In previous exercises you have used the computer programs to answer example problems associated with specific statistical tests. In this exercise you are given an example problem and you must decide which statistical test is appropriate to answer each question. You should explore each data set before you proceed with any tests. Use as many of the components of the program as possible. When you answer a question you should provide a fairly standard set of information. When we ask you to show all the steps, you should:

Read the questions thoroughly. One of the most common ways people lose marks is by providing a thorough answer to a question we didn't ask.

If you check your assumptions and transforms are needed, apply the appropriate transform, note the reason for the transform on your exam, and then assume that it worked. To save time for the exam (and only for this exam), you don't need to recheck to see if you can now meet the assumptions.

Look for key phrases: causation, prediction or functional relationship imply regression. Association or vary together imply correlation. Independence of variables or do the ratios vary imply contingency tests. Describe a regression relationship algebraically means tell us the equation of the line (don't forget to include any transforms in the equation). Describe the strength of a relationship means r². Summarize the differences between means should tell you to do a Tukey test.

Notes: The table lists, cumulatively downward, the tests appropriate for each level of measurement. Tests for lower levels of measurement may be applied to higher levels of measurement but these lower level tests generally have less power.

1. To test independence of two variables: c ² Contingency Test

2. To test prediction, causality or the functional dependence of one variable on another: Linear Regression

3. To test association or whether two items vary together: Correlation.

Assumptions:

All parametric tests require normality at the population level in some measure being tested.

When pooled variances are used in parametric tests, sample variances being pooled must be roughly equal.

All tests, both parametric and non-parametric, assume random sampling, with individuals within a sample being independently selected.

Try the following set of problems to review some of the ideas you have learned this year. This exercise is very similar to the lab exam you will write next week, although the lab exam will have fewer questions. You should not consider this set of questions to be all-inclusive, however. Any of the topics that we have dealt with in lab are fair game for the lab exam.

A study of clutch size (i.e. number of eggs laid per nest) in great blue herons (Ardea herodias) was undertaken in south coastal British Columbia. Nests were randomly sampled from three different colonies (populations) and the number of eggs in each nest were counted. The wing length (mm) of the female heron occupying each nest was also measured, as an index of body size. The data are stored in a file named CLUTCH in the shared directory.

a) Visually inspect the data from all three colonies and describe the distributions of clutch size and wing length. Test whether the distributions are normal.

b) Are there differences in average clutch size among the colonies? Are there differences in female wing length among the three colonies? If there are differences, summarize them.

c) In your analyses of part b, what were the powers of the tests to reject the null hypothesis, based on the default power estimates from your samples? What were the smallest sample sizes that would have allowed you to reject the null hypotheses?

d) The researchers observed that Douglas fir and alder trees were equally abundant at the three colony locations. Subsequently, they determined the frequency with which heron nests were located in different trees:

Are the ratios of herons nesting in Douglas fir to herons nesting in alder trees the same in all colonies? Considering only those herons nesting in Douglas fir trees, do herons prefer all colonies equally?

e) The average wing length of female great blue herons in North America is 500 mm and the average clutch size is 4. For the combined sample (i.e. assuming no differences among colonies in mean or variance for each variable), test whether or not clutch size of great blue herons in south coastal British Columbia is representative of great blue herons in North America. Carry out a similar analysis using female wing length.

f) For the combined sample (assuming no differences among colonies in mean or variance for each variable), decide whether or not it is possible to use female body size to predict clutch size in great blue herons from south coastal British Columbia. If so, describe the relationship algebraically.

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