewed distribution

obtained value

value of a test statistic value compared to critical value of a hypothesis test to make a decision when obtained value exceeds a critical value we decide to reject null hypothesis otherwise we retain null hypothesis

z score

value on x axis of a standard normal distribution numerical value of a z score specifies the distance or the number of standard deviation that a value is above or below the mean

random variable

variable obtained or measured in a random experiment unlike other mathematical variables a random variable is not the actual outcome of a random experiment but rather describes the possible as yet determined outcomes in a random experiment

mutually exclusive

when the 2 outcomes can't occur together the probability of 2 mutually exclusive outcomes occurring together is 0

standard normal transformation

z transformation formula used to any normal distribution with any mean and any variance to a standard normal distribution with a mean=0 standard deviation =1