stat midterm 2
s^2 pooled
(df1* s^2 sub1)/df total +(df2*s^2 sub 2)/df total
test statistic formula
(sample statistic-population parameter)/standard error of sample statistic; xbar-mew/Ssubxbar
variable criteria for rejecting null in independent sample
(xbar1-xbar2)cv= tcv(s subxbar1-sbar2)
steps recommended for evaluating the null hypothesis
1. decide level of significance 2. decide if its directional or nondirectional 3. outline the critical value of the test statistic and sample statistic
steps to interpreting results of sample statistic
1. decide whether or not statistically significant 2. if so, calculate the effect size, consider practical significance 3. state results in plain english
steps to eyeball estimate the confidence interval of a sample histogram
1. eyeball estimate the mean with balance point 2. eyeball estimate the standard deviation at points of inflection 3. eyeball estimate the standard error by dividing that by estimated sample size 4. for 95% confidence interval double the standard error in each direction to find the confidence interval
how is it different evaluating a test statistic in one tailed v two tailed tests?
1. hypothesis is one directional only 2. Level of significance all on one side 3. only one critical value and rejection region
4 steps of hypothesis testing
1. state the null and alternative hypothesis 2. establish criteria for rejecting the null 3. collect a sample of the sample statistic and compute the sample and test statistics 4. evaluate whether we should reject or fail to reject the null
z cv if apha is .05 and test is directioal
1.645
z cv if alpha is .05 and test is nondirectional
1.96
for sample size greater than _____ t subcv is about ____
10; 2
"rarely" in psychological statistics
5% of the the time
population proportion symbol and meaning
Pi-- same as population mean, mew
formula tobs independent sample
[xbar sub 1- xbar sub 2]/(s sub x1bar-x2bar); (sample 1 mean- sample 2 mean) divided by standard error of the difference of means
how far does the confidence interval extend for 95% confidence?
about two standard errors each direction from x bar
distribution of the variable shows...
all possible values of x if the null hypothesis is true
distribution of the test statistic shows
all possible values of z if the null hypothesis is true
probability of making type I error
alpha
level of significance
alpha or the setting of probability
95% statistical significance
alpha: .05
treatment variable
another name for independent variable which defines group memebership
effect of sample size on t obs
as sample size increases, t obs increases
why do we pool variances?
assuming the variances are equal under the null hypothesis the pooled variance of two samples is a more accurate point estimator of the population parameter
why might data in a one tailed test reject the null when it doesn't in a two tailed test? What about vice-versa?
bc the rejection region all lies on one side such the region there is bigger. but if
when do we set the critical value of the statistic?
before the experiment (research design)
probability of making type II error
beta
rejection region limits
beyond critical value(s) of the sample statistic
alternative name for type II error
bliindness error
effect size index
d= (xbar- mew)/standard deviation
dependent samples variable cv formula
dbar cv= tcrit*s sub d bar
practical significance
degree to which our result is actually important; how large the difference is in the real world importance
statistical significance
degree to which our result is not likely to be due simply to chance
descriptive statistics
describing statistical information in terms of a distribution
df pooled variance
dfpoooed is df1+df2
repeated measures design
experiment in which the same participant is measured multiple times
true experiment
experimenter has total control over the treatment variable and randomly assigns membership
if our hypothesis test falls within the null hypothesis we
fail to reject the null hypothesis and assume no change was made
standard error of the difference of two means is ___ than the ____ by a factor of ____
greater; standard error of the mean; sq rt 2
examples of what is not a true experiment
group membership defined by natural or uncontrolled occurrence such as gender, age, marital status, or height
alternative name for type I error
gullibility error
null hypothesis in a directional, single variable test
h0: x>a OR x<a
making the level of significance smaller makes the
harder to reject the null hypothesis
measure of effect size quantify
how far apart two curves are
how to tell between questions asking about confidence interval or hypothesis test?
hypothesis tests are in the form of yes/no while confidence interval might ask what could be inferred about a population when we only have sample mean
width of confidence interval is ____ to the ______ _______ the critical values
identical; width between
how do you use confidence interval for hypothesis testing?
