Psych 2005 exam 2 ACA Ch. 6-8 Vocab

difference between means foundation

(a) randomly select one mean from the distribution of means for the first group's population, (b) randomly select one mean from the distribution of means for the second group's population, and (c) subtract. (That is, take the mean from the first distribution of means and subtract the mean from the second distribution of means.) This gives a difference score between the two selected means. Then repeat the process. This creates a second difference score, a difference between the two newly selected means. Repeating this process a large number of times creates a distribution of differences between means. You would never actually create a distribution of differences between means using this lengthy method

3 new wrinkles for a t test for independent means

: (1) the comparison distribution is now a distribution of differences between means (this affects Step 2); (2) the degrees of freedom for finding the cutoff on the t table is based on two samples (this affects Step 3); and (3) your sample's score on the comparison distribution is based on the difference between your two means (this affects Step 4

difference scores

A distinguishing feature of a t test for dependent means is that ____________ must make sense.

accuracy of estimated pop variance (of t-test for independent means)

If one sample is larger than the other, the estimate it provides is likely to be more accurate (because it is based on more information). If both samples are exactly the same size, you could just take an ordinary average of the two estimates. On the other hand, when they are not the same size, you need to make some adjustment in the averaging to give more weight to the larger sample.

power for t-test of independent means

Power for a t test for independent means can be determined using a power table, a power software package, or an Internet power calculator. The power table shown gives the approximate power for the .05 significance level for small, medium, and large effect sizes, and one-tailed or two -tailed

Type I error (alpha error)

Rejecting the null hypothesis when it is true; getting a statistically significant result when in fact the research hypothesis is not true. False Positive "you've wrongly won, and the medicine's really no good.

controversy

The controversy is about how cautious to be and about how few is "only a few." One reason there is room for controversy is that, in most cases, the many comparisons being made are not independent; the chance of one coming out significant is related to the chance of another coming out significant.

N-1

The df for dependent means t test is...

0

The mean of the comparison distribution in a study of dependent means is ordinarily ___

s^2 pooled formula

The pooled estimate of the population variance is the degrees of freedom in the first sample divided by the total degrees of freedom (from both samples), multiplied by the population estimate based on the first sample, plus the degrees of freedom in the second sample divided by the total degrees of freedom multiplied by the population variance estimate based on the second sample.

Beta (β)

The probability of a Type II error.

Standard deviation of the distribution of differences between means (Sdifference) formula

The standard deviation of the distribution of differences between means is the square root of the variance of the distribution of differences between means.

t Score for the Difference Between the Two Actual Means

The t score is the difference between the two sample means divided by the standard deviation of the distribution of differences between means.

total degrees of freedom(dfTotal)

The total degrees of freedom for a t-test for independent means is the degrees of freedom in the first sample plus the degrees of freedom in the second sample. used to find the variance of distribution

Monte Carlo Method

The use of randomly generated numbers as part of an algorithm When Mathematics Becomes Just an Experiment, and Statistics Depend on a Game of Chance The result would be a more extreme level of significance than is truly warranted. There are special procedures for handling situations like this, called multilevel modeling, seriously distorts the results of a standard t test

Slightly smaller

The variance of a sample will generally be _________________ than the variance of the population from which it is taken.

null hypothesis of repeated measures design

There is no difference between the two groups of scores the mean of the population of the difference scores is 0

goal of t-test for independent means

To decide whether the difference between means of your two actual samples is a more extreme difference than the cutoff difference on this distribution of differences between means.

t test for independent means assumption

To do a t test for independent means, it has to be reasonable to assume that the populations the two samples come from have the same variance (which, in statistical terms, is called homogeneity of variance). (If the null hypothesis is true, they also have the same mean

logic of t-test for dependent vs. independent means

To remember which is which, think of the logic of each t test. The t test for dependent means involves difference scores. So, its comparison distribution is a distribution of means of difference scores. The t test for independent means involves differences between means. Thus, its comparison distribution is a distribution of differences between means.

test identical twins

What approach did Gosset (the beer-testing t test creator) suggest to greatly improve on the Lanarkshire milk experiment methodology?

use of estimated d

When you have the results of a completed study, you estimate the effect size as the difference between the sample means divided by the pooled estimate of the population standard deviation (the square root of the pooled estimate of the population variance). You use the sample means because they are the best estimate of the population means, and you use Spooled because it is the best estimate of σ

William Gosset

Who worked for Guinness and tested beer samples using the T test, after graduating from Oxford University with Math and Chem degrees, inventor of t-test?

