Inferential Statistics

Chapter 13

Inferential Statistics



Inferential stats

are used to determine whether we can make statements that the results found in the present experiment reflect a true difference in the entire population of interest and not just the sample used in the experiment.

 Therefore inferential statistics allow us to make predictions about the entire population based on the findings of sample groups.  Inferential statistics give a probability that the difference between the two means from the sample used in the experiment represents a true difference based on the manipulation of the IV, and not random error.

Null and Research Hypotheses

   

Null hypothesis

 states simply that the population means (after conducting the experiment) are equal and that any observed differences are due to random error.

Alternative hypothesis

population means are not equal and therefore the treatment or independent variable had an effect.  states that the

Statistical significance

 indicates that there is a low probability that the difference between the obtained sample was due to random error.

Alpha level

used to make a decision about statistical significance.

-pre-determined probability level

Probability and Sampling distributions

  

Probability

 likelihood of the occurrence or some event or outcome.

Statistical Significance —

is a matter of probability.

Sampling distribution

 probability distributions based on many different samples taken over and over and shows the frequency of different sample outcomes from many separate random samples.



Sampling distribution-

is based on the assumption that the null hypothesis is true.

 Critical Values are obtained from Sampling Distributions and they are calculations of probability based on sample size and degrees of freedom

Sample Size

 The sample size also has an effect on determining statistical significance.

 The more samples you collect, the more likely you are to obtain an accurate estimate of the true population value  Thus, as your sample size increases, you can be more confident that your outcome is actually different from the expectations of the null hyp

Differential Statistics

 T-tests and F-tests are differential statistics because they detect differences between groups.

 The sampling distribution of all possible t values has a mean of 0 and a standard deviation of 1  It reflects all the possible outcomes we could expect if we compared the means of two groups and the null hypothesis is correct

T-test

 The calculated

t

value is a ratio of two aspects of the data   The difference between the group means The variability within groups

Group difference

 • • Under the null hypothesis you expect this difference to be 0.

The value of

t

difference between your obtained means increases as the difference betweent your obtained sample means increases

Within-group variability

about the mean  the amt of variability of scores

T-test Formula

 t =       group difference within-group variability The numerator of the formula is the difference between the means of the two groups The denominator is the variance (s2) of each group divided by the number of Ss in the group, which are added together The square root of the variance divided by the number of subjects = standard deviation Finally, we calculate our obtained

t

mean difference by the SD value by dividing the You would then compare your obtained

t

to those listed in the

t

-table of critical values to determine if it is significant or not

One Tailed d vs. Two-Tailed Tests

A

one-tailed test

is conducted if you are interested only in whether the obtained value of the statistic falls in one tail of the sampling distribution for that statistic.

--This is usually the case when your research hypothesis is directional.

---Group one will score higher than group two.

---The critical region in a one-tailed test contains 5% of the total area under the curve (alpha = .05)

Two Tailed Test



Two-tailed test

if you wanted to know whether the new therapy was either better or worse than the standard method.

 

You need to check whether your obtained statistic falls into either tail of the distribution There are two critical region in a two-tailed test

•

To keep the probability at .05, the total percentage of cases found in the two tails of the distribution must equal 5%

• •

Thus each critical region must contain 2.5% of the cases So the scores required to reach statistical significance must be more extreme than was necessary for the one-tailed test

When to use a one vs. two tailed?

 Major implication - for a given alpha level, you must obtain a greater difference between the means of your two treatment groups to reach statistical significance if you use a two-tailed test than if you used a one-tailed test   The one-tailed test is more likely to detect a real difference if one is present (that is, it is more powerful) However, using a one-tailed test means giving up any info about the reliability of a difference in the other, untested direction  The general rule of thumb is: Always use a two tailed test unless there are compelling

F-test

 The analysis of variance or

F

the

t

test test is an extension of  When a study has only one IV,

F

identical —the

F

= t-squared and

t

are virtually  ANOVA is used when there are more than two levels of an independent variable  The F statistic is a ratio of two types of variance:  

Systematic variance

 the deviation of the group means from the grand mean or the mean score of all individual groups

Error variance

 the deviation of the individual scores in each goup from their respective group means  The larger the F value, the more likely the score is significant

Effect Size



Effect size

 quantifies the size of the difference between groups  If we have two grps, the effect size is the difference between the groups expressed in standard deviation units.

 Therefore, the effect size is between O and 1. The effect size indicates the strength of the relationship. The closer to one, the stronger the relationship.

 The advantage of the effect size is that it is not a function of the sample size

Type one and Type two errors



Type I error

 occurs when the researcher says that a relationship exists when in fact it does not  You have falsely rejected the null hyp 

Type II error

 occurs when the researcher says that a relationship does not exist, when in fact it does  You have falsely accepted the null hyp



True State of Affairs

Null is true Null is False Reject Null Type I error alpha Accept Null Correct Decision 1-alpha C Correct Decision 1-beta Type II error beta

Probability of Type II error

• If we set a low alpha level to decrease the chances of a Type I error (accepting a hypothesis that is true when it is not (e.g., p<.01), we increase the chances of a Type II error • True differences are more likely to be detected if the sample size is large.

• If the effect size is large, a Type II error is unlikely

Interpreting non-significant results

 Negative or nonsignificant results are difficult to interpret  There are several causes for nonsignificant results: • • • The instruction could be hard to understand Have a weak manipulation of the indep var Using an unreliable or insensitive dep measure • Sample size is too small.

Choosing a Sample Size: Power Analysis  Sample size can be based on what is typical in that particular area of research   Sample size can also be based on a desired probability of correctly rejecting the null hyp  This probability is called the

power

of the statistical test  the sensitivity of the statistical procedure to detect differences in your data Power = 1 –

p

(Type II error )  Power analysis-computer generated  Higher desired power demands a greater sample size  Researchers usually use a power between .70 and .90 to determine sample size