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Answer :
We are testing whether the data support the claim that more than $25\%$ of adults describe themselves as organized. Before carrying out the test, we must verify that the conditions for inference are met.
1. $$\textbf{Random Condition:}$$
A random sample of 100 adults is provided. This ensures that the sample is representative of the population and that the observations are independent.
2. $$\textbf{10\% Condition:}$$
The sample size must be less than 10% of the entire population. Since 100 adults is very small compared to the total population of adults, this condition is satisfied:
$$100 < 0.10 \times \text{(adult population)}.$$
3. $$\textbf{Large Counts Condition:}$$
We must check that the expected counts of successes and failures under the null hypothesis are at least 10. Under the null hypothesis, the population proportion is
$$p_0 = 0.25.$$
The expected count for successes (those who say they are organized) is:
$$ n \cdot p_0 = 100 \cdot 0.25 = 25, $$
and the expected count for failures is:
$$ n \cdot (1-p_0) = 100 \cdot 0.75 = 75. $$
Both counts are at least 10, fulfilling this condition.
In summary, the conditions for making an inference about a population proportion are:
- Random: The sample of 100 adults is random.
- 10\% Condition: 100 is less than 10% of the adult population.
- Large Counts: $$ n\,p_0 = 25 \quad \text{and} \quad n(1-p_0) = 75, $$ both of which are greater than or equal to 10.
Thus, all conditions for inference are met, and the test can proceed.
1. $$\textbf{Random Condition:}$$
A random sample of 100 adults is provided. This ensures that the sample is representative of the population and that the observations are independent.
2. $$\textbf{10\% Condition:}$$
The sample size must be less than 10% of the entire population. Since 100 adults is very small compared to the total population of adults, this condition is satisfied:
$$100 < 0.10 \times \text{(adult population)}.$$
3. $$\textbf{Large Counts Condition:}$$
We must check that the expected counts of successes and failures under the null hypothesis are at least 10. Under the null hypothesis, the population proportion is
$$p_0 = 0.25.$$
The expected count for successes (those who say they are organized) is:
$$ n \cdot p_0 = 100 \cdot 0.25 = 25, $$
and the expected count for failures is:
$$ n \cdot (1-p_0) = 100 \cdot 0.75 = 75. $$
Both counts are at least 10, fulfilling this condition.
In summary, the conditions for making an inference about a population proportion are:
- Random: The sample of 100 adults is random.
- 10\% Condition: 100 is less than 10% of the adult population.
- Large Counts: $$ n\,p_0 = 25 \quad \text{and} \quad n(1-p_0) = 75, $$ both of which are greater than or equal to 10.
Thus, all conditions for inference are met, and the test can proceed.
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