Question

Suppose the true population proportion werep= 0.5 and a researcher takes asimple random sample of sizen= 50.

(a) Find and interpret the standard deviation of the sample proportion ˆp.
(b) Calculate the probability that the sample proportion will be larger than 0.55 for a random sample ofsize 50.

Answer 1

answer is almost 0 i.e impossible event.

Explanation:

given that the the true population proportion were

p= 0.5

a) Std deviation of sample proportion =
`\sqrt{(pq)/(n) } \\=0.0707`

This is the std deviation for a single trial. We know in binomial variance is npq. Using this we can say for a single trial, variance would be pq/n and hence std deviation of proportion is square root of pq/n

b) The probability that the sample proportion will be larger than 0.55 for a random sample ofsize 50

= P(p>0.55)

=
`P(Z>(0.05)/(0.0707) )\\= P(Z>7.07)\\<0.00001`