Slide 1

Inferential Statistics

Chapter 13

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Slide 2 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.

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Slide 3 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 states that the population means are not equal and therefore the treatment or independent variable had an effect.

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

Alpha level -pre-determined probability level used to make a decision about statistical significance.

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Slide 4 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

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Slide 5 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

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Slide 6 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

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Slide 7 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 difference between your obtained means

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

• The value of t increases as the difference betweent your

obtained sample means increases

Within-group variability the amt of variability of scores

about the mean

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Slide 8 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 value by dividing the mean difference by the SD

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

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Slide 9 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)

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Slide 10 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

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Slide 11 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

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Slide 12 F-test

The analysis of variance or F test is an extension of the t test

When a study has only one IV, F and t are virtually identical—the F = t-squared

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

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Slide 13 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

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Slide 14 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

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Slide 15 True State of Affairs

Null is true Null is False

C

Reject Null Type I error

alpha

Accept

Null

Correct Decision

1-alpha

Correct Decision

1-beta

Type II error

beta

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Slide 16 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

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Slide 17 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.

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Slide 18 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

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