Question 1043433: A researcher wishes to estimate, with 99% confidence, the population proportion of adults who say chocolate is their favorite ice cream flavor. Her estimate must be accurate within 3% of the population proportion.
(a) No preliminary estimate is available. Find the minimum sample size needed.
(b) Find the minimum sample size needed, using a prior study that found that 28% of the respondents said their favorite flavor of ice cream is chocolate.
(c) Compare the results from parts (a) and (b).
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! The error must be +/-0.03
For 99% CI, z(0.995) is 2.576.
the SE is sqrt {0.5*0.5/n}. We use 0.5, because of all possible proportions, it gives the greatest number, which is the most error. For proportions that add to 1, it can be shown that the product is greatest when each proportion is 0.5
Therefore, for unknown proportion, the SE is sqrt (0.25/n) or 0.5/sqrt(n)
1.96*0.5/sqrt (n)=0.03
multiply by sqrt (n) and divide by 0.03,
0.98/0.03=sqrt (n)
square both sides
n=1067.111 or 1068
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When a proportion is known, the SE becomes sqrt {(0.28(0.72)/n}
This is 0.4490/sqrt(n)
Do the same operations
sqrt(n)=1.96*sqrt(0.4490)/0.03
That is 0.88/0.03=29.334
n=29.334^2=860.51 or 861.
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When a proportion can be estimated, and the estimate is some distance from 0.5, the sample size needed for a given error result will be significantly less than if the default 0.5 is used.
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