Psych Stats Exam 3
What are the three assumptions of the Independent Measures ANOVA?
1-Independent random sampling, normal distributions, homogeneity of variance
What three pieces are calculated in the independent samples t test?
1-df, 2- sample variance, 3- estimated standard error
What pieces are needed when computing for the F ratio?
All the SS, all the MS, and all the df
What is the main advantage of a repeated measures design?
Because it uses the same participants, there is no risk of individual differences between groups interfering with the measurements
What does the sample mean allow us to do with a hypothesis test?
It allows us to test the hypothesis about a population mean
What does pooled variance do for the estimated standard error in an independent measures t test?
It corrects the bias in the standard error by combining the two sample variances into a single value
Why is the symbol for estimated standard error sM if it is the standard error for the population?
It indicates that the estimated value is computed from sample data rather than from the actual population parameter
What is a limitation on the estimated standard error for the independent-measures t test?
It is limited to situations in which the two samples have the same sample size
What is the formula to calculate cohen's d in an independent samples t test?
M1 - M2/√ sp2
What is the formula to calculate the t statistic for a related samples t test?
MD - µD/ sMD
How do you calculate cohen's d for a related samples design?
MD/√s2
What is the formula for computing the F-ratio?
MSbetween/MSwithin
Notation: MS
Mean Square
Notation: MD
Mean difference- related samples
Notation: MSbet
Mean squares between treatments
What is the formula to compute dftotal?
N-1
What is the formula to compute dfwithin?
N-k
How is the t statistic for a repeated measures design different than the independent measures design?
Repeated uses the Difference scores (D) rather than the raw scores (X)
What is the formula for computing MSbetween?
SSbet/dfbet
What is the formula for computing MSwithin?
SSwithin/dfwithin
How is the difference scores calculated?
Subtract one score from the other, and you get the difference score- it's this score that is used in the rest of the calculations where the raw score is normally used
What is a level?
The individual conditions or values that make up a factor
Why are extra tests done after rejecting the null hypothesis?
To determine the size of the effect
What can a post hoc test do to help with the risk of type 1 errors?
Uses special methods to try to control for these errors.
Notation: α
alpha level
Notation: µD
population mean difference- related samples
How is sM calculated?
s / √n
What is the symbol for the estimated standard error?
sM
What is the goal of counterbalancing a repeated measures design?
to distribute any outside effects evenly over the two treatments
What is the formula to calculate SSbetween-treatments?
(∑T2/n ) - G2/N
What are the two ways the estimated standard error can be interpreted for an independent samples t test?
1- it is the standard or average distance between sample statistic and population parameter. 2- the estimate standard error produces a simplified version of the independent measures t statistic
What are three types of repeated measures used in the repeated measures design?
1-before/after measurements (pre and post test styles) 2- simultaneous measurement (same variable diff conditions in the same session) 3-successive measurements (same as simultaneous but conditions are successive and counterbalances)
What are two advantages of ANOVA?
1-it can be used to compare more than two treatments at a time, 2-impacts design and analysis (multiple hypothesis tests increase the chance of a type 1 error, with one test, it helps prevent this)
What are the three assumptions of the independent measures t test?
1-observations in each sample MUST be independent 2- the two populations from which the samples are selected must be normal 3- the two populations from which the samples are selected must have equal variances
What are the two assumptions for a repeated measures t test?
1-observations within each treatment condition must be independent, 2- population distribution of difference scores must be normal (not a concern unless the sample size is small)
What are three great advantages to a repeated measures design?
1-requires fewer subjects, 2- you can study changes over time, 3-reduces or eliminates the influence of individual differences
What three pieces of data do you need to compute for a related samples t test?
1-sample variance, 2-estimated standard error of difference, 3- the t statistic
What are the two big disadvantages of the repeated measures design for the t test?
1-threats to internal validity, 2-order effects
What is Tukey's HSD?
A single value that determines the minimum difference between treatment means that is necessary for significance
Describe the t statistic.
A statistic used to test hypotheses about an unknown population mean, μ, when the value of σ is unknown.
How do you calculate the degrees of freedom in an independent samples t statistic?
Add the df for the first sample plus the df for the second sample
What are post hoc tests?
Additional tests done to determine exactly which mean differences are significant, and which are not.
What is a repeated measures design?
