Question 533783
Solution using Binomial Distribution (Exact Solution)
X~Binomial(n=1000,p=0.8)
77% of 1000 = 770, use 769 as upper limit
probability that fewer than 77% of the resident in the sample are right-headed is equal to 
{{{sum( ( matrix( 2, 1,1000, x ))*(0.8^x)*(0.2^(1000-x)), x=0, 769) = highlight(0.00875))}}}
Solution using Normal Distribution (Approximate Solution)
{{{mu = np = 1000(0.8) = 800}}}
{{{ sigma^2 = npq = 1000(0.8)(0.2) = 160}}}
we use continuity correction by adding 0.5 to 769
{{{P(X<=769.5) = P(Z<=(769.5-800)/sqrt(160))}}}
               ={{{P(Z<=-2.4112)}}}
               = {{{highlight(0.00795)}}}

It is unusual to find that fewer than 77% of the resident in the sample are right-headed because the probability is less than 0.05.