5 steps of hypothesis testing:
1. Restate the question as a research hypothesis and a null hypothesis about the populations
2. Determine the characteristics of the comparison distribution (sample score or extreme score)
3. Determine the cutoff sample score on the comparison distributio
Statistically significant
(say when you reject the null)
Directional hypothesis
predict a particular direction of difference between populations
one-tailed
directional
two-tailed
non-directional
distribution of means
distribution of means of samples of a given size from a population
mean of a distribution of means
...
rules for characteristics of distribution of means
1. The mean of a distribution of means is the variance of the population of individuals divided by the number of individuals in each sample (Um=M)
2. The variance of a distribution of means is the variance of the population of individuals divided by the n
Standard error
the SD of the distribution of means, tells about how much means of samples differ from the population mean
Z-score for samples mean on the distribution of means
Z=M-Um/Om
Hypothesis testing with more than one person?
The comparison distribution is a distribution of means
Population mean unknown
best estimate is the sample mean, more likely to come from a population with the same mean than from any other population
Confidence interval
the range of scores that is likely to include the true population mean, the confidence interval changes around the true mean, 95% and 99% usually used
CL=M+-(Z(1-P/2)(Om)
Confidence level
upper and lower value of a confidence interval
95% confidence interval
range of values that you are 95% confident includes the population mean, estimated based on the scores in a sample
Basis for 95% confidence
lower confidence is limit is the point which a true population any lower would not have a 95% probability of including a sample with our mean, same for upper
theory vs hypothesis
plausible or scientifically acceptable general principle, expect to find between 2 or more variables
types of inferential statistics
classical or frequentist(based on sample data) and baysian
Decision errors
happen because you don't know the true population mean
Type 1 error
reject the null when it is really true (alpha, the lower the alpha the smaller chance of type 1)
Type 2 error
fail to reject the null when in fact it is false (beta, 100% minus power)
Effect size
standardized measure of difference (lack of overlap) between populations, effect size increases with greater differences between means
d= U1-U2/O
two factors determine effect size
1. difference between known and predicted population means (larger effect size=more power, less overlap, cutoff is greater)
2. population standard deviation (larger effect size=more power, distribution of means narrower and less overlap, cutoff is greater
Effect size conventions (Cohen's)
standard rules about what to consider a small, medium, and large effect size, helps compare the effect size to what is typically found in psychology research
Meta-analysis
statistical method for combining effect sizes from different studies, comes up with average effect sizes across studies and compares effect sizes for different subgroups of studies
standardized effect size
makes the results of studies using different measures comparable
Statistical-power
probability that the study will give a significant result 'if' the research hypothesis is true, helps configure how many participants needed and differ between statistically significant and not
power tables
hypothesis testing procedure showing the statistical power of a study for various effect sizes and sample sizes
Power
The bigger the difference that your theory or previous research says you should expect between the two populations, the more power there is in a study
Overlap
The less overlap between two distributions the more likely it is that the study will give a significant result
Sample size affects power
The larger the sample size, The smaller the SD of the distribution of means becomes
Narrow distribution of means
Population of individuals may have small SD or large sample size, makes the distributions of means narrower and less overlap
ways to increase power of a planned study
1. increase effect size by increasing the predicted difference between population means
2. increase effect size by decreasing the population SD
3. increase sample size*
4. use a less extreme level of significance (p<.10)
5. use one tailed test
6. use a mo
disadvantages of increasing power of planned study
1. more lenient significance level- increases chances of type 1 error
2. one tail- may not be appropriate and if results come out opposite=nonsignificant
statistically significant vs practically significant
ss means can be confident the effect would be unlikely to happen if null were true, does not mean large effect
when is a sample variance a biased estimate of the variance of the population the sample is taken from?
the sample variance will in general be slightly smaller than the variance of the population the sample is taken from
Single sample t test
� Known null population mean, population variance is unknown
� Biased estimate- estimate of a population parameter that is likely systematically to overestimate or underestimate the true value of the population parameter
� Degrees of freedom- number of sc
Dependent t-test
� Use when you have two scores for each participant and the population variance is unknown
� Repeated measure design- research strategy where each person is tested more than once; same as within subject designs
� Assume population mean is 0. You are compa
Independent t-test
� The goal of a t test for independent means is to decide whether the difference between the means of your two actual samples is a more extreme difference than the cutoff difference on this distribution of differences between means.
