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 in the sample. ‖ ‖ step 2 ⋮ Using the asksia-ll calculation list, the number of successes is $200 imes 0.48 = 96$. ‖ ‖ step 3 ⋮ Calculate the number of failures in the sample. ‖ ‖ step 4 ⋮ Using the asksia-ll calculation list, the number of failures is $200 imes (1 - 0.48) = 104$. ‖ ‖ step 5 ⋮ Determine if the sample size is large enough for both successes and failures. ‖ ‖ step 6 ⋮ Since both the number of successes (96) and failures (104) are greater than 5, the sample size condition is met. ‖ ∻Answer∻ ⚹ The conditions for constructing this confidence interval that their sample met are A, B, and C. ⚹ ∻Key Concept∻ ⚹Conditions for Constructing a Confidence Interval⚹ ∻Explanation∻ ⚹For a one-sample $z$ interval for a proportion, the sample must be random (A), and both the observed counts of successes and failures must be sufficiently large (B), typically at least 5. Additionally, individual observations should be independent (C), which is generally assumed for a random sample. In this scenario, all conditions are met.⚹◊What formula would be used to calculate the margin of error for the one-sample z-interval in this scenario?⍭ Generate me a similar question◊

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