Math QUestion?

The utility company wishes to estimate the percentage of people in the U.S. who use electricity to heat their homes. Past data indicate that 154 out of 700 customers surveyed used electricity to heat their homes. If the company wishes to be 98% confident to within 3%, how large a sample would be needed?





a. 516


b. 733


c. 854


d. 1036


e. 1266


f. 1503


g. 2066


h. none of these

Math QUestion?
The probability that k people in a sample of n people use electricity can be calculated using a binomial random variable..





P(K = k) = N-Choose-K p^k (1-p)^(n-k)





where p = 154/700 = .22


N-Choose-K = n! / [ (n-k)! * k!]





Let the lower limit be L1..


L1 = floor (.97 *.22 * n)





Let the upper limit be L2..


L2 = floor (1.03*.22*n)





Find the smallest n, such that


∑Pk ≥ .98


where the lower and upper limits to the summation are L1 and L2, respectively





When n=516, ∑Pk = .250


When n=733, ∑Pk = .344


When n=854, ∑Pk= .350


When n=1036, ∑Pk= .374


When n=1266, ∑Pk= .413


When n=1503, ∑Pk= .467


When n=2066, ∑Pk= .543





I calculated these in MathCad using this code:





N := 2066


L1 := floor(N*.22*.97)


L2 :=floor(N*.22*1.03)


pbinom(L2,N,.22) - pbinom(L1,N,.22) = .543





none of the answers came anywhere near .98.. however, when N=21091 =%26gt; .98 and when N=21090 =%26gt;.979





therefore N=21091


h) NONE OF THE ABOVE.