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Which of the following meets the success-failure condition when using the normal model for a sampling distribution?(1 point) Re...Asked by mula
Which of the following meets the success-failure condition when using the normal model for a sampling distribution?(1 point) Responses sample size=85; probability of success=11 percent sample size=85; probability of success=11 percent sample size=180; probability of success=95 percent sample size=180; probability of success=95 percent sample size=180; probability of success=96 percent sample size=180; probability of success=96 percent sample size=85; probability of success=12 percent
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GPT-4o mini
To determine whether the success-failure condition is met for using the normal model in a sampling distribution, we check if both \( np \) and \( n(1 - p) \) are greater than or equal to 10.
The success-failure condition states:
- \( np \geq 10 \)
- \( n(1 - p) \geq 10 \)
Let's evaluate each of the given scenarios:
1. **Sample size = 85; probability of success = 11% (0.11)**
- \( np = 85 \times 0.11 = 9.35 \)
- \( n(1 - p) = 85 \times (1 - 0.11) = 85 \times 0.89 = 75.65 \)
- Not satisfied because \( np < 10 \).
2. **Sample size = 85; probability of success = 11% (repeated)**
- Same as above. Not satisfied.
3. **Sample size = 180; probability of success = 95% (0.95)**
- \( np = 180 \times 0.95 = 171 \)
- \( n(1 - p) = 180 \times (1 - 0.95) = 180 \times 0.05 = 9 \)
- Not satisfied because \( n(1 - p) < 10 \).
4. **Sample size = 180; probability of success = 95% (repeated)**
- Same as above. Not satisfied.
5. **Sample size = 180; probability of success = 96% (0.96)**
- \( np = 180 \times 0.96 = 172.8 \)
- \( n(1 - p) = 180 \times (1 - 0.96) = 180 \times 0.04 = 7.2 \)
- Not satisfied because \( n(1 - p) < 10 \).
6. **Sample size = 180; probability of success = 96% (repeated)**
- Same as above. Not satisfied.
7. **Sample size = 85; probability of success = 12% (0.12)**
- \( np = 85 \times 0.12 = 10.2 \)
- \( n(1 - p) = 85 \times (1 - 0.12) = 85 \times 0.88 = 74.8 \)
- Satisfied since both \( np \geq 10 \) and \( n(1 - p) \geq 10 \).
**Conclusion**:
The only scenario that meets the success-failure condition is **sample size = 85; probability of success = 12 percent**.
The success-failure condition states:
- \( np \geq 10 \)
- \( n(1 - p) \geq 10 \)
Let's evaluate each of the given scenarios:
1. **Sample size = 85; probability of success = 11% (0.11)**
- \( np = 85 \times 0.11 = 9.35 \)
- \( n(1 - p) = 85 \times (1 - 0.11) = 85 \times 0.89 = 75.65 \)
- Not satisfied because \( np < 10 \).
2. **Sample size = 85; probability of success = 11% (repeated)**
- Same as above. Not satisfied.
3. **Sample size = 180; probability of success = 95% (0.95)**
- \( np = 180 \times 0.95 = 171 \)
- \( n(1 - p) = 180 \times (1 - 0.95) = 180 \times 0.05 = 9 \)
- Not satisfied because \( n(1 - p) < 10 \).
4. **Sample size = 180; probability of success = 95% (repeated)**
- Same as above. Not satisfied.
5. **Sample size = 180; probability of success = 96% (0.96)**
- \( np = 180 \times 0.96 = 172.8 \)
- \( n(1 - p) = 180 \times (1 - 0.96) = 180 \times 0.04 = 7.2 \)
- Not satisfied because \( n(1 - p) < 10 \).
6. **Sample size = 180; probability of success = 96% (repeated)**
- Same as above. Not satisfied.
7. **Sample size = 85; probability of success = 12% (0.12)**
- \( np = 85 \times 0.12 = 10.2 \)
- \( n(1 - p) = 85 \times (1 - 0.12) = 85 \times 0.88 = 74.8 \)
- Satisfied since both \( np \geq 10 \) and \( n(1 - p) \geq 10 \).
**Conclusion**:
The only scenario that meets the success-failure condition is **sample size = 85; probability of success = 12 percent**.
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