Answer :
To determine which condition for calculating a confidence interval for the population proportion [tex]\( p \)[/tex] has not been met, let's review the specific conditions:
1. Random Condition: The sample should be a simple random sample (SRS). According to the problem, Atoya conducted an SRS of 50 seniors, so this condition is met.
2. 10% Condition: The sample size should be no more than 10% of the population. In this case, the sample size is 50 and the population size is 175. We check if 50 is less than or equal to 10% of 175:
[tex]\[
10\% \text{ of } 175 = 0.1 \times 175 = 17.5
\][/tex]
Since 50 is greater than 17.5, the 10% condition is not met.
3. Large Counts Condition: This condition requires both [tex]\( n\hat{p} \geq 30 \)[/tex] and [tex]\( n(1-\hat{p}) \geq 30 \)[/tex], where [tex]\( \hat{p} \)[/tex] is the sample proportion. Let's calculate [tex]\( \hat{p} \)[/tex]:
[tex]\[
\hat{p} = \frac{14}{50} = 0.28
\][/tex]
Next, we compute:
[tex]\[
n\hat{p} = 50 \times 0.28 = 14
\][/tex]
[tex]\[
n(1-\hat{p}) = 50 \times (1 - 0.28) = 50 \times 0.72 = 36
\][/tex]
The Large Counts condition is not met since [tex]\( n\hat{p} = 14 \)[/tex], which is less than 30.
Given these checks, we can conclude that the 10% condition has not been met, as the sample size exceeds 10% of the population size. This explains why the correct statement is:
"The [tex]$10\%$[/tex] condition is not met because the sample size of 50 is greater than 10% of the population size (175) of seniors at the boarding school."
1. Random Condition: The sample should be a simple random sample (SRS). According to the problem, Atoya conducted an SRS of 50 seniors, so this condition is met.
2. 10% Condition: The sample size should be no more than 10% of the population. In this case, the sample size is 50 and the population size is 175. We check if 50 is less than or equal to 10% of 175:
[tex]\[
10\% \text{ of } 175 = 0.1 \times 175 = 17.5
\][/tex]
Since 50 is greater than 17.5, the 10% condition is not met.
3. Large Counts Condition: This condition requires both [tex]\( n\hat{p} \geq 30 \)[/tex] and [tex]\( n(1-\hat{p}) \geq 30 \)[/tex], where [tex]\( \hat{p} \)[/tex] is the sample proportion. Let's calculate [tex]\( \hat{p} \)[/tex]:
[tex]\[
\hat{p} = \frac{14}{50} = 0.28
\][/tex]
Next, we compute:
[tex]\[
n\hat{p} = 50 \times 0.28 = 14
\][/tex]
[tex]\[
n(1-\hat{p}) = 50 \times (1 - 0.28) = 50 \times 0.72 = 36
\][/tex]
The Large Counts condition is not met since [tex]\( n\hat{p} = 14 \)[/tex], which is less than 30.
Given these checks, we can conclude that the 10% condition has not been met, as the sample size exceeds 10% of the population size. This explains why the correct statement is:
"The [tex]$10\%$[/tex] condition is not met because the sample size of 50 is greater than 10% of the population size (175) of seniors at the boarding school."
