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Math
Statistics
Hypothesis Testing
PSYS 054 Final
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Terms in this set (179)
One-Way ANOVA
has 1 independent variable
Factors
Independent variables in an ANOVA
Levels
each possible value of a factor
Grand Mean
overall mean score of all participants across all groups
Omnibus Null Hypothesis
key significant test, represents the hypothesis that the population means of all groups are equal
ANOVA Assumptions
Homogeneity of variance; normality; independence of observations
ANOVA Variation
if there is a lot of between-group variation relative to within-group variation, then the ANOVA will be significant
F Ratio
if F is close to 1, retain null. if F is greater than 1, reject the null.
n
number of participants in each group
N
number of all participants
K
number of groups
Xbar
group mean
GM
grand mean
SStotal
SSgroup+SSerror
ANOVA Calculation
Calculate SStotal
- SSgroup
-SSerror
-MSgroup
-MSerror
-F ratio
Post-Hoc Tests
allows us to figure out which group means are significantly different from which other group means while keeping the alpha level constant
Liberal
more likely to find statistically significant differences
conservative
less likely to find statistically significant differences
Bonferroni Procedure
conduct multpile follow-up t-tests but adjust your alpha level for each one. take the desired alpha level (0.05)/number of tests and use that new number for the alpha level of each
Factorial ANOVA
ANOVA with more than one IV/factor and each factor has 2 or more levels
SSa, SSb, SSab
Main effect of first factor, main effect of second factor, interaction effect
SSab
SScells-SSa-SSb
SSerror
SStotal-SScells
Interaction Plot
graph of interaction where x-axis is one variable and different lines are another variable
Repeated Measures ANOVA
multiple assessments of the same participants on different levels
Between-Subjects Effects
Variability due to differences between levels of IV
Within-Subjects Effects
unique to RM-ANOVA, difference in single participant's score over time
t
number of time points
ANCOVA
one or more categorical IVs and 1 continuous DV but ask 1 or more continuous covariate
covariate
variable that might be predictive of the DV whose linear influence you wish to control for
ANCOVA Assumptions
normal distribution, equal variance, independent, DV and covariate association, any IV/covariate association is equal across groups
ANCOVA weaknesses
covariates can change meaning of DV, cannot equate groups that are naturally unequal
Nominal Scale
differently named categories
Ordinal Scale
numbers represent ranks for greater than/less than comparisons
Interval Scale
equal number representing equal intervals
Ratio Scale
True zero point that can indicate absence of measurement
Positive Skew
Data Curve is higher on the left. Mode < median < mean
Negative Skew
data curve is higher on the right. Mode > median > mean
mutually exclusive
events that cannot occur at the same time
collectively exhaustive
at least one of the possible events must occur
independent trials
trials are independent of each other. The occurrence of one does not affect the probability of another
Sampling Distribution
the probability distribution under repeated sampling from the population of a given statistic
Hypothesis Testing
start with assumption that your sample comes from a "non-special" population aka a null hypothesis
Null Hypothesis (Ho)
hypothesis of no difference between groups or no relationship between variables
Type I Error
Finding an effect when there isn't one
Type II Error
failing to find an effect when there is one
Central Limit Theorem
given a population with a mean u and a variance o2, the sampling distribution of the mean will have a mean equal to u, and a variance equal to o2/N
Z-Test
convert observed sample M to a z-score using central limit theorem to compare to the population M
T-Test
used for situations in which the population variance is unknown
One-Sample T-Test
comparing a sample mean to a hypothesized population value
Paired T-Test
used when we have "paired" or dependent observations
Independent Samples T-Test
most common in research, compared means with independent groups
Cohen's d
standardized mean difference for effect size
0.20 = small effect
0.50 = medium effect
0.80 = large effect
Effect Size
Numerical way of expressing the strength or magnitude of a relationship
magnitude of the effect under the alternate hypothesis
Cohens conventions for small/medium/large effect
0.20 - Small effect
0.50 - Medium effect
0.80 - Greater effect
Power
The probability of correctly rejecting a false H0
Factors that effect power level
-Alpha level
-effect size
-sample size
cutoff for power used in psych research
.80
Effect of alpha level on power level
increasing alpha decreases beta and increases power and probability of a type I error
Effect of sample size on power level
Increasing sample size decreases variance which decreases the over lap of the two distributions which increases power size