Aka within subjects design- a single sample of individuals is measured more than once on the same dependent variable
What are two items in the independent samples t test that are structurally different from the single sample t test?
All of the elements of the single sample t test are doubled, the independent measures uses the difference between two sample means to evaluate a hypothesis about the difference between two population means
What does ANOVA stand for?
Analysis of variance
What are order effects and what can it do to a repeated measures study?
Changes in scores that are caused by participation in an earlier treatment; can distort the mean differences
When making a decision for an independent measures t test, what numbers do you look at and when do you reject the null?
Compare the calculated t statistic to the critical value obtained for the selected alpha level- if t is bigger, reject the null.
In an independent measures t test, if we reject the null, what needs to be done after that?
Compute cohen's d and r2
What can you do if you think the homogeneity of variance assumption is violated?
Compute the Hartley's F-max test
Notation: CI
Confidence interval
What is the experimentwise alpha level?
Each comparison that is done adds a risk of type 1 error, and this added risk for the total experiment is the experimentwise alpha level
What are the two types of matched subjects that are used for repeated measures research designs?
Experimental- two separate samples are used, and Naturally Occuring pairs- husbands/wives, siblings, parent/child
What are fixed effects anova?
Experimenter is specifically interested in the levels chosen and concerned with extrapolating to untested levels
Notation: F
F ratio
When reporting in APA the results of the Anova, how do you put in the df?
F(dfbetween, dfwithin)
What is the formula for the Sheffe's test?
FAversusB = MSbet/MSwithin calculated with SS of two groups
What would a symbolized null hypothesis look like for an independent measures design look like?
H0 = µ1 - µ2 = 0
What would a symbolized alternative hypothesis look like for an independent measures design?
H0 = µ1 - µ2 ≠ 0
What is the symbolized null hypothesis test for a one way anova?
H0 = µ1 = µ2 = µ3
What is the symbolized null hypothesis for a repeated measures design?
H0 = µD = 0
What is the symbolized alternative hypothesis for a repeated measures design?
H0 = µD ≠ 0
Notation: F-max
Hartley's F-max test for homogeneity of variance
What does the F curve look like?
It is positively skewed, therefore it is expected to get high F values for statistical significance
What is the estimated standard error used for?
It is used as an estimate of the real standard error when the value of σM is unknown
Where does the sM go in the t statistic?
It is used in place of σM
What does the standard error (in the denominator of t formula) measure in an independent samples test?
It measures the amount of error when you use a sample difference to represent a population mean difference.
What does standard error provide us?
It provides us a measure of how well a sample mean approximates the population mean by describing how much difference is reasonable to expect between M and µ
What is a major problem with the z-score?
It requires more information than researchers typically have available
What is the advantage of using a t statistic rather than the z score?
It tests hypotheses about an unknown population mean when the value of σ is also unknown
If the null hypothesis is true, what will we expect for an F statistic?
It will be around 1
What is the formula to compute dfbeteen?
K-1
What is the formula for the t statistic?
M - µ / sM
What is pairwise comparison?
Post hoc compares two individual means at a time
What do we call the extra tests that we do after rejecting the null hypothesis?
Post hoc tests
Notation: p
Proportion
What is the estimated standard error formula for independent measures test?
S(M1-M2) = √s12/n1 + s22/n2
What is the formula for the estimated standard error in an independent measures test when the sample sizes are different?
S(M1-M2)= √ sp2/n1 + sp2/n2
What is the formula for sample variance?
SS / n-1
What is the formula for η2?
SSbet/SStot
What are the four steps to a hypothesis test for independent measures designs?
Same as others: 1- state the hypothesis and select alpha level 2-calculate df to determine critical region 3-obtain data and compute test statistic 4- Make a decision
How do you interpret the value of cohen's d in independent samples t tests?
Same as you interpret for other tests
Rewrite the t formula to reflect where the data comes from.
Sample mean - population mean (from the data) - (from the null hypothesis) / estimated standard error (computed from the sample data)
What is the t formula for the independent measures test?
Sample mean difference - population mean difference/estimated standard error AKA (M1 - M2) - (µ1 - µ2)/ S(M1-M2)
Notation: ∑
Summation
What is a testwise alpha level?
The alpha level set for each hypothesis test
What is the shape of the t distribution?
