Question
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=96 percent
sample size=180; probability of success=96 percent
sample size=180; probability of success=95 percent
sample size=180; probability of success=95 percent
sample size=85; probability of success=12 percent
sample size=85; probability of success=12 percent
Responses
sample size=85; probability of success=11 percent
sample size=85; probability of success=11 percent
sample size=180; probability of success=96 percent
sample size=180; probability of success=96 percent
sample size=180; probability of success=95 percent
sample size=180; probability of success=95 percent
sample size=85; probability of success=12 percent
sample size=85; probability of success=12 percent
Answers
GPT-4o mini
To use the normal model for the sampling distribution, we need to check the success-failure condition, which states that both the number of successes and the number of failures in the sample should be at least 5. This can be calculated using the following:
- Number of successes = \( n \cdot p \)
- Number of failures = \( n \cdot (1-p) \)
Where \( n \) is the sample size and \( p \) is the probability of success.
Now let's evaluate each option:
1. **sample size=85; probability of success=11 percent (0.11)**
- Successes: \( 85 \cdot 0.11 = 9.35 \) (not enough)
- Failures: \( 85 \cdot 0.89 = 75.65 \) (enough)
- Not valid as successes < 5.
2. **sample size=180; probability of success=96 percent (0.96)**
- Successes: \( 180 \cdot 0.96 = 172.8 \) (enough)
- Failures: \( 180 \cdot 0.04 = 7.2 \) (enough)
- Valid.
3. **sample size=180; probability of success=95 percent (0.95)**
- Successes: \( 180 \cdot 0.95 = 171 \) (enough)
- Failures: \( 180 \cdot 0.05 = 9 \) (enough)
- Valid.
4. **sample size=85; probability of success=12 percent (0.12)**
- Successes: \( 85 \cdot 0.12 = 10.2 \) (enough)
- Failures: \( 85 \cdot 0.88 = 74.8 \) (enough)
- Valid.
Based on the evaluations:
- The second, third, and fourth options meet the success-failure condition.
- The first option does not meet the condition.
Thus, the options that meet the success-failure condition are:
- **Sample size=180; probability of success=96 percent**
- **Sample size=180; probability of success=95 percent**
- **Sample size=85; probability of success=12 percent**
- Number of successes = \( n \cdot p \)
- Number of failures = \( n \cdot (1-p) \)
Where \( n \) is the sample size and \( p \) is the probability of success.
Now let's evaluate each option:
1. **sample size=85; probability of success=11 percent (0.11)**
- Successes: \( 85 \cdot 0.11 = 9.35 \) (not enough)
- Failures: \( 85 \cdot 0.89 = 75.65 \) (enough)
- Not valid as successes < 5.
2. **sample size=180; probability of success=96 percent (0.96)**
- Successes: \( 180 \cdot 0.96 = 172.8 \) (enough)
- Failures: \( 180 \cdot 0.04 = 7.2 \) (enough)
- Valid.
3. **sample size=180; probability of success=95 percent (0.95)**
- Successes: \( 180 \cdot 0.95 = 171 \) (enough)
- Failures: \( 180 \cdot 0.05 = 9 \) (enough)
- Valid.
4. **sample size=85; probability of success=12 percent (0.12)**
- Successes: \( 85 \cdot 0.12 = 10.2 \) (enough)
- Failures: \( 85 \cdot 0.88 = 74.8 \) (enough)
- Valid.
Based on the evaluations:
- The second, third, and fourth options meet the success-failure condition.
- The first option does not meet the condition.
Thus, the options that meet the success-failure condition are:
- **Sample size=180; probability of success=96 percent**
- **Sample size=180; probability of success=95 percent**
- **Sample size=85; probability of success=12 percent**