Midterm #2 Flashcards
single sample t-test vs. single sample z-test
single sample z-test –> known population mean
single sample t-test –> known population mean and stdev
when do you use an independent sample t-test
- used to compare two samples
- when the participants in each sample are independent of each other
performing an independent samples t-test: step 1
choose the appropriate test:
- an independent samples t-test is used to compare two samples when the participants in each sample are independent of each other
performing an independent samples t-test: step 2
check the assumptions:
- random samples –> each sample is a random sample from its population; robust
- independence of observations –> cases within a group are not influenced by other cases within the group; not robust
- normality –> the dependent variable is normally distributed in each population; robust, especially if N is large
- homogeneity of variance –> assumes the amount of variability in two populations that we would like to compare is equal; robust, especially if N is large
performing an independent samples t-test: step 3
state the hypothesis:
- null hypothesis:
H0: μ1 = μ2 OR μ1 - μ2 = 0
- alternative hypothesis:
H1: μ1 ≠ μ2 OR μ1 - μ2 ≠ 0
performing an independent samples t-test: step 4
decision rule:
- construct a sampling distribution of the differences between means (assuming the null is true) –> normal sampling curve
- calculate standard error by calculating the pooled variance, then the standard error of the difference
- specify alpha (0.05)
- determine the t-critical values using table (typically two tailed) –> DF = N-2, where N = total number of scores across both groups
performing an independent samples t-test: step 5
calculate the test statistic:
- t = (mean 1 - mean 2) / (standard error of the difference)
- compare calculated t-value to where it falls in reference to t-critical value
- above the t critical value (on the right) or lowere than the t-critical value (on the left) –> significant
performing an independent samples t-test: step 6
interpretation of effect size, and confidence interval:
- decide whether to reject or fail to reject the null hypothesis
- t-value in rare zone = reject null hypothesis
- t-value in common zone –> fail to reject null
- interpret the direction of effect - describe how the samples compare directionally on the dependent variable (i.e., which group scores are higher / lower)
- calculate and report the effect size (cohnen’s d)
- calculate and report the 95% confidence interval for the difference between means –> represents the range of mean differences that the true population mean difference is likely to lie within.
- lower and upper bound
paired-samples t-test vs. independent samples t-test
independent samples t-test:
- used to compare two samples when the participants in each sample are independent of each other.
paired samples t-test:
- if the participants in the two samples being compared are related to each other, you must use a paired samples t-test
- ex. the same participants were measured in both sample 1 and sample 2
performing a paired-samples t-test: calculate the difference in scores
score 1 - score 2 for participant one. this is done for the rest of the participants and calculated next to the table
benefits of the paired-samples t-test
- controls for individual differences (attributes that vary between individuals like age, gender, personality)
- participants are in each condition –> researcher is more confident that a difference between conditions is due to the IV and not because of differences in background characteristics
performing a paired-samples t-test: step 1
choose the correct test:
- a paired t-test is used to analyze the two samples when the participants in each condition are somehow related
performing a paired-samples t-test: step 2
check the assumptions:
- random samples –> the sample is a random sample from its population; robust
- independence of observations –> each case within a group or condition is independent of the other cases in that group or condition; not robust
- normality –> the population of difference scores is normally distributed; robust to violation
performing a paired-samples t-test: step 3
state the hypothesis:
- null hypothesis:
H0: μ1 = μ2 OR μD = 0
- alternative hypothesis:
H1: μ1 ≠ μ2 OR μD ≠ 0
performing a paired-samples t-test: step 4
decision rule:
- construct a sampling distribution of mean difference scores expected when the null hypothesis is true
- calculate the standard error of the sample distribution
- find the critical vlues
- specify alpha (0.05)
- two-tailed test
- df = N - 1 –> number of pairs of scores minus 1
performing a paired-samples t-test: step 5
calculate the test statistic:
- t value tells you the number above the mean
performing a paired-samples t-test: step 6
interpretation:
- decide whether to reject or fail to reject the null hypothesis
- describe the direction of the difference (i.e. which condition were the scores higher / lower on the DV)
- cohnen’s d is not an appropriate measure of effect size for a paired-samples t-test. instead, subtract the means:
- M1 - M2 = MD
- calculate 95% confidence interval
- this is the range where variable A compared with variable B falls within
between-subjects, one-way ANOVA described
analyzing the effect of an IV on a DV by comparing more than two samples
- compares all samples using a single test. does this by comparing:
- between group variability
- within group variability
between subjects, one-way anova vs. repeated-measures, one way anova
between subjects, one-way anova:
- between-subjects = participants in each sample are independent of each other
- one-way = there is one independent variable
- anova = more than two samples are being compared
repeated-measures, one way anova:
- repeated-measures = the participants in each sample are related to each other
- one-way = there is one independent variable
- anova = more than two samples are being compared
issues with comparing 2+ samples (anova): why not just perform multiple t-tests?
