Chapter 18
- 18.1 We use such tests when we have an ordinal dependent variable.
- 18.2
|
VARIABLE 1 |
VARIABLE 2 |
| OBSERVATION |
SCORE |
RANK |
SCORE |
RANK |
| 1 |
1.30 |
3 |
54.39 |
5 |
| 2 |
1.80 |
4.5 |
50.11 |
3 |
| 3 |
1.20 |
2 |
53.39 |
4 |
| 4 |
1.06 |
1 |
44.89 |
1 |
| 5 |
1.80 |
4.5 |
48.50 |
2 |
- 18.3
| OBSERVATION |
VARIABLE 1 RANK |
VARIABLE 2 RANK |
DIFFERENCE |
SQUARED DIFFERENCE |
| 1 |
3 |
5 |
−2 |
4 |
| 2 |
4.5 |
3 |
1.5 |
2.25 |
| 3 |
2 |
4 |
−2 |
4 |
| 4 |
1 |
1 |
0 |
0 |
| 5 |
4.5 |
2 |
2.5 |
6.25 |
- 18.4
- a. There is an extreme outlier, 139, suggesting that the underlying population distribution might be skewed. Moreover, the sample size is small.
- b. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (we chose to rank this way, but you could do the reverse, from 10 to 1).
- c. The outlier was 25 IQ points (139 − 114 = 25) behind the next-highest score of 114. It now is ranked 10th, compared to the next-highest score’s rank of 9th.
- 18.5 Nonparametric tests are performed on ordinal data, so any data that are scale must be converted to ordinal before we compute the nonparametric test.
- 18.6 To calculate T, we first take the difference between each person’s two scores. We then rank these differences and separately sum the ranks associated with positive and negative difference scores. The table shows the organized data:
| PERSON |
SCORE 1 |
SCORE 2 |
DIFFERENCE |
RANKS |
RANK FOR POSITIVE DIFFERENCES |
RANK FOR NEGATIVE DIFFERENCES |
| A |
2 |
5 |
−3 |
4 |
|
4 |
| B |
7 |
2 |
5 |
2 |
2 |
|
| C |
4 |
5 |
−1 |
5 |
|
5 |
| D |
10 |
3 |
7 |
1 |
1 |
|
| E |
5 |
1 |
4 |
3 |
3 |
|
The sum of the ranks for the negative differences is ΣR− = (4 + 5) = 9.
T is equal to the smaller of these two sums: T = ΣRsmaller = 6.
- 18.7 Step 1: We convert the data from scale to ordinal. The researchers did not indicate whether they used random selection to choose the countries in the sample, so we must be cautious when generalizing from these results. There are some ties, but we will assume that there are not so many as to render the results of the test invalid.
Step 2: Null hypothesis: Countries in which English is a primary language and countries in which English is not a primary language do not tend to differ in accomplishment-related national pride.
Research hypothesis: Countries in which English is a primary language and countries in which English is not a primary language tend to differ in accomplishment-related national pride.
Step 3: There are seven countries in the English-speaking group and seven countries in the non-English-speaking group.
Step 4: The cutoff, or critical value, for a Mann–Whitney U test with two groups of seven participants (countries), a p level of 0.05, and a two-tailed test is 8.
Step 5: (Note: E stands for English-speaking, and NE stands for non-English-speaking.)
| COUNTRY |
PRIDE SCORE |
PRIDE RANK |
ENGLISH LANGUAGE |
E RANKS |
NE RANKS |
| United States |
4.00 |
1 |
E |
1 |
|
| Australia |
2.90 |
2.5 |
E |
2.5 |
|
| Ireland |
2.90 |
2.5 |
E |
2.5 |
|
| South Africa |
2.70 |
4 |
E |
4 |
|
| New Zealand |
2.60 |
5 |
E |
5 |
|
| Canada |
2.40 |
6 |
E |
6 |
|
| Chile |
2.30 |
7 |
NE |
|
7 |
| Great Britain |
2.20 |
8 |
E |
8 |
|
| Japan |
1.80 |
9 |
|
NE |
9 |
| France |
1.50 |
10 |
NE |
|
10 |
| Czech Republic |
1.30 |
11.5 |
NE |
|
11.5 |
| Norway |
1.30 |
11.5 |
NE |
|
11.5 |
| Slovenia |
1.10 |
13 |
NE |
|
13 |
| South Korea |
1.00 |
14 |
NE |
|
14 |
Step 6: The smaller test statistic, 1, is smaller than the critical value, 8. We can reject the null hypothesis; it appears that English-speaking countries tend to have higher accomplishment-related national pride than do non-English-speaking countries.