1. What is the purpose of inferential statistics? With statistics, try to stay as simple as possible. Many times, the right statistics will jump out at you. The goal is to draw conclusions about the sample that can be inferred back to the population. There is always a chance of sampling error, but that error is lawful and predictable. Statistical tests take the sampling error into account, and they provide a probability that the sample observation is true for the population. The test is against a null hypothesis.

2. How do we test the null hypothesis? The null hypothesis states that there is no difference in the population. The test is run to determine the probability that the null is true. If the null being true is highly improbable, then the researcher can accept the alternative as true.

3. What types of errors can we make in testing the null? The researcher never knows if these types of errors have occurred, but he or she can know that they are at risk to do so. The first error is that the null would be rejected when it is true, and this is called Type I error. Type II error is accepting the null when it is false. The researcher can choose the level of significance to increase the odds for or against one of the types of errors. The significance level is the probability that what one sees is either chance or error. The smaller the level of significance, the lower the chance of Type I error but the higher the chance of Type II error. A lower significance level makes it more difficult to accept differences. The converse of this is also true with Type II errors. A higher level reduces the chance of a Type II error.

The issues are about power and the ability of the study/test to detect relationships and differences that exist. But there is a difference between statistical and practical difference. All the null hypothesis says is that there is no difference between two variables. When the null is true, the statistics may show a significant difference, but practical business says that the implementation of a particular procedure is not practical.

4. What are the major types of inferential statistics? First, the independent t-test can compare the difference between the population and sample mean to show that the sample is or is not a good representation of the population. The researcher can have two independent groups and can compare to see if one mean is significantly higher than the other. The dependent (non-independent) t-test can compare differences between matched subjects and can compare the difference between pretest and posttest scores. The steps for running this statistic are: (1) calculate the standard error of difference between two means (this is the expected difference), (2) calculate the observed difference, (3) divide the observed difference by the expected difference, and (4) use the degrees of freedom (total number of people in both samples minus 2) to look in the table for the probability.

The second statistic is the Chi-square test. The t-test looked at interval and ratio data, and this one looks at nominal and ordinal data. It compares differences among proportions of subjects. The steps for this test are: (1) calculate the expected frequency of subjects in each category, (2) calculate the observed frequency of subjects in each category, (3) subtract the expected frequency from the observed frequency, square the differences, and then, divide by the expected frequency, (4) add up the values for each category, and (5) use the degrees of freedom (K-1) to determine the significance of the value.

The third statistic is the Analysis of Variance (ANOVA), and it is a sophistocated t-test. It compares the difference of means among several groups rather than just two. The ANOVA is an expansion of the t-test, and it compares the variability within groups with the variability between groups. The steps are: (1) calculate the sum of squares between groups and add them together, (2) divide the sum by its degrees of freedom (K-1), (3) calculate the sum of squares within groups and add them together, (4) divide the sum by its degrees of freedom (N-K, which is total people minus the number of groups), (5) divide the mean square between by the mean square within, and (6) use the two values for degrees of freedom to obtain the probability from the table.

5. What assumptions are associated with each statistic? The t-test assumes: (1) random selection (talk about this in the statistical section), (2) the two groups have equal population variances (not two equal sample variances), and (3) distribution of the mean is approximately normal. With this test, the null is rejected when greater than one for the selected significance level. The Chi-square test assumes: (1) random sampling, (2) independence between groups (no one is in both groups), (3) mutually exclusive categories (not in both groups at the same time), and (4) observations are measured as frequencies rather than scores. The ANOVA test assumes: (1) random sampling and (2) population variances of all groups must be equal.

6. When would each type of inferential statistic be most appropriate? The t-test should be used when looking at "difference" types of research questions, equal-interval data, and with either one group or two group design. The Chi-square test should be used when looking at "difference" types of research questions, nominal data, and with either one group historical design, two group design (independent samples), or more than two group design (independent samples). The Chi-square test is also appropriate for "association" types of research questions when looking at nominal data. The ANOVA test should be used when looking at "difference" types of research questions, equal-interval data, and with more than two group design (mixed samples).

				Tom of Bethany

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