Correlation
The measure of the relationship between variables
Predictor Variable
The variable from which a prediction is made (independent)
Criterion variable
The variable to be predicted
Regression line
line of best fit through data points
Covariance
The degree to which the two variables vary together
Pearsons r
Measure the degree to which the covariance approaches its maximum
Residual
The difference between the obtained and predicted values of y
Regression sum of squares
Variability directly attributed to variability in the predictor
Data used with chi square test
Frequency data - how often things occur
Goodness of fit test
Test for comparing observed frequencies with theoretically predicted frequencies
Observed frequencies
The frequency you observe in the data
Expected frequency
Expected value for number of observations if the null hypothesis is true
Degrees of freedom for chi square test
k-1, categories minus one
Contingency table
2D table where observations are categorized on the basis of two variables
Calculating an expected frequencies table
multiply the row and column totals and divide by grand total
degrees of freedom for contingency table
(row-1)*(collumns-1)
Cohen's D
an effect size used to indicate the standardized difference between two mean
Relationship between power and type II error rate
As beta increases power decreases
1-beta
Regression to the mean
The tendency for predicted values to be less extreme (closer to the mean) than the score from which they are predicted
Delta
A value used in referring to power tables that combines cohens d and the sample size
Degrees of freedom and directionality for hypothesis test for rho (correlation coefficient)
n-2
always 2 tailed
Regression notation
Yhat
Standard error of the estimate
Standard deviation about the regression line
Error sum of squares
Variability not due to variability in the predictor
R^2 (pearsons r)
% of the variance in y is accounted
for by x
Hypothesis test for regression
same as correlation coefficient
if pearsons r is significant than the slop of the regression line is significant
Difference between simple regression and multiple regression equation
each variable has their own slope
significance of coefficients for multiple regression
represents the weight of that one variable to the whole
Difference between standardized and unstandardized coefficients for multiple regression
unstandardized are on different scales
Sampling Error
Variability of a statistic from sample to sample due to chance
Sampling distribution of the mean
Distribution of sample means from repeated sampling from the population of a statistic
Standard Error
Standard deviation of a sampling distribution
Research hypothesis
The hypothesis that the experiment was designed to investigate (H₁)
Null Hypothesis (H₀)
A hypothesis of no difference or relationship
Rejection level (significance level)
The probability with which we are willing H₀ (usually .05)
Rejection region
Any outcome who's probability under H₀ is less than or equal to the significance level, leading us to reject the H₀
Step by Step hypothesis testing
-State hypotheses (H₀,H₁)
-Determine directionality
-Set alpha level (.05)
-Collect data and compute statistics
-reject/retain null
Critical Value
The value of the test statistic at or beyond which we will reject H₀
Type I error
Rejecting the H₀ when it is true
Type II Error
The error of not rejecting the H₀ when it is false
α (alpha)
The probability of a type I Error
β (Beta)
The probability of a type II error
Non directional test
Reject H₀ if it is in the lowest or highest 2.5%
Directional test
Reject H₀ if it is the 5% in the expected directed
Central Limit theorem
The sampling distribution of the mean will:
-Have a mean equal to μ
-Have a variance equal to σ2 / N (and SD equal to σ / √N)
-Approach a normal distribution as N (number of samples) increases
µ (sub Xbar)
Mean of the sampling distribution of the mean
δ(sub Xbar)
Standard deviation of the sampling distribution of the mean - also known as standard error
Use of Z test
Comparing a sample mean to a population mean
Info needed for Z test
Population mean and SD, target sample mean
Z test concept
-Convert observed sample mean to a Z score
-Find probability of getting that Z score using area under the curve
Z test formula
Z=Xbar-µ/δ/√N
Z test steps
-Establish alpha level, rejection region and critical values
-Find standard error of population
-Convert observed sample mean to Z score
-Compare calculated Z score to the critical value
-Reject/retain null
Critical Values of Z
One tailed: 1.65 or -1.65 (depends on direction) (2.5%)
Two tailed ±1.96 (5%)
Use of one sample T test
Comparing target mean to hypothesized population mean
Info needed for one sample T test
sample Mean and SD, Hypothesized population mean
One sample t test concept
Same concept as computing a Z statistic except you are using S (sample SD) to estimate the population SD
One Sample t test formula
