Psych Stats Final

population

entire group we are interested in studying

sample

subset of the population that we actually study

samples should

represent the population

descriptive procedures

used to summarize, organize, and simplify sample data and result in statistics (designated by english letters)

inferential procedures

allow us to study samples and make generalizations about the population to estimate the population parameters (greek letters)

sampling

simple random (assign #s to ALL and pick randomly), quota/stratified random (first break into groups based on x characteristic and then choose a batch from each group), convenience sampling (using ‘easy’ population)

sampling error

natural discrepancy btwn a sample statistic and corresponding population parameter

bias in samples

selection bias (non-representative group), non-response bias (some groups are less likely to respond)

empiricism

scientists make predictions based off observations

theory

general statement about the causal relationship btwn variables

hypothesis

prediction about specific events derived from theories

scientific method steps

observe, predict, test, analyze, conclude, publish, restart

parsimony

given two competing theories, the simplest explanation is preferred

anecdotal evidence

based on personal experiences, often biased, least certain

correlational statistics

measure scores on two variables and determines potential relationship (should not imply causation)

experimental

researcher manipulates one variable and measures effect on another

independent variable

manipulated

dependent variable

measured

quasi-experimental

compare groups based on naturally occurring variables, no manipulation

scales of measurement: nominal

categorical, cannot be ranked (favorite color)

scales of measurement: ordinal

indicates rank or order but not magnitude of difference (places in a race)

scales of measurement: interval

equal intervals separate adjacent scores, no true zero (celsius or Fahrenheit temperature)

scales of measurement: ratio

true zero exists, with equal intervals separating adjacent scores (height, weight, etc)

bar chart

for nominal or ordinal data, bars have spaces

histogram

for interval or ratio data, no spaces

charts should

make information easier to understand and compare

normal distribution

bell-shaped, symmetrical

bimodal distribution

two peaks

skewed distribution

scores pile-up on one end with tail (positive skew= tail to +, negative skew = tail to -)

uniform / rectangular distribution

scores all have the same frequency

descriptive statistics

goal: summarize data in a clear and understandable way

types: mean, median, and mode

mode

score w/ highest frequency

only possible measure for nominal data

median

middle value, 50th percentile score

good for ordinal data, highly skewed data, and open ended distributions

mean

average

mu = population mean, ex bar = sample mean

preferred measure of central tendency

variability

do the scores cluster about the center point or do they spread out?

measures of variability

range, interquartile range, standard deviation and variance

range

difference between highest and lowest scores in distribution, very reliant on outliers

interquartile range

range between the first and third quartile (Q1-Q3)

75th percentile score - 25th percentile score = IQR

helps to eliminate outliers

mean absolute deviation (MAD)

average absolute deviation of a data set

variance

average squared deviation (s²)

Variance = σ² = Σ(x - μ)² / N

variance = s² = Σ(x - xbar)² / N-1

standard deviation

average deviation from the mean = s

sqrt [Σ(x - xbar)² / N-1]

standard deviation is

still impacted by outliers

variance and standard deviation are

always >0

only used for interval and ratio data

can have same mean w/ different variance

can have different mean w/ same variance

for normal distribution

68% of scores fall in the region ± 1 SD from the mean

deviation score

X - Mean

z-score

number of standard deviations above or below the mean

z = (X - μ) / σ

standard normal distribution

has a mean of 0 and a standard of 1

the larger the absolute value of the z-score

the less frequently it occurs

the area under the curve

corresponds to z-scores, is proportional to the frequency or scores

finding a score from a percentile rank

convert to z-value and convert z to raw score

three uses of z-scores (units = SDs)

examine the relative status of an individual score in a normal distribution

compare scores coming from different variables

examine the relative status of an entire sample in a normal population

analytic view of probability

number of occurrences of the event divided by the total number of occurrences of all events

frequency/N

frequentistic view of probability

probability in terms of past performance

subjective view of probability

based on personal belief in the likelihood of an outcome

probability notation

a proportion, fraction, or percentage

multiplicative law

used for probability of joint occurrence of two or more events (&)

additive law

used for the probability of occurrence of one or more mutually exclusive events (or)

sampling distribution of the means

the frequency distribution of all possible sample means drawn from a population

sampling distribution of the means attributes

sample means clump together around the true population mean

a different sampling distribution is obtained with each sample size (N)c

central limit theorem

1. sampling distribution will approach a normal distribution, even if the population distribution does not

