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A mobile phone company wants to be 90% confident its estimated proportion of text-messaging customers aged 50+ is within EBP = 0.03 of the truth, but has no prior estimate of p'. Why does the sample-size formula use p' = q' = 0.5 rather than a guess closer to zero or one?
A0.5 is always the true proportion in real populations
Bp'q' = 0.25 is the largest possible product, giving the largest safe n
CUsing 0.5 makes the margin of error disappear entirely
DRegulations require every survey to assume a 50-50 split
Answer & Solution
Correct answer: B. p'q' = 0.25 is the largest possible product, giving the largest safe n
1. The sample-size formula for a proportion contains the product p'q' in its numerator.
2. Trying other splits, such as (0.6)(0.4)=0.24 or (0.3)(0.7)=0.21, all give smaller products than (0.5)(0.5)=0.25.
3. Since a larger p'q' calls for a larger sample size, using 0.5 for both guarantees the resulting n is large enough no matter the true proportion.
4. Option A is false in general; 0.5 is simply the worst-case choice for sample-size planning, not a claim about any specific population.
_Source: OpenStax Introductory Statistics (CC BY 4.0), Ch 8 "Confidence Intervals", section 8.3 A Population Proportion_
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