One common type of association study is one in which two or more different groups of phenomena are suspected of having significantly different values for some co-existing quantitative property. For example, one may suspect:

• men and women (a qualitative property) may have different drinking habits ("number of weekend drinks," which is a quantitative property); or
• Democrats, Republicans, and Independents (a qualitative property) might have different television watching habits ("daily minutes of watching TV news," which is a quantitative property).

As with any association study, a relationship between two co-existing properties is determined to exist if individual phenomena having a particular value for the one property (X) are also likely to have a particular value for the co-existing property (Y). Here, the qualitative property is the "independent" property (X), and the quantitative property is the "dependent" property (Y).

To test such suspected associations, we would conduct what is said to be a "comparison of means." That is,

• We collect a set of observations of phenomena having the two co-existing properties of interest. This is said to be our sample.
• For the phenomena in our sample, we sort the phenomena into groups according to their values for the qualitative property (X).
• We then characterize the typical value of the quantitative property for the individuals in each group by calculating the mean and standard deviation of the quantitative property (Y) for that group.
• We can then assess the differences among the typical values found for the different groups:
• If the differences among the groups are small or even moderate, we typically judge those differences to be within the range of normal variability and NOT due to the grouping. In this case, we would say the observed differences were not significant and the two properties ARE NOT RELATED.
• On the other hand, if the differences among the groups are large, we would typically judge those differences NOT TO be within the range of normal variability. In this case, we would say the observed differences among the groups are significant and the properties PROBABLY ARE RELATED.

Where the qualitative property (X) has only two values (for example, "Men" and "Women"), we assess the "significance" of the differences between the two groups using what is said to be the t-Test for the comparison of means.

Where the qualitative property (X) has more than two values (for example, "Republicans," "Democrats," and "Independents"), we assess the "significance" of the differences among the group means by assessing the "significance" in the differences in their standard deviations. This assessment is said to be the Analysis of Variance, or ANOVA.

IN THIS ASSIGNMENT, WE WILL FOCUS OUR ATTENTION ON CASES IN WHICH THE QULAITATIVE PROPERTY OF INTEREST HAS ONLY TWO VALUES.

The idea behind the t-Test can be best understood in terms of Sampling Distributions. We establish a hypothetical probability distribution representing what we would expect to find where the two properties of interest ARE NOT RELATED. We then compare our actual observations to those expectations. That is, we compare the ACTUAL DIFFERENCE BETWEEN THE TWO MEANS with the HYPOTHETICAL, EXPECTED DIFFERENCE BETWEEN THE TWO MEANS.

Prospectively,

• Small differences between the actual observations (ACTUAL DIFFERENCE) and the expected (EXPECTED DIFFERENCE) values are expected to occur with a large probability.
• Moderate differences between the actual observations (ACTUAL DIFFERENCE) and the expected (EXPECTED DIFFERENCE) values are expected to occur with a moderate probability.
• Large differences between the actual observations (ACTUAL DIFFERENCE) and the expected values (EXPECTED DIFFERENCE) are expected to occur with a small probability.

Retrospectively,

• If the difference we have observed is assessed as having a large probability (p-value) of having occurred, we judge the difference to be not significant.
• If the difference we have observed is assessed as having a moderate probability (p-value) of having occurred, we judge the difference to be not significant.
• If the difference we have observed is assessed as having a small probability (p-value) of having occurred (less than 0.05), we judge the occurrence to be remarkable and improbable. We therefore assess the difference to be sufficient for us to REJECT our assumption that the two properties are NOT RELATED and draw the inference that the two properties ARE RELATED. We would say the differences observed between the two groups are "statistically significant."

This Assignment, Part A

For this assignment, you are to design and execute a research project in which you suspect two properties of some set of phenomena of interest to be related in an association.

• One of the properties is to be qualitative. This is to be the independent property (X), and this property is to have only two attainable values.

• The other is to be quantitative. This property, (Y), will be the dependent property.

Property X Name	Value 1 (Group 1)	Value 2 (Group 2)

Property Y Name	Units of Measurement

Collect observations of 50 phenomena. The number of phenomena in each of the two groups need not be the same (e.g., 25 and 25), but should not be too different (e.g., 10 and 40). List the observations it the following table:

Observation number	Property X (Group) Value	Property Y Value
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50

Sort the observations into groups according to Property X. and find the mean value for each group:

Property X, Value 1 (Group 1)

Property Y Value

Add together all the observed Y values for this group (A):

Count the number of observations for this group (B)

Calculate the Mean by dividing (A) by (B):

Property X, Value 2 (Group 2)

Property Y Value

Add together all the observed Y values for this group (C):

Count the number of observations for this group (D)

Find the Mean by dividing (C) by (D):

Compare the group Means:

Group 1 Mean
Group 2 Mean

Are the group means similar or different?