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PSY 138: EXAM 2
Descriptive Statistics
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Summarize tenancies and characteristics of data
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Inferential Statistics
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Use sample statistics to make inferences about the population from which the samples were drawn
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What are Inferential Statistics used for?
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Hypothesis testing
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distribution of sample means
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collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population (aka sampling distribution of means/M) (ex. sampling distribution)
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When can the Distribution of Sample Means be normal?
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1. If the original population is normal
2. If the sample size n>30
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Standard Error of the Mean
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is the SD of the DSM (SD/SqrRt(n))
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Central Limit Theorem definition
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For any population with a mean and SD, the distribution of sample means for sample size n will approach a normal distribution with a mean and standard error as n approaches infinity
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central limit theorem meaning
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established statistical rule that tells us that if we were to take an infinite number of sample size n from a population of N members, the means of these samples would be normally distributed
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Hypothesis test
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is a test of the null hypothesis (Ho)
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Alternative Hypothesis
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Is the hypothesis the researcher believes in.
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Correct Decision
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Ho is true and retained it
Ho is false and you rejected it
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Type II Error
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Null Hypothesis is false and you retained it.
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Type I Error
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Null Hypothesis is true and you rejected it.
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Alpha |
The probability of making a Type I error when the null hypothesis is true.
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Z score
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Standard score specifies how many standard deviations an observation is above or below the mean. (sample mean - mean of DSM)/ standard error.
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Beta
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Probability of making a Type II Error
Retain Ho when false
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1-tailed test
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H1 specifies the direction of the effect
maen of group A = Mean of group B
X and Y are correlated
look for "more, better, improve,increase, less, worse"
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2-tailed test
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H1 has no direction
X and Y are correlated
mean of Group A not = to mean of group B
IV has "an effect"
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Hypothesis Testing
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State null and alternative hypothesis
Set you decision criteria (alpha)
collect data
compute test statistics
reject or retain Ho
State conclusion in ordinary language
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Null Distribution
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The population distribution from which the sample is drawn if Ho is true.
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Alternative Distribution
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The population distribution from which the sample is drawn if Ho is false
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Z-Test answers...
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How likely is the sample mean came from a population
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P-value
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What the probability that the sample mean happened by chance if the null hypothesis is true.
Reject Ho when p < Alpha
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Normality Assumption
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DSM is normal if:
If n< 30;Population must be normal
If n> 30; DSM is normal
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Independence Assumption
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each member of the sample is selected independently
Random selection
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Statistically Significant
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That is a result in unlikely to occur merely by chance
rejected Ho
different than meaningful
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Z-test
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How likely is a sample w/ a mean of __ cam from a population w/ a mean of __ and a SD of __
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Critical Z
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Where the critical region(s) begin
depends on Alpha
1-tailed critical region = bottom 5%
reject Ho when Observed Z < Critical Z
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Observed Z
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The actual score
(Xbar - sample mean)/ standard error
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Statistical Power
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Probablility of correctly rejecting the null hypothesis when false; ONLY has power when Ho is false
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Power |
1-B
Lower power the higher the chance of retaining Ho
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The alpha lvl
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larger Alpha more power
lower alpha lower power
However, if its too big its error prone; Too much power= too much data
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Sample size (n) and power
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larger n = smaller the standard error= more power
however, larger n takes more time and money
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Effect size and power
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Larger effect = more power (distance from 1 mean to the next, makes them skinnier)
Give a bigger dose to increase, less overlap
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1-tailed v 2tailed
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1-tailed is better
More power if in correct direction
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SD and power
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Larger SD =Larger standard error = LESS POWER
the smaller the SD the more power
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Point Estimate
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A single number that is used to estimate a population parameter.