The exact shape of a t distribution changes with degrees of freedom. As df gets very large, the t distribution gets closer in shape to a normal z-score distribution
Describe the structure of the t statistic formula.
The formula for the t statistic has the same structure as the z-score formula, except that the t statistic uses the estimated standard error in the denominator
What is a factor?
The independent (or quasi-independent) variable that designates the groups being compared
What happens to the sample variance as the df increases?
The larger the df, the better the sample variance represents the population variance, and the better the t statistic approximates the z score
What is the major difference between a non-directional and a directional hypothesis test for independent measures designs?
The null and alternative hypothesis are set up with < or > or ≥ or ≤ and you conduct a one tailed not a two tailed test
What does η2 compute?
The percentage of variance accounted for by the treatment conditions
In an independent measures test situation where the sample sizes are different, what formula can be used to fix this?
The pooled variance: sp2= SS1 + SS2/df1 + df2
What does the sample mean M approximate?
The population mean µ
For the tests of effect size, which one is different from the t test that is used in anova?
The r2 for the t test is conceptually the same as the η2
What is the general difference in shape from the normal distribution and the t distribution?
The t distribution tends to be flatter and more spread out, whereas the normal z distribution has more of a central peak
How is the t distribution table read?
The two rows at the top of the table show proportions of the t distribution contained in either one or two tails, depending on which row is used. The first column of the table lists degrees of freedom for the t statistic. Finally, the numbers in the body of the table are the t values that mark the boundary between the tails and the rest of the t distribution.
What is the symbolized alternative hypothesis for a one way anova?
There are so many combinations, we simply state: There is at least one mean difference among the populations
What is the purpose of an independent measures research design?
To evaluate the mean difference between two populations (or treatment conditions)
Why do we use the z-score statistic?
To quantify our inferences about the population (comparing the obtained sample mean with the hypothesized population mean)
Notation: HSD
Tukey's Honestly Significant Difference
When the population variance is not known, what information can be used in its place?
Use the corresponding sample value in its place
Describe the within treatments variance and that I measures.
Variability that exists within each sample; measures this individual differences inside each sample; measures differences caused by random unsystematic factors
Describe the between treatments variance and what it measures.
Variability that results from general differences between the treatment conditions; measures differences among the sample means; measures differences caused by systematic treatment effects and random, unsystematic factors
In words, what is the formula for the F ratio for a one way anova?
Variance between sample means/variance expected with no treatment effect
Describe the denominator of the F ratio in one way anova (variance no treatment and why its used).
Variance in the denominator measure the mean differences that ould be expecte if there is no treatment effect
Describe the numerator of the F ratio in one way anova (the variance and why its used).
Variance is used to define and measure the size of differences among the sample means
What is a matched-subjects research design and why is it used?
When subjects are matched based on various factors (age, IQ, gender, etc), used to try to approximate the advantage of a repeated measures design
What are random effects anova?
When the levels of the independent variable are selected at random from a larger set
What is the information that leads to the problem with z scores?
Z scores require knowledge of the population standard deviation, and researchers usually have only the sample data available
Give a generalized APA formatted results sentence, with all the computations needed.
__ appears to have a statistically significant effect on __, t(dftot) = 1.00, p <.05, 95% CI: 3.01 - 12.04, Cohen's d = 1.00, r2 = .01. individual numbers vary, as well as what is reported.
What is a factorial design?
a study that combines two or more factors
Notation H1
alternative hypothesis
Define the estimated standard error.
an estimate of the real standard error, σM , when the value of σ is unknown. It is computed using the sample variance or sample standard deviation and provides an estimate of the standard distance between a sample mean, , and the population mean, μ.
Notation: d
cohen's d
Notation: tcrit
critical value for the t statistic
Notation: q
critical value in the studentized range distribution
Notation: df
degrees of freedom
Notation: dfbet
degrees of freedom between treatments
Notation: dfw
degrees of freedom within treatments
How do you calculate the degrees of freedom for a related samples design?
df = n-1
What is symbol for degrees of freedom and how do you calculate it?
df; n-1
Notation: D
difference score- related samples
Notation: η2
effect size for ANOVA (like the r2)
Notation: sM
estimated standard error for sample
Notation: σM
estimated standard error for the population
Notation: s(M1-M2)
estimated standard error for two independent samples
Notation: sMD
estimated standard error with related samples
Notation: f
frequency
Notation: G
grand total of all scores in the study
Notation: M1
indicates which sample mean
Notation: n1
indicates which sample the number of scores (n) is for
Notation: df1
indicates which sample's degrees of freedom
Notation: SS1
indicates which sum of squares
Describe why we use sample variance to calculate the estimated standard error.
it is an unbiased statistic; on average, the sample variance provides an accurate and unbiased estimate of the population variance . Therefore, the most accurate way to estimate the standard error is to use the sample variance to estimate the population variance.
