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From the values you have taken it could be observed  that
you take the proportional value P = 0.5, Q = 1-P = 0.5. Therefore the estimayed  sample
standard deviation is sqrt (PQ/n)  for the sample size n  is sqrt((0.5)(0.5)/n) =
sqrt(0.25/n)

Let the observed value of the sample
proportion  be p

|p-P| = 3% of P= 0.03*(1/2) =
0.015.

So,   (Observed value, p - population Prportion
P)/sqrt(0.5^2/n is a  normal variate  p with mean P and variance PQ/n =
0.25/n

Therefore Pr( 0.5-0.015 < p <
0.5+0.015)  = 0.98 Or

Pr( Z < 0.515) <
0.99.

Therefore  (0.515-0.5)/sqrt(0.5^2/n) = 
2.3266.

Or

(0.015)(sqrtn)/0.25)
= 2.3266

n = 2.3266^2*0.25/0.015^2 =  6015
.

So I differ with your book also. The reason is that
instead of the sample proportion to vary 0.03 % of population  proportion on either
side, the work out shows p-P = 0.03 which amounts 0.03/(1/2)*100 % = 6% on either side
of the population proportion P =1/2.