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(Please correct me if I'm wrong)
Here is my logic:
Let n be the sample size.
Let p' be the sample mean.(which is different from true mean).
Let S = np' be the no of smokers in the sample.
Let e = error rate in %(which is given as 0.005 in the question)
Concept of error
The error, if given as a percentage like 0.5% of the sample population n (=0.005), tells us about the standard deviation in terms of n.
Since, standard deviation of the binomial distro is,
The absolute value of the error is then 0.005n.
In this question, we are concerned with the error of the sample mean(np' - np).
Hence, its.
And we can set up the inequality:
(np - ne) < S < (np + ne)
with which to start solving the question numerically.
Concept of Confidence Interval
The range of S which we defined earlier must constitute 95% of the probabilities. The range of S is huge, stretching from 0 smoker to n (=100%) smokers. But the tail ends are not likely. Only the ones centred around the mean (plus or minus the standard deviation) matters. In fact, they constitute 95% of all probabilities = having a 95% chance of containing the true mean = 95% confidence interval.
Plotted on the standard normal curve, its the area within the range -1.96 to 1.96. These values and area are INVARIANT for the confidence interval of 95%. In fact, we use it so often we call it "within 2 standard deviations".
How to calculate sample size
We are given a fixed error rate (a number we cannot exceed but can go down), and a fixed confidence interval, but we can vary the n. We find a relationship between e and n:
So as n -->, e --> 0.
(can anyone help me out here?...I 'm trying to find an expression for e < n)
