EXAMPLE 19 Calculating the test statistic for the rank correlation test
femaleliteracy
The fertility rate is the mean number of children born to a typical woman in the country, and the female literacy rate is the percentage of women at least 15 years old who can read and write. The table contains the female literacy rate (in percent) and the fertility rate (in numbers of children) for a random sample of 10 countries.
| Country | Female literacy | Fertility |
|---|---|---|
| Afghanistan | 21 | 6.69 |
| India | 48 | 2.73 |
| Sudan | 51 | 4.7 2 |
| Saudi Arabia | 71 | 4.00 |
| South Africa | 86 | 2.20 |
| China | 87 | 1.73 |
| Israel | 94 | 2.41 |
| Italy | 98 | 1.28 |
| United States | 99 | 2.09 |
| Poland | 100 | 1.25 |
Calculate the test statistic for the rank correlation test, using the following steps:
- Rank the values of the first variable (female literacy) from lowest to highest.
- Rank the values of the second variable (fertility) from lowest to highest.
- For each subject (country), find the difference in ranks, , and square the difference in ranks to get . Add up the -values to get .
Complete the calculation of the test statistic
where represents the sample size (number of matched pairs).
Solution
Table 15 contains the calculations needed to find .
14-46
| Country | Female literacy |
Fertility | Literacy rank |
Fertility rank |
Difference |
|
|---|---|---|---|---|---|---|
| Afghanistan | 21 | 6.69 | 1 | 10 | −9 | 81 |
| India | 48 | 2.73 | 2 | 7 | −5 | 25 |
| Sudan | 51 | 4.72 | 3 | 9 | −6 | 36 |
| Saudi Arabia | 71 | 4.00 | 4 | 8 | −4 | 16 |
| South Africa | 86 | 2.20 | 5 | 5 | 0 | 0 |
| China | 87 | 1.73 | 6 | 3 | 3 | 9 |
| Israel | 94 | 2.41 | 7 | 6 | 1 | 1 |
| Italy | 98 | 1.28 | 8 | 2 | 6 | 36 |
| United States | 99 | 2.09 | 9 | 4 | 5 | 25 |
| Poland | 100 | 1.25 | 10 | 1 | 9 | 81 |
Because there are countries, the value of the test statistic is given by
We now present the steps for performing the rank correlation test.
NOW YOU CAN DO
Exercises 9–12.