if the mean of the null hypothesis lies in the confidence interval of the observed hypothesis
robust
insensitive to violations of the assumptions meaning that in
what is the standard error of the distribution of means?
it's the standard deviation of a single x bar divided by square root n
standard effect size; formula
magnitude of a result's distance from null hypothesis given in standard deviations; d= lxbar-mewl/sigma
raw effect size
magnitude of a result's distance from null hypothesis in the scale of original experiment
effect size
magnitude of our statistical result
making the level of significance smaller effects the rejection region how?
makes it further from the mean and thus the criterion for rejection more stringent
null hypothesis in independent sample test (two ways)
mu1=mu2 OR mu1-mu2=0
for the same sample size and same critical value, the graph of the confidence interval is ____ for known sigma vs an unknown sigma
narrower
degrees of freedom in dependent samples
number of difference scores-1
degrees of freedom
number of observations, minus one
raw effect size formula
numerator of the null hypothesis in question
difference between the critical value of sample statistic and that of the test statistic?
only the scaling such that the test statistic is standarized
sample proportion (symbol and statistical equivalent)
p-- same as sample mean, x bar
the _____ variance is always _______ variance x sub 1 and variance xsub 2
pooled; between
rejection region
region of z axis beyond critical value
proportional standard deviation (symbol and equation) for yes and no answers
s sub p= square root(p*(1-p)/n-1) where p= "proportion of no's (or yes')"
standard error of the differences of means symbo
s sub xbar1-xbar2
equation sample standard error
s/square root n
formula pooled variance with different sample sizes
s^2 (sub pooled)_= [(s^2 sub 1)(dfsub1)+(s^2 sub 2) (dfsub2)]/[f total; pooled variance is variance1 times degrees of freedom1 plus variance 2 times degrees of freedom 2 all divided by the degrees of freedom1 plus degrees of freedom
formula s^2xbar, s^2ybar in independent samples; in words
s^xbar=s^2pooled/nx, s^ybar=s^2pooled/ny; standard deviation of the mean xbar= pooled variance divided by sample size of x, variance of mean ybar= pooled variance/sample size y
tcv for independent test
same as tcv for any test: define alpha, look at TOTAL degrees of freedom and correct column for one tailed or two tailed test
effect of sample size on effect size
sample size doesn't effect effect size
inferential statistics
seeks to draw conclusions about the world from statistics
distribution of the sample statistic
shows all possible values of the sample mean x bar if the null hypothesis is true
equation population standard error
sigma/square root n
recommended steps to "state null and alt hypothesis"
sketch 3 distributions: that of the null hypothesis, the distribution of x bar, distribution of the test statistic
___ standard deviation makes it more likely to reject the null
smaller
larger the sample size and smaller the s sub D, the _____ the required ____ ______ Dbar obs and ____
smaller; distance between; zero
the standard error is ___ than the standard deviation by a factor of ____
smaller; sq rt n
formula for standard error of the difference of means
sq root (ssubxbar1+ssubxbar2) OR s sub x1bar-x2bar=square rt (s^2pooled/n1)+sq root (s^2pooled/n2); standard error of the difference of means is: variance of sample1 plus variance of sample 2 all square rooted
formula standard deviation of the difference of means
sq rt (s^2pooled/n1 + s^2pooled/n2)
standard error of the difference of means conceptually
standard deviation of the distribution that would occur from creating a sample of means of the difference between xbar1 and xbar2 with infinite samples
why do we use standard error to calculate confidence interval as opposed to standard deviation? How does it change the equation? Why?
standard error is narrower than standard deviation because the distribution of means is narrower than that of a single sample according the central limit thrm; we use this narrower range for the confidence interval because we want to find the population mean not the population range of values and we believe by the central limit thrm it will fall between about 2 standard errors of any sample mean we pull
denominator of independent sample
standard error of the difference between means (s sub xbar1-sbar2)
equation for xbar sub obsv
summationX/n
how does t observed relate to t cv?
