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Answer :
To determine if the conditions for inference are met, let's evaluate each of the conditions required for conducting a hypothesis test.
1. Random Condition: The sample needs to be random. In this scenario, we are given a random sample of 100 adults. This condition is satisfied as we have a random sample of 100.
2. 10% Condition: The sample size should be less than 10% of the population to ensure independence of the sample. Since 100 adults is less than 10% of a much larger adult population, this condition is met.
3. Large Counts Condition: This condition checks whether the sample size is large enough for the normal approximation to the binomial distribution to be appropriate. It requires both [tex]\(n \times p_0\)[/tex] and [tex]\(n \times (1 - p_0)\)[/tex] to be at least 10, where [tex]\(n\)[/tex] is the sample size and [tex]\(p_0\)[/tex] is the hypothesized population proportion.
- Calculate [tex]\(n \times p_0\)[/tex]:
[tex]\[
n \times p_0 = 100 \times 0.25 = 25.0
\][/tex]
- Calculate [tex]\(n \times (1 - p_0)\)[/tex]:
[tex]\[
n \times (1 - p_0) = 100 \times (1 - 0.25) = 100 \times 0.75 = 75.0
\][/tex]
Both calculations (25.0 and 75.0) are greater than or equal to 10, so the Large Counts condition is satisfied.
In conclusion, all the conditions for inference are met: the sample is random, the sample size is less than 10% of the population, and the required counts for the normal approximation are sufficient. Therefore, you can proceed with the hypothesis test to determine if more than 25% of adults would describe themselves as organized.
1. Random Condition: The sample needs to be random. In this scenario, we are given a random sample of 100 adults. This condition is satisfied as we have a random sample of 100.
2. 10% Condition: The sample size should be less than 10% of the population to ensure independence of the sample. Since 100 adults is less than 10% of a much larger adult population, this condition is met.
3. Large Counts Condition: This condition checks whether the sample size is large enough for the normal approximation to the binomial distribution to be appropriate. It requires both [tex]\(n \times p_0\)[/tex] and [tex]\(n \times (1 - p_0)\)[/tex] to be at least 10, where [tex]\(n\)[/tex] is the sample size and [tex]\(p_0\)[/tex] is the hypothesized population proportion.
- Calculate [tex]\(n \times p_0\)[/tex]:
[tex]\[
n \times p_0 = 100 \times 0.25 = 25.0
\][/tex]
- Calculate [tex]\(n \times (1 - p_0)\)[/tex]:
[tex]\[
n \times (1 - p_0) = 100 \times (1 - 0.25) = 100 \times 0.75 = 75.0
\][/tex]
Both calculations (25.0 and 75.0) are greater than or equal to 10, so the Large Counts condition is satisfied.
In conclusion, all the conditions for inference are met: the sample is random, the sample size is less than 10% of the population, and the required counts for the normal approximation are sufficient. Therefore, you can proceed with the hypothesis test to determine if more than 25% of adults would describe themselves as organized.
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