precise but almost always wrong
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Interval Estimate
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an interval of numbers around the point estimate, within which the parameter value is believed to fall
"95% chance Meu is btw 2 and 8"
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Confidence Interval
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z-test rearranged
an interval calculated using sample statistics to contain the population parameter within a certain degree of confidence
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1.96
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Z-score of Alpha in a 1-tailed hypothesis
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Slope |
(x)(slope)+(intercept)
change in Y/ change in X
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Intercept |
The value of y when x=0
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Y-hat
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The predicted value of Y
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Residual error
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vertical distance between Y and line
e=Y - Yhat
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Variance |
correlation coefficient squared r^2
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3 Measures of Error
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Variance not explained by X=1-r^2
Sum of the squared residuals
Standard error of the estimate
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r^2
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Percent variance in Y accounted for by X
Regression
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Standard error of the estimate
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average distance of Y from the predicted value of Y
SD of errors= Standard error
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confidence interval
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an interval estimate calculated using sample statistics to contain the population parameter, within a certain degree of confidence
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Margin of error
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The distance from the point estimate to the upper/lower bound of a confidance interval
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Cohen's d
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The distance between 2 means in SD units
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b1 |
slope |
b0 |
intercept
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Sum of the Squared Errors
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sum of the squared differences between each score on the criterion variable and its predicted score
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In the Conditional Distribution
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Mean= Yhat
SD= Standard error of the estimate
What Y would look like if people scored X
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One-tailed & Ho>H1
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Reject Ho when Observed Z < Critical Z
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One-tailed & H1>Ho
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Reject Ho when Observed z > Critical z
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Nominal |
A set of categories without numerical values
eg. colors
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Ordinal |
A set of category with numerical order but unequal distances
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Interval
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A set of categories with numerical order and equal distances.
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Ratio |
A set of categories with numerical value, equal intervals and a TRUE ZERO.
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Skew |
Asymetric distribution of a certain kind
Tail points in (+/-) direction.
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Mode |
Most frequently occuring thing/score in a data set
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Parameters
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descriptive statistics used to define apopulation
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Statistics |
describe samples
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Central Tendancy
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A representative # from a distribution
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If positivly skewed...
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Mode < Median <Mean
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If negativly skewed
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Mean <Meadian <Mode
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Variability
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Degree of spread (scatter in a distribution).
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4 Roles of variability
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clarify precision/ representativness of central tendancies
Identify outliers
compute other stats
theoretically interesting on its own
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Range |
(high score)-(low score)
only two most extreme scores measured
very suseptable to outlires.
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Variance |
SD^2
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Z Score
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How many SDs the raw score is above or below the mean.
SD=1
M=0
maintains shape of original distribution
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Restricted Range Problem
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True nature of correlation is hidden due to restricted range.
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Effect Size
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Standadized measure of the magnitude of a variable's effect on another (Cohen's D)
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Cohen's d
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Distance of 2 means in terms of SD
0.2 small
0.5 medium
0.8 large
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Degrees of freedom
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the number of scores that are free to vary when estimating a population parameter from a sample
df=N-1
Higher # df bigger the peak
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t-distribution
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Shape depends on df
Peaked at top (critical regions)
fatter in tails
as n...> infinity t-dist becomes normal
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1-Sample t-Test
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Use if population is unknown but SD is known
critical t> critical z
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Within-Person Pairs
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Longitudinal (same variable different times)
Within-Person Contrast (different variables same person)
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Dyads |
Same variable different people
must be comparable or have defined relationship
must be distiguishable |
2 Types of Dyads
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Distinguishable (particular role; parent/child)
Indistinguishable (don't have differences; couples, coworkers)
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In Expected Mean Difference Scores...
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If Ho is true then m1 = m2
Therefore Mean Difference is zero
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Paired Sample t-Test
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is a 1-sample t-test on difference scores
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Assumptions of t-Tests
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Normality Assumption
Independent assumption
Homogeneity of Variance Assumption (Independant t-Test)
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Normality assumption
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That the DSM is normally distributed
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Independanct Assumption
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No scores are related to other scores
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Homogeneity of Variance Assumption
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Whether two populations have equal variances (levenes test)
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