In a particular experiment, an event E is believed to have probability 0.60. Sup

Question

In a particular experiment, an event E is believed to have probability 0.60. Suppose the experiment is run 570 times. Find a 95% Confidence Interval for the number of times E occurs. (That is, find a CI for the mean, and then multiply by 570.) A random variable U is believed to be roughly normally distributed. and to have a standard deviation of 5.3 or less. The experimenters want to estimate the mean, with 95% confidence, to the nearest 0.5. That is, they want to choose a number of repetitions such that, for the average a from the experiment, they have 95% confidence that the true mean is between a - 0.5 and a + 0.5. Since experiments require money, time and effort, they wish to find a suitable n that is smallest necessary. Find n.

Explanation / Answer

3.
Confidence Interval For Proportion
CI = p ± Z a/2 Sqrt(p*(1-p)/n)))
x = Mean
n = Sample Size
a = 1 - (Confidence Level/100)
Za/2 = Z-table value
CI = Confidence Interval
No. of success(x)=342
Sample Size(n)=570
Sample proportion = x/n =0.6
Confidence Interval = [ 0.6 ±Z a/2 ( Sqrt ( 0.6*0.4) /570)]
= [ 0.6 - 1.96* Sqrt(0) , 0.6 + 1.96* Sqrt(0) ]
= [ 0.56,0.64]

4.
Compute Sample Size ( n ) = n=(Z/E)^2*p*(1-p)
Z a/2 at 0.05 is = 1.96
Sample Proportion = 0.6
ME = 0.5
n = ( 1.96 / 0.5 )^2 * 0.6*0.4
= 3.688 ~ 4

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