t=xbar-µ/s/√N
Degrees of freedom
The number of independent pieces of information remaining after estimating one or more parameters
One Sample T test degrees of freedom
N-1
T test steps
-Calculate your t-statistic using formula
-Calculate your degrees of freedom
-Use Appendix E.6 to find critical value(s)
-Compare your t-statistic to critical value
-If t is more extreme than critical value reject the H₀
Use of paired T test
Comparing mean differences between related or dependent scores
Info needed for paired T test
Mean and the SD of difference scores
Difference Score
D=(x₁-x₂)
-If time based do earlier test as x₁
-If not time based x₁ will be specified
Paired T test formula
t=Dbar/S(d)/√N(d)
Reporting T tests APA style
-Italicize notation letters but not numbers
-Include degrees of freedom in parenthesis
-If score falls in the rejection region write p<.05
Paired T test degrees of freedom
N(d)-1
Use of Independent samples t test
Comparing mean difference between unrelated or independent scores
Info needed for independent samples T test
Mean, Variance and N of each independent sample
Degrees of Freedom independent samples T test
N₁+N₂-2
Independent samples T test Formula
T=Xbar₁-Xbar₂/√(S²₁/N₁)+(S²₂/N₂)
Confidence intervals
Calculate a range of potential population means
Confidence interval of a one sample T test
confidence interval of a population mean
CI=Xbar±T.₀₅S/√N
Confidence interval of a independent samples t test
Confidence interval of the population mean difference
CI=(Xbar₁-Xbar₂)±t.₀₅S(subx1-x2)
S(subx1-x2) is the standard error of the difference
-√(S²₁/N₁)+(S²₂/N₂)
Alpha level and confidence interval
1-α=CI
α=.05 - when rejecting null
CI=.95 - When retaining null
Confidence Interval APA style
-lower limit and upper limit separated by a coma
-enclosed in a bracket
-specify with CI as abbreviation
Descriptive statistics
Describes data (central tendency & variability)
Inferential statistics
Infers information about a population based on information from a sample
Uses of Frequency tables
See ordering in a set of data, examine data for outliers or impossible values, Increase familiarity with distributions
Uses of Histograms
Represent the frequency of observations in each interval
Skew
Placed to one side; asymmetrical
Kurtosis
How peaked or flat the distribution looks
Leptokurtic
More peaked
Platykurtic
More flat
Unimodal
A distribution having one distinct peak
Bimodal
A distribution having two distinct peaks
Mode
Most frequent value, highest bar in a bar graph
Mode Strengths
Actual number in the data set, unaffected by outliers, applicable to any scale
Mode Weaknesses
may be unrepresentative of data sets, no mode for a flat distribution
Median
Middle score of distribution (N/2) In a frequency table where cumulative percent crosses 50%
Median Strengths
unaffected by outliers, unaffected by internal distances among scores
Median Weaknesses
Not easily used in complex equations, relatively unstable from sample to sample
Mean
Arithmetic average of a distribution
Mean of sample
X(bar)=∑x/n
Mean of population
µ=∑x/n
Mean strengths
Easily used in complex formulas, relatively stable from sample to sample, used to estimate population mean from sample
Mean weaknesses
Very sensitive to outliers, Often not actual value on data set
measure of central tendency to scale of measurement
nominal - mode
ordinal - median,mode
interval - mode, median, mean
ratio - mode, median, mean
Central tendency & skew
0 skew - mean=median=mode
positive skew - mode<median<mean
negative skew - mode>median>mean
Trimmed mean
Mean that results from trimming away a fixed percentage of extreme observations
Range
Difference between highest and lowest score
focus on extremes and outliers
Interquartile range (IQR)
one quarter above and below median
75th percentile score minus 25th percentile score
Boxplots
Means of identifying outliers
Winsorized mean
replace the extreme observations after trimming 20% with the new highest and lowest values
Winsorized Variance
variance of the winsorized sample
Normal Distribution
Symmetrical bell shaped distribution
mean=median=mode
Area under the curve of normal distribution
probability of scoring in that range
Area under curve from 1 SD and above
34%
Area under curve from 1 SD to 2 SD
14%
Area under curve from 2 SD to 3 SD
2%
Z score
Number of standard deviations above or below the mean
Z=x-µ/δ
Z=x-x(bar)/s
outcome
possible result
events
subject of the outcome interested in (E)
sample space
total set of possible outcomes (N)
probability
p(E)=E/N
Mutually exclusive event
cannot occur at the same time (one trial)
Collectively exhaustive events
At least one of the events must occur (one trial)
Advanced addition rule
non mutually exclusive events (1 trial)
p(A or B)=p(a)+p(b)-p(A and B)
Advanced multiplication rule
Two dependent trials
p(A and B)=p(A)*p(B|A)
Conditional probability
The chance of something happening given a certain situation
p(B|A)
Probability of B if A occurs
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