Answer :
To determine if the conditions for inference are met in this scenario, we need to check three main criteria: Random, 10% Condition, and Large Counts. Let's walk through each condition:
1. Random Condition:
- This criterion confirms that the sample is random. We are given a random sample of 100 adults.
2. 10% Condition:
- This condition requires that the sample size (100 adults) is less than 10% of the entire population.
- Since 100 is significantly less than 10% of the total adult population, this condition is met.
3. Large Counts Condition:
- This condition checks if the sample size is large enough to ensure a reliable normal approximation. We do this by calculating:
- [tex]\( n \times p_0 \)[/tex] and [tex]\( n \times (1 - p_0) \)[/tex], where [tex]\( n \)[/tex] is the sample size and [tex]\( p_0 \)[/tex] is the hypothesized proportion of the population.
- With [tex]\( n = 100 \)[/tex] and [tex]\( p_0 = 0.25 \)[/tex], we calculate:
- [tex]\( n \times p_0 = 100 \times 0.25 = 25 \)[/tex]
- [tex]\( n \times (1 - p_0) = 100 \times 0.75 = 75 \)[/tex]
- Both values (25 and 75) are greater than or equal to 10. Therefore, the Large Counts condition is met.
Since all three conditions for inference are satisfied, we can proceed with the hypothesis testing. The data provide sufficient evidence to conduct a test to determine if more than 25% of adults would describe themselves as organized.
1. Random Condition:
- This criterion confirms that the sample is random. We are given a random sample of 100 adults.
2. 10% Condition:
- This condition requires that the sample size (100 adults) is less than 10% of the entire population.
- Since 100 is significantly less than 10% of the total adult population, this condition is met.
3. Large Counts Condition:
- This condition checks if the sample size is large enough to ensure a reliable normal approximation. We do this by calculating:
- [tex]\( n \times p_0 \)[/tex] and [tex]\( n \times (1 - p_0) \)[/tex], where [tex]\( n \)[/tex] is the sample size and [tex]\( p_0 \)[/tex] is the hypothesized proportion of the population.
- With [tex]\( n = 100 \)[/tex] and [tex]\( p_0 = 0.25 \)[/tex], we calculate:
- [tex]\( n \times p_0 = 100 \times 0.25 = 25 \)[/tex]
- [tex]\( n \times (1 - p_0) = 100 \times 0.75 = 75 \)[/tex]
- Both values (25 and 75) are greater than or equal to 10. Therefore, the Large Counts condition is met.
Since all three conditions for inference are satisfied, we can proceed with the hypothesis testing. The data provide sufficient evidence to conduct a test to determine if more than 25% of adults would describe themselves as organized.
