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Answer :
Sure! Let's solve the problem step-by-step to check if the conditions for calculating a confidence interval for the proportion [tex]\( p \)[/tex] are met.
### Step 1: Check the Random Condition
Latoya conducted a Simple Random Sample (SRS) of 50 students. In statistics, an SRS is a sample in which each member of the population has an equal chance of being selected. This means that the sample is random, and so the random condition is met.
### Step 2: Calculate the Sample Proportion ([tex]\(\hat{p}\)[/tex])
The sample proportion, denoted as [tex]\(\hat{p}\)[/tex], is calculated by dividing the number of students who like the cafeteria's food by the total number of students in the sample.
[tex]\[
\hat{p} = \frac{\text{Number of students who like the food}}{\text{Total number of students in the sample}} = \frac{14}{50} = 0.28
\][/tex]
### Step 3: Check the Large Counts Condition
The Large Counts condition helps ensure that the sampling distribution of [tex]\(\hat{p}\)[/tex] is approximately normal. To check this condition, calculate:
- [tex]\(n \hat{p}\)[/tex]
- [tex]\(n(1 - \hat{p})\)[/tex]
where [tex]\( n \)[/tex] is the sample size.
1. Calculate [tex]\(n \hat{p}\)[/tex]:
[tex]\[
n \hat{p} = 50 \times 0.28 = 14
\][/tex]
2. Calculate [tex]\(n (1 - \hat{p})\)[/tex]:
[tex]\[
n (1 - \hat{p}) = 50 \times (1 - 0.28) = 50 \times 0.72 = 36
\][/tex]
For the Large Counts condition to be satisfied, both [tex]\(n \hat{p}\)[/tex] and [tex]\(n (1 - \hat{p})\)[/tex] must be at least 10. Here, [tex]\( n \hat{p} = 14 \)[/tex] and [tex]\( n (1 - \hat{p}) = 36 \)[/tex], so both are greater than 10.
### Conclusion
Since both the Random and Large Counts conditions are met, it is appropriate to calculate a confidence interval for the population proportion [tex]\( p \)[/tex].
Thus, the conditions for inference are met.
### Step 1: Check the Random Condition
Latoya conducted a Simple Random Sample (SRS) of 50 students. In statistics, an SRS is a sample in which each member of the population has an equal chance of being selected. This means that the sample is random, and so the random condition is met.
### Step 2: Calculate the Sample Proportion ([tex]\(\hat{p}\)[/tex])
The sample proportion, denoted as [tex]\(\hat{p}\)[/tex], is calculated by dividing the number of students who like the cafeteria's food by the total number of students in the sample.
[tex]\[
\hat{p} = \frac{\text{Number of students who like the food}}{\text{Total number of students in the sample}} = \frac{14}{50} = 0.28
\][/tex]
### Step 3: Check the Large Counts Condition
The Large Counts condition helps ensure that the sampling distribution of [tex]\(\hat{p}\)[/tex] is approximately normal. To check this condition, calculate:
- [tex]\(n \hat{p}\)[/tex]
- [tex]\(n(1 - \hat{p})\)[/tex]
where [tex]\( n \)[/tex] is the sample size.
1. Calculate [tex]\(n \hat{p}\)[/tex]:
[tex]\[
n \hat{p} = 50 \times 0.28 = 14
\][/tex]
2. Calculate [tex]\(n (1 - \hat{p})\)[/tex]:
[tex]\[
n (1 - \hat{p}) = 50 \times (1 - 0.28) = 50 \times 0.72 = 36
\][/tex]
For the Large Counts condition to be satisfied, both [tex]\(n \hat{p}\)[/tex] and [tex]\(n (1 - \hat{p})\)[/tex] must be at least 10. Here, [tex]\( n \hat{p} = 14 \)[/tex] and [tex]\( n (1 - \hat{p}) = 36 \)[/tex], so both are greater than 10.
### Conclusion
Since both the Random and Large Counts conditions are met, it is appropriate to calculate a confidence interval for the population proportion [tex]\( p \)[/tex].
Thus, the conditions for inference are met.
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