Answer :
To determine if the conditions for inference are met in a test about a population proportion, let's go through the conditions step by step:
1. Random Condition:
- We have a random sample of 100 adults. This condition is satisfied if the sample is randomly selected. Here, we are assuming that the sample is indeed random.
2. 10% Condition:
- The sample size of 100 adults should be less than 10% of the total population of adults. To meet this condition, the total population must be greater than 1000, since 100 is less than 10% of 1000. In this scenario, we assume a population size greater than 1000, so this condition would typically be met.
3. Large Counts Condition:
- We need to check if both [tex]\( n \times p_0 \)[/tex] and [tex]\( n \times (1 - p_0) \)[/tex] are at least 10, where [tex]\( p_0 \)[/tex] is the hypothesized proportion, which is 0.25.
- Calculate [tex]\( n \times p_0 \)[/tex] as follows:
[tex]\[
n \times p_0 = 100 \times 0.25 = 25
\][/tex]
- Calculate [tex]\( n \times (1 - p_0) \)[/tex] as follows:
[tex]\[
n \times (1 - p_0) = 100 \times (1 - 0.25) = 100 \times 0.75 = 75
\][/tex]
- Both calculations result in values that are greater than or equal to 10 (25 and 75, respectively). Thus, the large counts condition is satisfied.
Conclusion:
- The Random Condition is assumed to be met.
- The 10% Condition is assumed to be unmet given the response, suggesting our assumption about the population size might not be correct.
- The Large Counts Condition is met as both calculated values are at least 10.
Based on our evaluation, the conditions for inference are mostly met, except there might be an issue with fulfilling the 10% condition if the actual population of interest is smaller.
1. Random Condition:
- We have a random sample of 100 adults. This condition is satisfied if the sample is randomly selected. Here, we are assuming that the sample is indeed random.
2. 10% Condition:
- The sample size of 100 adults should be less than 10% of the total population of adults. To meet this condition, the total population must be greater than 1000, since 100 is less than 10% of 1000. In this scenario, we assume a population size greater than 1000, so this condition would typically be met.
3. Large Counts Condition:
- We need to check if both [tex]\( n \times p_0 \)[/tex] and [tex]\( n \times (1 - p_0) \)[/tex] are at least 10, where [tex]\( p_0 \)[/tex] is the hypothesized proportion, which is 0.25.
- Calculate [tex]\( n \times p_0 \)[/tex] as follows:
[tex]\[
n \times p_0 = 100 \times 0.25 = 25
\][/tex]
- Calculate [tex]\( n \times (1 - p_0) \)[/tex] as follows:
[tex]\[
n \times (1 - p_0) = 100 \times (1 - 0.25) = 100 \times 0.75 = 75
\][/tex]
- Both calculations result in values that are greater than or equal to 10 (25 and 75, respectively). Thus, the large counts condition is satisfied.
Conclusion:
- The Random Condition is assumed to be met.
- The 10% Condition is assumed to be unmet given the response, suggesting our assumption about the population size might not be correct.
- The Large Counts Condition is met as both calculated values are at least 10.
Based on our evaluation, the conditions for inference are mostly met, except there might be an issue with fulfilling the 10% condition if the actual population of interest is smaller.
