Below is the detailed step-by-step solution.
Let's denote the weights in the groups as follows:
Group A: 14.6, 15.9
Group B: 11.2, 9.2
Step 1: Calculate the mean weight for each group.
To calculate the mean for Group A ([tex]\(\bar{x}_A\)[/tex]):
[tex]\[ \bar{x}_A = \frac{14.6 + 15.9}{2} = \frac{30.5}{2} = 15.25 \][/tex]
To calculate the mean for Group B ([tex]\(\bar{x}_B\)[/tex]):
[tex]\[ \bar{x}_B = \frac{11.2 + 9.2}{2} = \frac{20.4}{2} = 10.2 \][/tex]
Step 2: Calculate the difference between the mean weights of Group A and Group B.
[tex]\[ \bar{x}_A - \bar{x}_B = 15.25 - 10.2 = 5.05 \][/tex]
Results for each randomization:
1. After the first randomization:
- [tex]\(\bar{x}_A\)[/tex] is 15.25,
- [tex]\(\bar{x}_B\)[/tex] is 10.2,
- [tex]\(\bar{x}_A - \bar{x}_B\)[/tex] is 5.05.
2. After the second randomization:
- [tex]\(\bar{x}_A\)[/tex] is 15.25,
- [tex]\(\bar{x}_B\)[/tex] is 10.2,
- [tex]\(\bar{x}_A - \bar{x}_B\)[/tex] is 5.05.
3. After the third randomization:
- [tex]\(\bar{x}_A\)[/tex] is 15.25,
- [tex]\(\bar{x}_B\)[/tex] is 10.2,
- [tex]\(\bar{x}_A - \bar{x}_B\)[/tex] is 5.05.
Summarized in boxes:
1. After the first randomization:
- [tex]\(\bar{x}_A\)[/tex] is [tex]\(\boxed{15.25}\)[/tex],
- [tex]\(\bar{x}_B\)[/tex] is [tex]\(\boxed{10.2}\)[/tex],
- [tex]\(\bar{x}_A - \bar{x}_B\)[/tex] is [tex]\(\boxed{5.05}\)[/tex].
2. After the second randomization:
- [tex]\(\bar{x}_A\)[/tex] is [tex]\(\boxed{15.25}\)[/tex],
- [tex]\(\bar{x}_B\)[/tex] is [tex]\(\boxed{10.2}\)[/tex],
- [tex]\(\bar{x}_A - \bar{x}_B\)[/tex] is [tex]\(\boxed{5.05}\)[/tex].
3. After the third randomization:
- [tex]\(\bar{x}_A\)[/tex] is [tex]\(\boxed{15.25}\)[/tex],
- [tex]\(\bar{x}_B\)[/tex] is [tex]\(\boxed{10.2}\)[/tex],
- [tex]\(\bar{x}_A - \bar{x}_B\)[/tex] is [tex]\(\boxed{5.05}\)[/tex].