2. the mean of the sampling distribution always equals the mean of the pop. distribution (mu sub xbar = mu)

3. the standard deviation of the sampling distribution is determined by: the standard deviation of the population and the sample size (σ_xbar=σ / √n)

standard error of the mean

the standard deviation of the sampling distribution of the means

becomes smaller as N increases

how do you assess relative standing of a single sample mean to all samples from population?

transform sample mean into z-score

observed effects can be due to

systematic effects or chance or some combination

errors in decision making

type 1: reject H₀ when it is actually true (probability = alpha = 0.05) (probability of avoiding = 1-alpha)

type 2: retain H₀ when it is false (probability = beta, power = 1=beta [probability of rejecting it when it is false])

things that alter power

larger N increases power, variability decreases power, strength of effect (stronger increases power, weaker decreases power)

when to use z-test

comparing sample to population, sd is knownw

when to use 1 sample t-test

comparing sample to population, sd is unknown

when to use independent samples t-test

comparing two groups that are measured independently

when to use related samples t-test

comparing twp groups that are measured in some related way (repeated measures, matched samples, or natural pairs)

when to use one-way anova

comparing multiple groups with 1 independent variable with >2 levels

when to use two-way anova

comparing multiple groups with 2 independent variables

between subjects anova

independent samples

within subjects anova

related samples

when to use Pearson correlation coefficient [r]

when identifying a relationship between variables

two ways to estimate population mean

point estimation (exact value)

interval estimation (using confidence interval)

confidence interval

xbar ± margin of error

CI_.95=x̄ ± (S_x-) (t_0.05)

we can say with a probability of .95 that the interval between x- and x+ includes the true mean x for —.

effect size

how big is the significant difference?

cohen’s d= units of standard deviations = mean1-mean2/ SD

small= 0.2, medium = 0.5, large = 0.8

statistical hypothesis

z-test: H₀: mu=pop mean, H₁: mu does not equal pop mean

1 sample t-test: H₀: mu=pop mean, H₁: mu does not equal pop mean

related and independent samples t-tests: H₀: mu_d=0, H₁: mu_d does not equal 0

one-way ANOVA: H₀: mu₁=mu₂=mu₃=mu_n, H₁: mu₁ does not equal mu₂ does not equal mu_n

two-way ANOVA: mu₁=mu₂=mu₃=mu_n, H₁: mu₁ does not equal mu₂ does not equal mu_n (for both main effects) & H₀= no interaction, H₁= interaction

P’s CC: H₀: rho=0, H₁: rho does not = 0

all t-tests look at

ratio of systematic variance to chance variance

reporting results - parentheses = df

z-test: z= -2.38, p<0.05

1 sample t-test: t(24) = 2.75, p< 0.05

related and independent samples t-tests: t(11) = 4.95, p<0.05

one-way and two-way ANOVA: f(2,9) = 6.45, p=0.018, n²=0.59

P’s CC: r(4) = 0.902, p=0.014

advantages with related samples designs

avoids participant to participant variability, only analyzes difference scores, requires fewer participants

disadvantages with related samples designs

order effects (one condition may produce varied results from being first or second- practice, fatigue, sensitization)

carry-over effects (effect of earlier treatment linger)

homogeneity of variance

the variance of each population being represented is similar (for independent sample t-tests)

for independent samples t-tests

having very different ns (number of people in each group) results in lower power

factors

independent variables in ANOVA

residuals in JASP

error values

k

number of condition

why use ANOVA instead of lots of t-tests?

more t-tests increases the odds of type 1 errors, ANOVA reduces

familywise error rate

with multiple t-tests, it is much larger than alpha. ANOVA keeps it equal to alpha

post-hoc tests

reveal which values are significantly different

ANOVA assumptions

must use interval or ratio data

scores in each population are normally distributed

all groups have similar variance (homegeneity of variance)

partitioning variance

ANOVA looks at ratio of within group variance to between group variance (MS_error vs MS_group)

Mean Squared Deviation

MS, similar to variance

MS_error

within group variance

MS_group

variance between groups

MS_group+MS_error= total variance

variability among all the scores in a data set

when null is true

F_obtained =1w

when null is false

F_obt>1

F_obt=

(treatment effect + chance)/ chance

ANOVA summary table

Sum of squares: Σ(x -x̄)²

df_group= k-1

df_error= k(n-1)

df_total= N-1

MS_group= SS_group/df_group

MS_error= SS_error/df_error

F_obt= MS_group/ MS_error