Out of 5000 students at a university, 1000 get picked at random (with replacemen

Question

Out of 5000 students at a university, 1000 get picked at random (with replacement). These students get asked if they know the name of the dean. Let p be the proportion of students among the 5000 which know the name of the dean. Out of the 1000 surveyed, 800 know the name of the dean.

a) Estimate p.
b) Give a 95% confidence interval for p.
c) Assume that the school would have 100000 students instead of 5000. Assume again that we pick 1000 at random (from the 100000) and that 800 know the name of the dean. Give a 95%- confidence interval for p. What do you notice?
d) If we survey 100 times more people, how much smaller does the 95%-confidence interval get? Why?

Explanation / Answer

a)

p = 800/1000 = 0.8

b)

standard error = sqrt( p * (1-p)/n) = sqrt( 0.8 * 0.2/5000) = 0.00566

margin of error = 1.96 * 0.00566 = 0.0111

lower bound = 0.8 - 0.0111 = 0.789

upper bound = 0.8 + 0.0111 = 0.811

c)

standard error = sqrt( p * (1-p)/n) = sqrt( 0.8 * 0.2/10000) = 0.00127

margin of error = 1.96 * 0.00127 = 0.0025

lower bound = 0.8 - 0.0025 = 0.7975

upper bound = 0.8 + 0.0025 = 0.8025

d)

it will almost equal to 0.8 ,since we have bigger data set we can predict more accurately

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