Answer :
Sure! Let's go through the conditions for inference step by step:
1. Random Condition:
- We have a random sample of [tex]\(\square\)[/tex]. This part of the condition is often stated in the problem. Assuming the sample is random ensures that the results are representative.
2. 10% Condition:
- This condition checks if the sample size is less than 10% of the population size to ensure independence. Here, it's given that the sample size is 100.
- We assume the population size to be 1,000, resulting in:
[tex]\[
10\% \text{ of the population size is } 0.1 \times 1000 = 100
\][/tex]
- So, since our sample size (100) is not less than 10% of the population (100), this condition is not met.
3. Large Counts Condition:
- This condition involves ensuring that the expected counts are large enough. This is checked using the formulas [tex]\( n \times p_0 \)[/tex] and [tex]\( n \times (1 - p_0) \)[/tex].
- Assuming a hypothetical probability, [tex]\( p_0 = 0.5 \)[/tex]:
[tex]\[
n \times p_0 = 100 \times 0.5 = 50
\][/tex]
[tex]\[
n \times (1 - p_0) = 100 \times (1 - 0.5) = 50
\][/tex]
- Both values are required to be at least 10, which is true in this case as both 50 and 50 are greater than or equal to 10.
In summary:
- The 10% condition is not met, as the sample size must be less than 10% of the population.
- The Large Counts condition is met, as both calculated values are greater than 10.
1. Random Condition:
- We have a random sample of [tex]\(\square\)[/tex]. This part of the condition is often stated in the problem. Assuming the sample is random ensures that the results are representative.
2. 10% Condition:
- This condition checks if the sample size is less than 10% of the population size to ensure independence. Here, it's given that the sample size is 100.
- We assume the population size to be 1,000, resulting in:
[tex]\[
10\% \text{ of the population size is } 0.1 \times 1000 = 100
\][/tex]
- So, since our sample size (100) is not less than 10% of the population (100), this condition is not met.
3. Large Counts Condition:
- This condition involves ensuring that the expected counts are large enough. This is checked using the formulas [tex]\( n \times p_0 \)[/tex] and [tex]\( n \times (1 - p_0) \)[/tex].
- Assuming a hypothetical probability, [tex]\( p_0 = 0.5 \)[/tex]:
[tex]\[
n \times p_0 = 100 \times 0.5 = 50
\][/tex]
[tex]\[
n \times (1 - p_0) = 100 \times (1 - 0.5) = 50
\][/tex]
- Both values are required to be at least 10, which is true in this case as both 50 and 50 are greater than or equal to 10.
In summary:
- The 10% condition is not met, as the sample size must be less than 10% of the population.
- The Large Counts condition is met, as both calculated values are greater than 10.
