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Answer :
We first check that the conditions for inference in testing a proportion are satisfied.
1. Random Condition:
The sample of 100 adults is stated to be random, which satisfies this condition.
2. 10% Condition:
The sample of 100 adults is less than 10% of the adult population. This ensures that the sample is small relative to the population, satisfying the 10% condition.
3. Large Counts Condition:
We need to check that the expected counts under the null hypothesis are large. The null hypothesis states that the proportion is
[tex]$$p_0 = 0.25.$$[/tex]
The expected count of adults who are organized is
[tex]$$n p_0 = 100 \times 0.25 = 25,$$[/tex]
and the expected count of adults who are not organized is
[tex]$$n (1-p_0) = 100 \times 0.75 = 75.$$[/tex]
Both values, 25 and 75, are greater than 10, so the large counts condition is satisfied.
In summary, the conditions for inference are met because:
- The sample is random.
- The sample size (100) is less than 10% of the population.
- Both [tex]$n p_0 = 25$[/tex] and [tex]$n(1-p_0) = 75$[/tex] are at least 10.
Thus, all conditions required for using the normal approximation in this test are satisfied.
1. Random Condition:
The sample of 100 adults is stated to be random, which satisfies this condition.
2. 10% Condition:
The sample of 100 adults is less than 10% of the adult population. This ensures that the sample is small relative to the population, satisfying the 10% condition.
3. Large Counts Condition:
We need to check that the expected counts under the null hypothesis are large. The null hypothesis states that the proportion is
[tex]$$p_0 = 0.25.$$[/tex]
The expected count of adults who are organized is
[tex]$$n p_0 = 100 \times 0.25 = 25,$$[/tex]
and the expected count of adults who are not organized is
[tex]$$n (1-p_0) = 100 \times 0.75 = 75.$$[/tex]
Both values, 25 and 75, are greater than 10, so the large counts condition is satisfied.
In summary, the conditions for inference are met because:
- The sample is random.
- The sample size (100) is less than 10% of the population.
- Both [tex]$n p_0 = 25$[/tex] and [tex]$n(1-p_0) = 75$[/tex] are at least 10.
Thus, all conditions required for using the normal approximation in this test are satisfied.
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