- each time you perform a statistical test using sample data, there is a chance of making a type I error
- each time you perform a statistical test, there is a 5% chance of making a type I error, and performing multiple tests increases these chances
why aren’t all scores within a condition the same
- individual differences (all things about the individuals that make them different from one another) –> dependent on age, experience, interests, etc.
why do sample means between conditions differ
- individual differences (differences in people) and treatment effect (if there is an effect, we would see that reflected in the means)
define within-group variability
the variability among all the scores within each sample.
- this variability is due to individual differences, which are characteristics that vary between individuals
define between-group variability
the variability among the scores between each sample
- this variability is due to individual differences plus treatment effect
- treatment effect refers to the effect of the IV on the DV
what is the ratio for between-subjects one-way anova
F = (between group variability) / (within group variability)
meaning,
F = (individual differences + treatment effect) / (individual differences)
between-subjects, one-way ANOVA: if there was no effect of the IV on the DV, what will F equal
F = 1 –> no treatment effect
F greater than 1 –> treatment effect
between-subjects, one-way ANOVA: step 1
choose the correct test:
- between-subjects, one-way ANOVA compares:
- the effect of one IV on a DV
- when there are more than two samples
- the participants in each sample are independent of one another
between-subjects, one-way ANOVA: step 2
check the assumptions:
- random samples –> each sample is a random sample from its population; robust
- independence of observations –> each case is not influenced by other cases in the sample; not robust
- normality –> the dependent variable is normally distributed in each population; robust, especially if N is large and the n’s are about equal
- homogeneity of variance –> assumes the amount of variability in two populations that we would like to compare is equal; robust, especially if N is large and the n’s are about equal
between-subjects, one-way ANOVA: step 3
state the hypothesis:
- null hypothesis:
H0: μ1 = μ2 = μ3
- alternative hypothesis:
H1: at least one population mean is different from the others
between-subjects, one-way ANOVA: step 4
decision rule:
- the sampling distribution for an anova is an f-distribution made up of the F-ratios that would occur when the null is true
- the distribution is positively skewed (big on left small on right).
- sampling distribution has this property because it can only have positive values (variance is always positive)
- calculate the F value (differentiates common zone from rare zone)
- specify alpha (0.05)
- find the F-critical value on an F-critical value table using:
- alpha and degrees of freedom
- numerator degrees of freedom (between group variability)
- (k (the number of groups) - 1)
- denominator degrees of freedom (within group variability)
- (N (total number of scores ex. 3 groups each with 4 participants) - k)
- use table to find the critical value based on what was calculated prior
between-subjects, one-way ANOVA: step 5 (overview)
calculate the test statistic:
- the formula for calculating the test-statistic is F = (MS between) / (MS within)
how do you calculate MS between
numerator
- (SS between) / (df between)
formula for SS between:
- create a table:
- first column: mean of each group separately
- take the mean of all the means (Mgrand)
- second column: mean group (first column) - Mgrand
- square Mgroup - Mgrand
- multiply (Mgroup - Mgrand)^2 by Ngroup (number of participants in each group)
- add all of these values together to get sum of squares between
- calculate the degrees of freedom –> k (number of groups being compared) - 1
to solve:
- (SS between) / (df between)
how to calculate sum of squares within
column 1: line up all of the scores in a column for each condition.
column 2: the mean of each group
column 3: column one - column 2 (its okay to have negatives)
column 4: square column three (values should be positive)
- add all of these values together to get SS within
to calculate:
MS within = (SS within) / (df –> N (total participants) - k (number of groups))
use these values to calculate F
between-subjects, one-way ANOVA: step 6
interpretation:
- decide whether to reject or fail to reject the null hypothesis
- place the F value calculated on the graph and see where it compares to the F critical value (if F value is bigger than F critical value, the data is significant).
- if the omnibus test (overall anova) is significant, H0 can be rejected
- determine the effect size (r^2)
how do you know which test is specifically significant
post-hoc test
how to determine r^2
(SS between) / (SS total –> SS between + SS within) *100
- produces a percentage which tells you the effect size
- small effect –> approximately 1%
- medium effect –> approximately 9%
- large effect –> approximately 25%
between-subjects, one-way ANOVA: step 7
- if the omnibus test (the overall anova) is significant, that tells us that at least one mean is different from another mean
- perform the tukey hsd post-hoc test
describe the tukey hsd post-hoc test
q * sqrt((Ms within) / (n))
to find q, use the post-hoc table –> the top tells you the number of groups you are comparing and the side tells you the degrees of freedom of MS within
when # of groups comparing goes up, the q value goes up, too.
n is the ammount of participants in ONE group
- the HSD is the amount by which two means must differ in order to be significantly different from one another
- subtract every mean by each other using the absolute value (ex. Mean1 - mean2, mean2 - mean3, mean1 - mean3)
- if the obtained subtracted mean values are bigger than the HSD value, than there is a significance
