Statistics for Behavioral Sciences: Chapter 9
Type I error
rejecting the null hypothesis when it is actually true; the probability of this error is controlled by your choice of α level *this is the worst kind of error to make*
How do you determine effect size of a one-sample t-test?
1.start by finding Cohen's d using the following equation: d=(X⁻-μ₀)/ŝ Where: X⁻=mean of the sample μ₀=mean specified by the null hypothesis ŝ=estimate of the standard deviation of the population based on the sample 2. interpret d
How do you create the null and alternative hypotheses for this test?
1st, you must keep in mind that you're trying to determine if a known population and the sample population are the same population It is logical to think that they would be, so you should therefore *formulate the alternative hypothesis first, stating the obvious of what you think is true*: Hₐ:μ₀≄μ₁ Then state the null hypothesis: H₀:μ₀=μ₁ Where: H₀=null hyp μ₀=known population μ₁=*population that your sample is drawn from*
How does NHST provide an exact probability of data?
1st: calculate the sampling distribution of differences 2nd: determine the probability of obtaining any particular difference
So, once you've completed your one-sample t-test and you know whether or not your null hyp is true, how do you formulate your conclusion?
1st: state the hypothesis in terms of whether or not there was a significant difference as determined by p 2nd: state the direction of the data (toward the null or the alternative) 3rd: provide the effect size of the data
When do you reject or retain null hypothesis in terms of probability and alpha?
If p ≤ α, reject H₀ If α = 0.05, and p ≤ 0.05, reject H₀ If α = 0.05, and p > 0.05, retain H₀
What is the rejection region?
The area of a sampling distribution that corresponds to test statistic values that lead to rejection of the null hypothesis; it includes all the differences that have a probability equal to or less than α; *any event in rejection region leads to rejection of the null hypothesis*
null hypothesis statistical testing
a.k.a NHST; a process that produces probabilities that are accurate when the null hypothesis is true
one tailed vs. two tailed tests
a *one-tailed test* is a *directional* hypothesis; it tells whether the μ₀ is greater than, or less than or equal to μ₁ a *two tailed test* is a *non-directional* hypothesis that just determines whether μ₀ is equal to or not equal to μ₁
what does NHST result in?
a conclusion about the relationship b/w μ1 and μ₀
Is the null hypothesis a statement about a statistic or a parameter?
a parameter!
When analyzing how the null hyp is written: "H₀:μ₀=μ₁" what kind of statistic is it?
a probability!
two tailed test of significance
a statistical test for a difference in population means that can detect positive and negative differences
one sample t-test
a statistical test of the hypothesis that determines if a sample with mean, X⁻, came from a population with mean, μ: X⁻=μ
What can you do to reduce the probability of these errors occurring?
analyze your one-sampled t-test under a smaller α value! *lowering the α level to a probability, i.e., 0.01 or 0.001, reduces the probability of uncertainty*
At what point do you decide between rejecting the null hypothesis and retaining it?
by setting alpha or establishing a significance level
What is statistically significant?
data is statistically significant when the difference is so large that chance is not a plausible explanation for the difference; occurs when Ho is rejected
Type II error
failure to reject the null hypothesis when it's false; this is only possible if you retain the null hyp.; the probability of this error is controlled by β ex.) When you don't have enough evidence in your sample to say the null hypothesis is wrong, although it is known within the population that the null hypothesis is false
How many alternative hypotheses can data have?
for any NHST problem, there are *three* possible alternative hypotheses; a researcher chooses 1 before the data are collected; the choice of a specific alternative hyp helps determine conclusions that are possible
sampling distribution of differences
found by subtracting the mean from each sample mean
What does NHST allow us to makes statements on?
if the data permit, NHST allows us to make strong statements of support for the hypothesis of difference b/w μ1 & μ₀, *but it cannot result in strong support for a hypothesis of equality, regardless of how the data come out*
What does the particular difference mean?
if the probability of the obtained particular difference is very small when the null hypothesis is true, we have evidence that the null hypothesis is not correct and should be rejected; if the probability is large, there's evidence that is consistent with the null hypothesis... *but large probabilities do not permit adoption or acceptance of the null hypothesis because large probabilities are also consistent w/ hypotheses other than the null hypothesis... they only retain the null hypothesis as one among many hypotheses that the data support*
What do we say if the null hyp is wrong? right?
if wrong: "we reject" if right: "we fail to reject"
alpha and probability
in an experiment, the researcher sets α, gathers data, & uses a sampling distribution to find the probability (p) of the data; *this p value is only correct when the null hypothesis is true*
How does NHST begin?
it *always* begins with a claim about a parameter!
critical values
numbers from a sampling distribution that determine whether the null hypothesis is rejected; they separate the rejection region from the acceptance region
determining probability of obtaining any particular difference
pick the particular difference you're interested in (ex. The difference b/w the sample mean and the null hypothesis mean)
β
probability of a Type II error; α & β are inversely related; as α gets smaller, you increase the chances of committing a type II error... but Type II is better to commit than Type I
Interpreting d
small effect=0.20 medium effect=0.50 large effect=0.80 *if effect size is in between two kinds of effect, call it an effect between the two*. ex.) d=0.3 is a small-medium effect size *knowing that p is large or small doesn't tell you anteing about d!!!*
What is a null hypothesis?
symbol: H₀; the hypothesis about a population or the relationship among populations; for any NHST, there is a null hypothesis
alternative hypothesis
symbol: H₁; the hypothesis about population parameters that is accepted if the null hypothesis is rejected
alpha
symbol: α; the probability of a Type I error; setting alpha is the *choice* of a probability value; *the 0.05 mark is the widely accepted position to set the break between the continuum of rejecting or retaining the null hypothesis*
What is the formula for the one sample t-test?
t = (X⁻ - μ₀)/sᵪ₋ Where X⁻=mean of the sample μ₀=hypothesized mean of the population sᵪ₋=standard error of the mean *the algebraic sign of the t-value doesn't matter*
What kind of sampling distribution is used for NHST?
t distribution!
What t-distribution corresponds to the rejection region?
t values more extreme than + or - 2.37 lead to rejecting Ho
What is the true name of the hypothesis of difference?
the alternative hypothesis
μ₀
the known population mean
What is the true name of the hypothesis of equality?
the null hypothesis
significance level
the probability (α) chosen as the criterion for rejecting the null hypothesis; establishing this level is the same as setting alpha
What is a p value?
the probability of finding "t" if the null hypothesis is true ex.) p= 0.00003 is the probability of finding a hobbit as tall as Gandolf
μ1
the symbol for the unknown population mean; this value can be determined with the use of NHST and sample values
Are probability figures usually provided?
they are, but statistical tables don't provide probability figures for every outcome of a statistical test... instead, they provide stat test values that correspond to commonly chosen α values, or critical values
critical value
tₐ; find it using α for a one or two-tailed distribution, by looking up the value in the back of the book
NS
when the difference is not significant; occurs when Ho is retained
Is there another way to test the hypothesis besides contracting an entire sampling distribution?
yes! the conventional practice is to analyze the data using the *one-sample t-test*
What do you do with the critical value?
you compare it to the absolute value of the t-value to see which one is bigger. Based off of which is bigger, you can compare this to your null hyp and see whether or not you can reject it
How can you get support for the hypothesis of difference?
you discredit the hypothesis of equality!; the hypothesis of equality produces predictions ab outcomes, & NHST tests these predictions against actual outcomes
After determining the one sampled t-test value, what else must you do?
you must determine the critical value