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20160406021858class_6_comparing_two_means._3.28.16.pptxHypothesis Testing/ Differences between two Means
April 4th, 2016
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Testing The Hypothesis
State in operational terms, the relationships that should be observed if the research hypothesis is true
State the null hypothesis.
Gather the data
Determine if evidence is sufficient to accept or reject the null hypothesis
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Hypothesis Testing Cont’d
Rejection of research hypothesis after empirical testing does not mean that the study was failure.
Unconfirmed hypothesis are apart of scientific research and still add to the body of knowledge
Hypothesis are never proved or disproved
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Null Hypothesis
A researcher finds the following differences in Self-Esteem scores between boys and girls.
Girls m=70.00 Boys m=80.00
Do boys have higher self-esteem or are the results due to chance factors?
We use inferential tests to answer the question.
Null Hypothesis:
The true difference between the means (in the population is zero)
H0: u1-u2=0
Type I Error
Symbolized by a Greek lowercase level alpha (α).
Rejection of a true null hypothesis.
States that the results of the study were not due to chance.
Researcher incorrectly concludes that significant differences WERE found or that a relationship between the variables exists.
Considered to be more serious than Type II error.
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Type II Error
Symbolized by a Greek lowercase level beta (β).
Investigator retains a false null hypothesis
Researcher incorrectly concludes that significant differences were NOT found or that a relationship between the variables does not exist.
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The Strategy of Inferential Statistics
Null Hypothesis
Type I error: reject true null hypothesis
Type II error: accept false null hypothesis
| Innocent | Guilty | |
| Not guilty verdict | Justice | Type II Error β |
| Guilty verdict | Type I Error α | Justice |
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Level of Significance
Investigators must weigh the consequences of a Type I or Type II error before the conducting the experiment.
Level of significance: is the level at which the null hypothesis will be rejected.
Probability that the investigator is willing to risk in rejecting a null hypothesis.
Most common used levels are .05 and .01
.05 level, willing to be wrong 5 times in 100 in rejecting the null hypothesis
.01 level, willing to be wrong 1 time in 100 in rejecting the null hypothesis (rejecting the null with more confidence compared to .05).
.001 level, willing to be wrong 1 time in 100 in rejecting the null hypothesis.
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Consequence of Type I and Type II Errors in Education
Type I errors can lead to changes in the educational system that are not necessary such as:
Best practices
Curriculum modifications
Teacher training programs
Type II errors can lead to the lack necessary changes being made. Maintenance of the status quo.
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Rationale to the t-test
| t | = | observed difference between sample means | − | expected difference between population means (if null hypothesis is true) | |
| estimate of the standard error of the difference between two sample means |
t-test
Dependent t-test
Compares two means based on related data.
E.g., Data from the same people measured at different times.
Data from ‘matched’ samples.
Test re-test
Independent t-test
Compares two means based on independent data
E.g., data from different groups of people
Boys vs. Girls.
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Rational for the t-test
Two samples of data are collected and the sample means calculated. These means might differ by either a little or a lot.
If the samples come from the same population, then we expect their means to be roughly equal. Although it is possible for their means to differ by chance alone, we would expect large differences between sample means to occur very infrequently.
There is no effect and sample means in our population fluctuate a lot and we have, by chance, collected two samples that are atypical of the population from which they came.
The two samples come from different populations but are typical of their respective parent population. In this scenario, the difference between samples represents a genuine difference between the samples (and so the null hypothesis is incorrect).
.
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Rational for t-test
As the observed difference between the sample means gets larger, the more confident we become that the second explanation is correct (i.e. that the null hypothesis should be rejected). If the null hypothesis is rejected, then we gain confidence that the two sample means differ because of the different experimental manipulation imposed on each sample.
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T-test Cont’d
p<.05
t test yields a low probability that the null hypothesis is true
Researchers will reject the null hypothesis
Larger Samples=less likely that the difference between two means is due to sampling errors.
Less sampling error, less change that the null hypothesis is correct
Larger observed difference between means = less likely that the difference was created by sampling errors.
T-test Cont’d
Smaller the variance among subjects: less likely the difference between two means is due to sampling error.
Smaller the variance: more likely the null hypothesis will be rejected
No sampling error
More variation leads to observed differences attributed to sampling error.
Statistical Tests
t test for independent samples
t test for dependent samples
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Assumptions of the t-test
Both the independent t-test and the dependent t-test are parametric tests based on the normal distribution. Therefore, they assume:
The sampling distribution is normally distributed. In the dependent t-test this means that the sampling distribution of the differences between scores should be normal, not the scores themselves.
Data are measured at least at the interval level.
The independent t-test, because it is used to test different groups of people, also assumes:
Variances in these populations are roughly equal (homogeneity of variance).
Scores in different treatment conditions are independent (because they come from different people).
Effect Size: Cohen’s D
Magnitude of the difference between variables
Experimental Group-Control Group/SD of Control Group
Mean for group 1 is 9 tenths of a standard deviation above the control group or group 2.
| Mean | SD | |
| Group 1 | 110 | 10 |
| Group 2 | 100 | 11 |
| Effect Size | 110-100=10 | 10/11=.90 |
When Assumptions are Broken
Dependent t-test
Mann-Whitney Test
Wilcoxon rank-sum test
Independent t-test
Wilcoxon Signed-Rank Test
SPSS
Example:
1) Independent and Dependent Sample
Spider anxiety
2). Calculate the effect size
3). Practice Exercise
