Stats Unit 2

Degrees of freedom

-df=N-1 where N is sample size

How many samples are in a paired samples t test?

two (same participants)

New assumption for independent samples t test

two populations from which the samples are selected must have equal variances

Independent samples t test reporting the statistics

-t(df)=t calc, p (<>) .05

A hypothesis test has a power of 0.7, what is the probability that the test will result in a type II error?

0.3 because you do 1-beta, which is 0.3

Medium effect size

0.5

When n is small, the t distribution is _____ and _______ spread out than normal distribution

flatter, more spread out

p>.05

there is a difference between means

95% confidence interval

-constructed around H0: where we expect the sample mean to fall 95% of the time -constructed around H1: where the value of the true mean is expected to fall 95% of the time

What might be an advantage of using a within samples t test compared to an independent samples t test?

-distribution of mean differences rather than the distribution of differences between means -sample size -df -power

How do we increase power?

-increase alpha (makes critical values less extreme) -turn a two tailed test into a one tailed test -increase sample size -decrease standard deviation in the population (reduce variability) -increase the difference between the means

Paired samples t test power calculation

-use z distribution for calculations -find raw score of critical value under H0 -compute z scores for that value under H1

r squared small effect size

0.01

r squared medium effect size

0.09

Small effect size

0.2

r squared large effect size

0.25

Large effect size

0.8

Calculating Confidence Intervals

1.) Draw a picture of a distribution that will include confidence intervals 2.) Indicate the bounds of the CI on the drawing 3.) Determine the z stats that fall at each line marking the middle 95% 4.) Turn the z stat back into raw means 5.) Check that the CIs make sense

Calculating the estimated population SD

1.) calculate the sample mean 2.) use sample mean in the corrected SD formula (s formula)

Calculating Power

1.) determine information needed to calculate (pop mean, pop SD, sample mean, sample size, standard error based on sample size) 2.) determine a critical z value and raw mean to calculate power 3.) calculate power: the % of the distributions of the means for population 2 that falls above the critical value

Calculating the CI for single sample t test

1.) draw a picture of the distribution 2.) indicate the bounds 3.) look up the t stat 4.) convert the t value into a raw mean

Independent samples t test calculating CI

1.) draw normal curve with sample difference between means in center 2.) put bounds of CI on ends, write percents under segments of curve 3.) look up t values for upper and lower ends of CIs in the t table 4.) convert t values into raw differences 5.) check -(Mx-My) plus or minus t*(Sdiff)

Paired samples t test steps for calculating CI

1.) draw normal curve with sample difference between means in the center 2.) indicate bounds on curve 3.) look up t values for upper and lower CI in the t table 4.) convert t scores into raw differences 5.) check answer -95% CI: Mdiff plus or minus t*(Smdiff)

Steps to calculate paired samples t test

1.) populations, distributions, and assumptions 2.) null (mew 1 = mew 2 or mew 1 -mew 2 = 0) and research hypothesis 3.) characteristics of comparison distribution 4.) critical values based on df, number of difference scores - 1 5.) test stat 6.) make decision

What is the measure of effect size that assesses the difference between two means in terms of standard deviation?

Cohen's d

Calculating effect size

Cohen's d: estimates effect size; assesses the difference between means using standard deviation instead of standard error

What is the comparison distribution for a single samples t test?

Distribution of means

What is the comparison distribution for a z test?

Distribution of means

True or False: the null hypothesis states that the sample mean (after treatment) is equal to the population mean

False

True or False: we construct our 95% CI around the null hypothesized mean value

False, it is around the sample mean

True or False: the larger the sample variance, the greater the likelihood of rejecting the null hypothesis

False, larger variance means smaller t stat which means smaller likelihood of rejecting the null

True or False: for a paired samples t test, df = N1-N2 - 2

False, that is for an independent samples t test

How many samples are in a single samples t test?

One

How many samples are in a z test?

One

What is the difference between z and t tests?

