A school administrator is curious what proportion of the 2,400 students at the school regularly eat breakfast. They take an SRS of 150 of these students, and 104 of them report that they regularly eat breakfast. The administrator wants to use this data to construct a one-sample $z$ interval for a proportion.

Which conditions for constructing this confidence interval did their sample meet?

Choose all answers that apply:
A The data is a random sample from the population of interest.
B The observed counts of successes and failures are both sufficiently large.
c Individual observations can be considered independent.

Question

A school administrator is curious what proportion of the 2,400 students at the school regularly eat breakfast. They take an SRS of 150 of these students, and 104 of them report that they regularly eat breakfast. The administrator wants to use this data to construct a one-sample $z$ interval for a proportion.

Which conditions for constructing this confidence interval did their sample meet?

Choose all answers that apply:
A The data is a random sample from the population of interest.
B The observed counts of successes and failures are both sufficiently large.
c Individual observations can be considered independent.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ The first condition to check is whether the data is a random sample from the population of interest. The problem states that the administrator takes a Simple Random Sample (SRS) of 150 students from the 2,400 students. This satisfies condition A. ‖ ‖ step 2 ⋮ The second condition to check is whether the observed counts of successes and failures are both sufficiently large. The rule of thumb is that both $np$ and $n(1-p)$ should be at least 10. Here, $n = 150$, $p = \frac{104}{150} \approx 0.693$. Thus, $np = 150 imes 0.693 \approx 104$ and $n(1-p) = 150 imes (1 - 0.693) \approx 46$. Both values are greater than 10, so condition B is satisfied. ‖ ‖ step 3 ⋮ The third condition to check is whether individual observations can be considered independent. Since the sample size (150) is less than 10% of the population size (2,400), the individual observations can be considered independent. This satisfies condition C. ‖ ∻Answer∻ ***A, B, C*** ∻Key Concept∻ ⚹Conditions for constructing a confidence interval for a proportion⚹ ∻Explanation∻ ⚹The conditions include having a random sample, sufficiently large counts of successes and failures, and independent observations.⚹◊What is the formula for calculating the margin of error in a one-sample z interval for a population proportion?⍭ Generate me a similar question◊

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