Sample Size Assumption: Success/Failure Condition This condition is satisfied since the two groups have 120 and 36 expected successes and 30 and 84 expected failures, all at least 10. This condition is satisfied since the two groups have 120 and 36 observed successes and 30 and 84 observed failures, all at least 10. This condition can not be checked without knowing the hypotheses to calculate the expected success and failures. This condition is not satisfied as the expected successes and failures for both samples are all below 10.
The Correct Answer and Explanation is:
The correct answer is: This condition is satisfied since the two groups have 120 and 36 expected successes and 30 and 84 expected failures, all at least 10.
Explanation:
In hypothesis testing, particularly when performing tests for proportions, the success/failure condition refers to the requirement that both the expected number of successes and failures should be at least 10 in each group. This condition helps ensure that the sampling distribution of the test statistic approximates a normal distribution, which is a key assumption for valid inference when using methods like the z-test for proportions.
The success/failure condition can be broken down as follows:
- Expected successes: These are the anticipated number of successes in the sample based on the population proportions under the null hypothesis. For a proportion test, the expected number of successes is calculated as n×pn \times pn×p, where nnn is the sample size, and ppp is the assumed population proportion.
- Expected failures: These are simply the complement of the expected successes, calculated as n×(1−p)n \times (1 – p)n×(1−p).
In this case, the provided information mentions two groups with the following expected successes and failures:
- Group 1: 120 expected successes and 30 expected failures.
- Group 2: 36 expected successes and 84 expected failures.
All of these values are greater than or equal to 10, which satisfies the success/failure condition. This ensures that the sampling distribution of the test statistic is sufficiently normal, making the use of a z-test appropriate.
The other options provided are incorrect:
- Option 2 incorrectly describes observed successes and failures. The condition should focus on expected values.
- Option 3 incorrectly claims that the success/failure condition cannot be checked without knowing the hypotheses. In reality, the condition relies solely on sample size and proportions.
- Option 4 is incorrect because, according to the provided numbers, all expected successes and failures are above 10, which satisfies the condition.
Therefore, the correct answer is that the success/failure condition is satisfied since all expected values are at least 10.
