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