� Focus now on the dis
Dependent vs. independent
� Dependent- compare two sets of scores from a single group of people
� Independent- compare two sets of scores, one from each of two entirely separate groups of people
Z test vs. T test
� T test- We don't know the population variance
� Z test- when we know the population variance
� Estimate using sample variance
� Comparison distribution is a t distribution
� Use a single sample t-test for comparing sample from population
5 steps for independent t test
1. Restate the question as a research hypothesis and a null hypothesis about the populations
2. Determine the characteristics of the comparison distribution
a) Estimated population variances: S2=SS/(N-1)
b) Pooled estimate of population variance:
S2pooled
5 steps for dependent t test
1. Restate the question as a research hypothesis and a null hypothesis about the populations
2. Determine the characteristics of the comparison distribution:
a. Make each persons into difference scores
b. Find mean of difference scores
c. Assume mean of d
5 steps for single sample t test
1. Restate the question as a research hypothesis and a null hypothesis about the populations
2. Determine the characteristics of the comparison distribution:
a. Estimated population variance: S2=SS/df
b. Variance of distribution of means S2m=S2/N
c. SD of
differences in hypothesis testing for independent vs. dependent
1. the comparison distribution for independent means is a distribution of differences between means
2. the degrees of freedom for independent means is the sum of the degrees of freedom for the two samples
3. the t score for a independent means is based on
effect size for independent means
d=M1-M2/S-pooled
assumptions for the t test for independent means
1. the two populations are normally distributed
2. the two populations have the same variance- you make a pooled estimate of the population variance
what is the assumption of independent scores
the scores within and between the two groups cannot be matched or paired in any way
ANOVA
analysis of variance, hypothesis testing procedure for studies with three or more groups, used when comparing means of samples from more than one population
F ratio
ratio of the between groups population variance estimate to the within groups population variance estimate
within groups population variance estimate
based on the variation among the scores in each of the samples, not affected by null because the variation within each population is not affected by whether the population means differ
between groups population variance estimate
based on the variation among the means of the samples, larger when null is false because the variation among the means of the populations is greater when the population means differ
two sources of variation contribute to the between groups population variance estimate
1.variation among the scores in each if the populations
2.varation among the means of the populations
why F ratio 1 when null true, larger than 1 when null false?
both estimates are based on entirely on the same source of variation... when null is false influences by variation among the means of the populations, the between groups estimate will be bigger than the within groups estimate
steps for analysis of variance:
1.restate hypothesis
2.determine characteristics
-DFbetween (Ngroups-1) and DFwithin(df1+df2+...dflast)
3.determine cutoff sample score
4.determine sample score on comparison distribution
--estimate variance for dist. of means S2m=E(M-GM)2/DFbetween
a.fig
3 assumptions for ANOVA
populations are assumed each to be normally distributes with equal variance(homogeneity) and the scores are independent of each other
equal variance assumption
to justify averaging the estimates from each sample into an overall within groups population variance
rule when violations of the equal variance assumption likely to lead to inaccuracies?
when the variance estimate from the group with the largest estimate is more than 4 or 5 times the smallest variance estimate
Planned comparison
a focused comparison of two groups in an overall analysis of variance that the researcher planned in advance of the study,more likely to be of theoretical/practical interest than overall difference among means
-bonferroni correction- with multiple planned
Post Hoc comparison
comparisons figured after an analysis of variance, exploratory procedure to see the patterns among populations
-don't use bonferroni correction because the corrected significance level is too extreme to find a significant result.
-Scheffe test- for any nu
The purpose of ANOVA is to distinguish between two alternatives:
1.Differences between groups are simply due to chance.
2.The differences between groups are significantly greater than can be expected by chance alone
causes variance
- Error (e.g., measurement)(between)
- Random Variability (within)
- Variables of Interest (within)
F distribution
� Distribution of ratios of variance estimates
� Variance is always positive
� Therefore, F will always be positive (F > 0) - Floor effect!
- Always "one tailed"
� Large area near 1 - Why?
- If null is true...
between group ? within group
F?1
factorial research design
the effect of two or more grouping variables is examined at once by making groupings of every combination of the variables, it is more efficient also makes it possible to see if there are interaction effects
main effect vs interaction effect of factorial design
main- the effect of one go the grouping variables. ignoring the pattern of results on the other grouping variables
interaction- is the different effect of of one grouping variable according to the level of the other grouping means