Answer :
To use a confidence interval for a proportion:
- Sample must be random (Latoya's is).
- Successes (np) and failures (n(1-p)) must be at least 10 each.
- Latoya's situation (a) meets both conditions, while (b) doesn't due to insufficient success/failure counts.
Conditions for calculating a confidence interval for p:
1. Random sample: The sample needs to be a simple random sample (SRS), meaning each individual has an equal chance of being selected, and the selections are independent of each other.
2. Large counts: Both the number of successes (n * p) and failures (n * (1 - p)) in the sample should be at least 10. This ensures the normal approximation for the sampling distribution of the sample proportion is valid.
Evaluating Latoya's situation:
a. Meets conditions:
- Random: Latoya interviews an SRS of 50 students, which satisfies the randomness condition.
- Large counts:
- Successes: 14 students like the food (p = 14/50 = 0.28).
- Failures: 36 students don't like the food (1 - p = 0.72).
- Both counts (14 and 36) are greater than 10, satisfying the large counts condition.
b. Fails conditions:
- Random: Same as in (a).
- Large counts:
- Successes (14) and failures (36) are less than 10, violating the large counts condition.
c. Incorrect: This option directly states that the conditions are not met, but based on our analysis in (a), the conditions are actually met in this scenario.
d. Incorrect: As mentioned in (a), Latoya uses an SRS, fulfilling the randomness condition.
Conclusion:
- Scenario (a): The conditions (random sample and large counts) are met. Latoya can proceed with calculating a confidence interval for p.
- Scenario (b): The conditions (large counts) are not met due to the small number of successes and failures. Latoya cannot reliably calculate a confidence interval in this case.
Q-Latoya wants to estimate p = the proportion of all students at her large boarding high school that like the cafeteria's food. She interviews an SRS of 50 of the students living in the dormitory and finds that 14 think the cafeteria's food is good. Check to see if the conditions for calculating a confidence interval for p are met
a. Random? Latoya interviews an SRS of 50 students √
Large Counts? 50(14/50) = 14 ≥ 10 and 50(36/50) = 36 ≥10 √
b. Random? Latoya interviews an SRS of 50 students √
Large Counts? 50(14/50) = 14 < 10 and 50(36/50) = 36 < 10 √
c. Fails Large Counts
d. Not Random