The SD of difference scores tends to be smaller

Why do repeated measures studies often have much larger effect sizes (and more power) than other kinds of research designs?

effect size

a standardized measure of difference (lack of overlap) between populations. increases with greater differences between means. how much something changes after a specific intervention how much two populations are separated due to experimental procedure

weighted average

average in which the scores being averaged do not have an equal influence on the total, as in figuring the pooled variance estimate in a t-test for independent means. what proportion of the total degrees of freedom each sample contributes and multiply that proportion by the population variance estimate from that sample. Finally, you add up the two results, and that is your weighted, pooled estimate.

variance of a distribution of means(S^2m)

based on an estimated population variance is the estimated population variance divided by the number of scores in the sample.

SD of the distribution of means Sm

based on an estimated population variance is the square root of the variance of the distribution of means based on an estimated population variance.

Pooled estimate of pop variance S^2pooled

best estimate of pop variance for Bothe populations

A t test for independent means is usually described in a research article

by giving the means (and sometimes the standard deviations) of the two samples, plus the usual way of reporting any kind of t-test

assumption

condition, such as a population's having a normal distribution, required for carrying out a particular hypothesis-testing procedure; a part of the mathematical foundation for the accuracy of the tables used in determining cutoff values.

small effect size

d = 0.2

Medium effect size

d = 0.5

Large effect size

d = 0.8

difference scores

difference between a person's score on one testing and the same person's score on another testing; often an after-score minus a before-score, in which case it is also called a change score

distribution of differences between means

distribution of differences between means of pairs of samples such that, for each pair of means, one is from one population and the other is from a second population; the comparison distribution in a t-test for independent means A distribution of differences between means is, in a sense, two steps removed from the populations of individuals: First, there is a distribution of means from each population of individuals; second, there is a distribution of differences between pairs of means, one of each pair of means taken from its particular distributions of means.

comparison distribution for t-test for independent means

distribution of means of difference scores not difference scores

assumptions for t-test for independent means

each of the population distributions is assumed to follow a normal curve. In practice, this is only a problem if you have reason to think that the two populations are dramatic same for any t-test -two pops have the same variance (homogeneity of variance) -scores are independent of each other

Biased estimate

estimate of a population parameter that is likely systematically to overestimate or underestimate the true value of the population parameter. Ex: SD^2 would be __ of the population variance( it would systematically underestimate it.)

unbiased estimate of the population variance(S^2)

estimate of the population variance, based on sample scores, which has been corrected so that it is equally likely to overestimate or underestimate the true population variance; the correction used is dividing the sum of squared deviations by the sample size minus 1, instead of the usual procedure of dividing by the sample size directly.

T-test accurate

even when there are fairly large differences in the population variances, particularly when there are equal or near equal numbers of scores in the two samples.

Robustness

extent to which a particular hypothesis-testing procedure is reasonably accurate even when its assumptions are violated

Type II error (beta error)

failing to reject the null hypothesis when in fact it is false; failing to get a statistically significant result when in fact the research hypothesis is true. False negative you wrongly lose, and the patient won't get something she should."

t test for independent means used for

focuses on the difference between the means of the two groups when the scores in one group are for different people than the scores in the other group, what you can compare is the mean of one group to the mean of the other group.

t test for a single sample

hypothesis-testing procedure in which a sample mean is being compared to a known population mean and the population variance is unknown

t-test

hypothesis-testing procedure in which the population variance is unknown; it compares t scores from a sample to a comparison distribution called a t distribution.

t test fir dependent means

hypothesis-testing procedure in which there are two scores for each person and the population variance is not known; it determines the significance of a hypothesis that is being tested using difference or change scores from a single group of people.

t test for independent means

hypothesis-testing procedure in which there are two separate groups of people tested and in which the population variance is not known

t-test misleading

if (a) the scores in the samples suggest that the populations are very far from normal, (b) the variances are very different, or (c) there are both problems. In these situations, there are alternatives to the ordinary t test procedure,

S^2 pooled mistake

if it does not come out between the two estimates of the population variance. (You also know you have made a mistake if it does not come out closer to the estimate from the larger sample.

