Answer: The minimum number of students you need if you want the margin of error to be 5% IS 278.
Explanation:
Cochran’s Sample Size Formula gives the minimum number of students as [tex]n= \frac{z^{2}pq }{e^{2} } [/tex]
Where:
e is the desired level of precision (i.e. the margin of error),
p is the (estimated) proportion of the population which has the attribute in question and q is 1 – p.
The z-value for 95% confidence interval is found to be 1.96 in a Z table.
Assuming that half of the teenagers favor the elimination of a curfew: this gives us maximum variability. So p = 0.5 and q=0.5.
Then [tex]n= (\frac{1.96^{2}*0.5*0.5 }{0.05^{2} } )[/tex]
[tex]n= frac{0.9604}{0.0025} } [/tex]
[tex]n= {384.16 } [/tex]
Rounding up, [tex]n= \frac{385 } [/tex]
But considering that 1000 is a small population, we can modify the sample size we calculated above formula by using this equation:
[tex]s = \frac{n}{1 + \frac{n - 1}{N} }[/tex]
Where s is the adjusted sample size, n is the original sample size we calculated and N is the population size.
[tex]s = \frac{385}{1 + \frac{385 - 1}{1000} }[/tex]
[tex]s = \frac{385}{1 + \frac{384}{1000} }[/tex]
[tex]s = 278[/tex]