t observed is the value we calculate which shows up either in the rejection region (beyond the t cv) or not showing us whether or not too reject the null hypothesis
critical value of the statstic on standard scale (symbol and definition)
t sub cv; beginning of the rejection region such that if our observed value lies beyond this we reject the null hypothesis
to pool is to
take weighted average of variances
directional test
testing an alternative hypothesis which is either greater than or less than the null hypothesis, thus having direction of change
nondirectional test
testing an alternative hypothesis without a claim of any particular direction
which sample statistic do you use in a test statistic?q
the best point estimate of the population parameter of interested
x1 and x2 are strongly correlated in dependent samples test; effect of this
the differences are about equal throughout; this makes the s sub d and s sub dbar large such that t obs will be large
null hypothesis
the hypothesis that nothing has changed
alternative hypothesis
the hypothesis that things have changed
sigma subx1bar-x2bar in relation to sigma sub xbar
the independent sample standard error of the means is larger by a factor of sq rt two than the standard error of population means
which population parameter do we use in the test statistic?
the null hypothesis parameter
xbar1-xbar2 cv (concept)
the point(s) beyond which the rejecction region lies in a pooled independent sample: tcv*(s sub xbar1-xbar2)
statistical power
the probability of correctly rejecting the null hypothesis; 1-beta (either we are correct or we make a type II error)
alpha
the probability of making a type I error
beta
the probability of making a type II error
level of significance
the probability we accept in order to reject the null hypothesis
difference score
the score for the difference between two matched samples
which sample standard deviation do you use in a test statistic?
the sd of the sample statistic we used in the numerator
standard error of the mean of differences conceptually
the standard deviation of the samples from two populations making dbars again and again infinitely
if d=0
the two curves overlap (observed and null hypothesis)
t observed conceptually
the value we observe for test statistic of the sample representing the population data as expressed on the standard scale
what distributions can (or cannot) have rejection regions?
the variable of sampling statistic or the variable of test statistic (not the dist of the variable under the null hypothesis)
what is the purpose of a confidence interval
to try to find mew within a range of probability
which error type is worse to make? why?
type one error because it makes claims of change that don't occur; type two error usually does not result in a significant paper just in subsequent experimentation
dependent variable
variable of primary interest in a study
independent variable
variable that defines group membership
where is mew along the confidence interval?
we don't know, but we hope it falls on the interval
if our hypothesis test falls within the alternative hypothesis
we reject the null hypothesis and assume change occurs because of our variable
when do we use test statistic z vs. test statistic t?
we use test statistic z when sigma is known we use test statistic t when sigma is unknown and thus we calculate s
ways to be correct in hypothesis testing
when we fail to reject the null hypothesis and it actually makes no difference in nature; if we reject the null hypothesis and change actually occurs in nature
type 2 error
when we fail to reject the null hypothesis but it was actually false (change made in nature); concluding no effect exists when it actually does
type 1 error
when we reject the null hypothesis but it actually is true (makes no change); concluding that an effect exists when it actually doesn't
for the same (small) sample size and same alpha, the graph of the confidence interval is ____ for unknown sigma vs a known sigma
wider
the t distribution is ___ than the z distribution
wider and shorter
how do we reject the null hypothesis?
with the critical values or confidence interval
what's the center-most point of the confidence interval?
x bar
observed value of the statistic (symbol and definition)
x bar sub obsv; a computed statistic of our sample (mean?) distribution
calculation confidence interval
xbar+/- tcrit*s sub x bar; point estimate of mew plus or minus the standard critical value multiplied by the standard error of the point estimate
standardized effect size independent samples
xbar1- xbar2/ spooled; mean1-mean2 all divided by the pooled standard deviation
standard effect size of independent samples
xbar1-xbar2/spooled
equation for critical value of the sample statistic
xbarsubcv= mew+ z subcv* sigma subxbar
x cv calculation
xcv= tcv(s sub xbar) -/+ mu (if repeated or independent mu is assumed zero)
what is the difference between looking up a critical value in z distribution and t distribution? When do you use each?
you use the z you look up the z value and when you use t you look up the degrees of freedom; you use z when sigma (population standard deviation) is known and t when only have sample standard deviation
what is the difference numerically between zcv and tcv?
zcv is always the same wheras tcv is dependent upon df
