Answer :
We begin by checking each condition required for using an inference procedure for a population proportion.
1. **Random Condition:**
The problem states that the sample of 100 adults is a random sample. Therefore, the random condition is satisfied.
2. **10% Condition:**
The sample size should be less than 10% of the population of adults. Since 100 adults is much less than 10% of the entire adult population, this condition is met.
3. **Large Counts Condition:**
Under the null hypothesis ($p_0 = 0.25$), we calculate the expected number of successes and failures.
- The expected number of successes is
$$
n p_0 = 100 \times 0.25 = 25.
$$
- The expected number of failures is
$$
n(1-p_0) = 100 \times 0.75 = 75.
$$
Both expected counts ($25$ and $75$) are at least $10$, so the large counts condition is also satisfied.
Thus, all the conditions for the inference are met:
- Random: We have a random sample of 100 adults.
- 10%: 100 adults < 10% of the population.
- Large Counts: $n p_0 = 25$ and $n(1-p_0) = 75$, and both are at least $10$.
1. **Random Condition:**
The problem states that the sample of 100 adults is a random sample. Therefore, the random condition is satisfied.
2. **10% Condition:**
The sample size should be less than 10% of the population of adults. Since 100 adults is much less than 10% of the entire adult population, this condition is met.
3. **Large Counts Condition:**
Under the null hypothesis ($p_0 = 0.25$), we calculate the expected number of successes and failures.
- The expected number of successes is
$$
n p_0 = 100 \times 0.25 = 25.
$$
- The expected number of failures is
$$
n(1-p_0) = 100 \times 0.75 = 75.
$$
Both expected counts ($25$ and $75$) are at least $10$, so the large counts condition is also satisfied.
Thus, all the conditions for the inference are met:
- Random: We have a random sample of 100 adults.
- 10%: 100 adults < 10% of the population.
- Large Counts: $n p_0 = 25$ and $n(1-p_0) = 75$, and both are at least $10$.
