A market analyst is curious what proportion of Los Angeles residents have a landline telephone. A survey of 200 randomly selected Los Angeles residents shows that $48 \%$ of those selected have a landline telephone. The analyst 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 market analyst is curious what proportion of Los Angeles residents have a landline telephone. A survey of 200 randomly selected Los Angeles residents shows that $48 \%$ of those selected have a landline telephone. The analyst 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 ⋮ Calculate the number of successes using the percentage and the sample size: $200 imes 48\% = 96$. ‖ ‖ step 2 ⋮ Calculate the number of failures using the percentage and the sample size: $200 imes 52\% = 104$. ‖ ‖ step 3 ⋮ Check the conditions for constructing a one-sample $z$ interval for a proportion: ‖ ‖ step 4 ⋮ Condition A is met because the data is stated to be a random sample from the population of interest. ‖ ‖ step 5 ⋮ Condition B is met because the observed counts of successes ($96$) and failures ($104$) are both greater than or equal to $10$. ‖ ‖ step 6 ⋮ Condition C is met if we assume that the sample size is less than $10\%$ of the entire population, which is reasonable given the population size of Los Angeles. ‖ ∻1 Answer∻ ***A, B, C*** ∻Key Concept∻ ⚹Conditions for constructing a one-sample $z$ interval for a proportion⚹ ∻Explanation∻ ⚹To construct a one-sample $z$ interval for a proportion, the sample must be random, the observed counts of successes and failures must be sufficiently large (typically at least 10), and individual observations must be independent. If the sample size is less than $10\%$ of the population, the independence condition is generally satisfied.⚹◊What is the formula for calculating the margin of error for the one-sample z interval for a proportion?⍭ Generate me a similar question◊

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