Standard deviation of the distribution of differences between means (Sdifference)

in a t test for independent means, square root of the variance of the distribution of differences between means

Pooled estimate of the population variance (S^2pooled)

in a t test for independent means, weighted average of the estimates of the population variance from two samples (each estimate weighted by the proportion consisting of its sample's degrees of freedom divided by the total degrees of freedom for both samples).

decision errors

incorrect conclusions in hypothesis testing in relation to the real (but unknown) situation, such as deciding the null hypothesis is false when it is really true possible in hypothesis testing because you are making decisions about populations based on information in samples.

t distribution

mathematically defined curve that is the comparison distribution used in a t test

Cohen's d formula

mean difference/standard deviation d=(μ1−μ2)/σ

degrees of freedom(df)

number of scores free to vary when estimating a population parameter; usually part of a formula for making that estimate—for example, in the formula for estimating the population variance from a single sample, the degrees of freedom is the number of scores minus 1. df=N-1

t score

on a t distribution, number of standard deviations from the mean (like a Z score, but on a t distribution).

variance of a distribution of differences between means(S^2difference)

one of the numbers figured as part of a t test for independent means; it equals the sum of the variances of the distributions of means associated with each of the two samples. Population 1's distribution of means plus the variance of Population 2's distribution of means. (This is because, in a difference between two numbers, the variation in each contributes to the overall variation in their difference. It is like subtracting a moving number from a moving target.)

scores independent from each other

other words, none of the scores within each group or between the groups can be paired or matched up in any way (as this would make the scores dependent on each other to some degree). (Of course we can allow that those in each group are similar, in that they have the same experimental condition or the same characteristic that put them into the two groups

Alpha (a)

probability of making a Type I error; same as significance level.

Statistical Power

probability that the study will give a significant result if the research hypothesis is true.

repeated measures design

research strategy in which each person is tested more than once; same as within-subjects design.

Effect size for the t test for independent means

same as usual The effect size is the difference between the population means divided by the population's standard deviation.

6 key details of t-test for independent means

six key details: (1) the mean of the distribution of differences between means, (2) the estimated population variance, (3) the variance of the two distributions of means, (4) the variance and standard deviation of the distribution of differences between means, (5) the shape of the distribution of differences between means, and (6) the t score for the difference between the two means being compared.

Estimated population SD formula

square root of the estimated population variance

meta-analysis

statistical method for combining effect sizes from different studies. a statistical technique that averages the results of two or more studies to see if the effect of an independent variable is reliable

steps for a t test for dependent means

step 1: restate the question as a research hypothesis and a null hypothesis about the populations (the null difference mean should be 0). step 2: determine the characteristics of the comparison distribution. - make each person's two scores into a difference score. do all the remaining steps using these difference scores. - figure the mean of the difference scores. - assume a mean of the distribution of means of difference scores of 0 (μ = 0). - the shape is a t distribution with df = N - 1. - the standard deviation of the distribution of means of difference scores is figured as follows: a. figure the estimated population variance of difference scores (S² = SS/df) b. figure the variance of the distribution of means of difference scores (S²M = S² / N) c. figure the standard deviation of the distribution of means of difference scores (SM = √S²M) step 3: determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected. - decide the significance level and whether to use a one-tailed or a two-tailed test. - look up the appropriate cutoff in a t table step 4: determine your sample's score on the comparison distribution: (t = (M - μ) / SM step 5: decide whether to reject the null hypothesis: compare the scores from steps 3 and 4

Population variances

t test allows us to overcome not knowing...

power table

table for a hypothesis testing procedure showing the statistical power of a study for various effect sizes and sample sizes

t table

table of cutoff scores on the t distribution for various degrees of freedom, significance levels, and one- and two-tailed tests.

main difference between t-test for dependent vs. independent means

that the focus is now on the difference between means, so the comparison distribution is a distribution of differences between means.

Comparison of t-tests

the population variance is not known for each test, and the shape of the comparison distribution for each test is a t distribution. The single sample t test is used for hypothesis testing when you are comparing the mean of a single sample to a known population mean. However, in most research in psychology, you do not know the population mean. With an unknown population mean, the t test for dependent means is the appropriate t test when each participant has two scores (such as a before-score and an after-score) and you want to see if, on average, there is a difference between the participants' pairs of scores. The t test for independent means is used for hypothesis testing when you are comparing the mean of scores from one group of individuals (such as an experimental group) with the mean of scores from a different group of individuals (such as a control group)

t test for independent means estimate pop variance (S^2pooled)

the sum of squared deviation scores divided by the degrees of freedom (the number in the sample minus 1)

estimated population variance (S^2) formula

the sum of the squared deviation scores divided by the number of scores minus 1.

t test for dependent means used for

with scores from pairs of research participants, considering each pair as if it were one person, and figuring the difference score for each pair.

Null hypothesis for t-test for independent means

μ1=μ2 if null is true the two population means are the same if null is true, distribution of differences between means has a mean of 0.