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1

central limit theorem

the theorem that specifies the nature of the sampling distribution of the mean

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central limit theorem definition

given a population with a mean μ and a variance σ², the sampling distribution of the mean will have a mean equal to μ and a variance equal to σ²/N.

The distribution will approach the normal distribution as N, the sample size, increases (this sentence is true regardless of the shape of the population distribution)

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if you increase the sampling size,

the distribution becomes more normal (bell-shaped)

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4

5 factors that affect whether the t-value will be statistically significant

the difference between X̄ and μ

N (sample size)

S (standard deviation)

α

one-tailed vs two-tailed tests

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the difference between X̄ and μ

the greater the difference, the more likely you'll have statistical significance

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N

the greater the sample size, the more likely you'll have statistical significance

critical value becomes smaller

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7

S

the greater the standard deviation, the less likely you'll have statistical significance

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α

the greater the alpha (the larger the rejection region), the more likely you'll have statistical significance

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one-tailed vs two tailed tests

if you're using a one-tailed test and you're right about the direction, you're more likely to have statistical significance because you've doubled the area of the rejection region

if you're using a one-tailed test and you're wrong about the direction, you'll never have statistical significance because the area of the rejection region is 0

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effect size

the difference between 2 populations divided by the standard deviation of either population

it is sometimes presented in raw score units (and sometimes in standard deviations)

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effect size in terms of standard deviations

d^= X̄ - μ / S

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guidelines for interpreting d^

trivial, small, medium, large effect size

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< 0.2

trivial effect size

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≥ 0.2 & < 0.5

small effect size

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≥ 0.5 & < 0.8

medium effect size

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≥ 0.8

large effect size

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confidence interval

an interval, with limits at either end, having specified probability of including the parameter being estimated

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Confidence Interval

X̄± t_df * S/√N

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what makes a confidence interval wider?

smaller α (0.05 to 0.01)

larger s

smaller N

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20

what makes a confidence interval narrower?

bigger α

smaller s

bigger N

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21

related samples

an experimental design in which the same subject is observed under more than one treatment

ex: comparing students' two sets of quiz scores (DV) after they used two different studying strategies

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repeated measures

data in a study in which you have multiple measurements for the same participants

ex: measuring a person's cortisol level before and after a competition

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matched samples

an experimental design in which the same subject is observed under more than one treatment

ex: asking a husband and wife to each provide a score on marital satisfaction

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d‾

the difference of the means

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S_D

the standard deviation of the difference scores

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difference scores (gain scores)

the set of scores representing the difference between the subjects' performance on two occasions

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advantages of related samples

it avoids the problem of person-to-person variability

it controls for extraneous variables

it requires fewer participants than independent samples designs to have the same amount of power

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disadvantages of related samples

order effect

carry-over effect

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order effect

the effect on performance attributable to the order in which treatments were administered

ex: you want to know if stimulant A or B improves reaction time more

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carry-over effect

the effect of previous trials (conditions) on a subject's performance on subsequent trials

ex: you want to know if Drug A or B improves depression more

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independent-sample t tests

are used to compare two samples whose scores are not related to each other, in contrast to related-samples t tests

ex: comparing male and female grades on a language verbal test

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two assumptions for an independent-samples t test

homogeneity of variance

the samples come from populations with normal distributions

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homogeneity of variance

the situation in which two or more populations have equal variances

the variance of a sample is similar to another

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heterogeneity of variance

a situation in which samples are drawn from populations having different variances

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homogeneity assumption is for

population variances, not sample variances

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the samples come from populations with normal distributions

however, independent-samples t tests are robust to a violation of this assumption, especially with sample sizes less than 30

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pooled variance

a weighted average of separate sample variances

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standard error of difference between means

the standard deviation of the sampling distribution of differences between means

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sampling distribution of differences between means

the distribution of the differences between means over repeated sampling from the same populations

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degrees of freedom for one-sample t test

N-1

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degrees of freedom for related-samples t test

N-1

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degrees of freedom for independent-samples t test

n1+n2-2 or N-1

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formula for pooled variance

^

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formula for independent-samples t test

^

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counterbalancing

how to solve the problems of order and carry-over effect (disadvantages)

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