T tests use estimated SD while z tests uses actual SD

True or False: For a hypothesis test using the t stat, the bounds of CI will change if sample size changes

True

True or False: If the 90% CI for mew is from 40 to 50, then the sample mean is M=45

True

True or False: When a population variance or SD is unknown, you must use a t stat instead of a z stat during hypothesis testing

True

True or False: a type I error occurs when a treatment has no effect but the decision is to reject the null hypothesis

True

True or False: the null is stated in terms of the population even though the data comes from the sample

True, the null is always stated in terms of the population

True or False: two samples from the same population with the same size and mean will have different t statistics

True, they will likely have different standard deviations and therefore different t stats

If there is a small effect size, what is the likely outcome for a hypothesis test evaluating the treatment?

Type II error because you are unlikely to reject the null hypothesis

Interval estimate

a range of sample statistics we would expect if we repeatedly sampled from the same population

Sample SD is a ________ estimator

biased; it underestimates the population's SD

How do you calculate effect size for a single sample t test?

d=(M-mew)/s

Paired samples t test effect size

d=(Mdiff - mewdiff)/s diff

Statistically significant

data differs from what we would expect by chance if there was no difference; does not mean the same thing as meaningful

Increasing effect size

decrease the amount of overlap between 2 distributions -their means are farther apart -the variation within each population is smaller

Critical values in the single sample t test are based on

degrees of freedom

What is the comparison distribution for a independent samples t test?

distribution of differences between means

What is the comparison distribution for a paired samples t test?

distribution of mean difference scores

The t distribution is a

distribution of means

T-distribution

distribution of means when the parameters are not known; means you estimate a population's SD from a sample SD

Order effects (paired samples t test)

how a participants behavior changes when the dependent variable is presented for a second time

Interpretation of the CI

if we were to sample 5 students from the same population over and over, the 95% CI would include the population mean 95% of the time

Effect size

increasing sample size will make us more likely to find a statistically significant effect -smaller variability in the sampling distribution -more likely that a value will fall within the rejection range

Confidence interval (usually 95%)

interval estimate that includes the mean we would expect for the sample stat a certain percentage of the time if we were to sample from the same population repeatedly -range around the mean when we add and subtract a margin of error -confirms findings of hypothesis testing and adds more detail

What would make a small treatment effect be statistically significant?

large sample size and small variance

How do you write the null hypothesis for a single sample t test?

mew=x

Counter balancing

minimizes order effects by varying the order presentation of different levels of the independent variable from 1 participant to the next -can reduce order effects in within-groups research designs

Dependent samples t test generally results in a ______ powerful stat test bc dependent samples t test procedures reduce the variability in the sampling distribution

more

p<.05

no difference between means

As sample size increases, the distribution looks more ______, and the critical value

normal; would decrease or move closer to zero

Variance accounted for

r squared

Independent samples t test proportion of variance explained

r squared = n(M-mew)^2/s^2(n-1)+(M-mew)^2

Independent samples t test proportion of variance

r squared= t squared obs/ t squared + df

Paired samples t test variance accounted for

r^2=tobs^2/tobs^2 + df

What is effect size?

size of a difference that is unaffected by sample size; standardization across studies -less overlap: larger effect size and less variability

Point estimate

summary statistic where 1 number serves as an estimate of the population (mean)

Statistical power

the measure of our ability to reject the null hypothesis given that the null is false; the probability that we will reject the null when we should; the probability that we will avoid a type II error

How many samples are in a independent samples t test?

two (different participants)

Paired samples t test

two sample means and a within groups design (reduced variability) -we must create difference scores for every participant

Independent samples t test effect size

use cohen's d -d=(Mx-My)-(mew x - mew y)/s pooled

Calculating standard error for the t stat

use the estimated standard deviation -t=Sm

Independent samples t test

used to compare two means in a between-group design (each participant is in only one condition)

The t statistic

when sample size increases, s approaches SD and t and z become more equal

Smaller sample size means a

wider, flatter t distribution

We always use _____ scores when doing our power calculations

z