What do you do if your df is not on the t table?
look up the critical t value for both the one smaller and the one larger and then use the larger t value. If your sample t statistic is greater than the larger value listed,
Notation: µlower
lower interval estimate- related samples
Notation: LRL
lower real limit
Notation: MSw
mean squares within treatments
Notation: H0
null hypothesis
Notation: N
number of scores in a population set
Notation: n
number of scores in the sample set
Notation: k
number of treatment conditions
Define counterbalancing in a repeated measures study.
participants are randomly divided into two groups, with one group receiving treatment 1 followed by treatment 2, and the other group receiving treatment 2 followed by treatment 1
Notation: X
particular variable score
Notation: r2
percentage variance accounted for by treatment
Notation: sp2
pooled variance
Notation: µ
population mean
Notation: σ
population standard deviation
What is the formula for Tukey's HSD?
q√MSwithin/n
What is the symbol for sample variance?
s2 (s squared)
What is the formula for the Hartley's F-max test?
s2(largest)/s2(smallest) if it's a large value, then the assumption is violated
What is the formula to compute the estimated standard error in a related samples t test?
sMD = √s2/n
How do you calculate r2 for a related samples design?
same as for independent design: t2/ t2 + df
What is the formula to compute sample variance for a related samples t test?
same as normal: SS/n-1
Notation: M
sample mean
Notation: s
sample standard deviation
Notation: s2
sample variance
Notation: T
sum of scores for each treatment
Notation: SS
sum of squares
Notation: SSbet
sum of squares between-treatments
Notation: SStot
sum of squares total
Notation: SSw
sum of squares within treatments
Notation: t
t-statistic
What is the formula for calculating r2 in an independent samples t test?
t2/t2 + df
Describe the t distribution.
the complete set of t values computed for every possible random sample for a specific sample size (n) or a specific degrees of freedom (df). The t distribution approximates the shape of a normal distribution, especially for large samples or samples from a normal population.
Describe degrees of freedom.
the number of scores in a sample that are independent and free to vary. Because the sample mean places a restriction on the value of one score in the sample, there are degrees of freedom for a sample with n scores
What is the Sheffe's test?
the safest of all post hoc, uses an F ratio to evaluate significance of the difference between two treatment conditions.
What is the primary disadvantage of a repeated measures design?
the structure of the design allows for factors other than the treatment effect to cause a participant's score to change from one treatment to the next
What is the difference between the t statistic and the z-score formulas?
the z-score uses the actual population variance, σ2 (or the standard deviation), and the t formula uses the corresponding sample variance (or standard deviation) when the population value is not known.
If you are calculating a t, and the table doesn't list the df you have, why would you use the larger critical value listed?
then you can be certain that the data are in the critical region, and you can confidently reject the null hypothesis
How many t distributions are there?
there is actually a different sampling distribution of t for each possible number of degrees of freedom.
Notation: dftot
total degrees of freedom
Notation: N
total number of scores
Notation: µupper
upper interval estimate- related samples
Notation: URL
upper real limit
What is the formula to compute the s2 in a related samples test?
variance of D = s2 = SS/df
What do we do in a hypothesis testing situation with the t statistic?
we begin with a population with an unknown mean and an unknown variance, often a population that has received some treatment. The goal is to use a sample from the treated population (a treated sample) as the basis for determining whether the treatment has any effect.
Notation: z
z-score
What is the formula to construct the confidence interval estimate for the independent measures test?
µ1 - µ2 = (M1 - M2) - tcrit s(M1-M2) then the same formula only add the means to the t(s) values
What is the formula to construct the confidence interval estimate in a repeated measures t test?
µlower = MD - tcritsMD the upper is the same only plus instead of minus
What is the formula to calculate SSwithin-treatments?
∑SSinside each treatment
What is the formulat to calculate SStotal?
∑X2 - G2/N