Solution Found!
a. Show that the standard error of an estimated proportion
Chapter , Problem 23(choose chapter or problem)
a. Show that the standard error of an estimated proportion is largest when p = 1/2.
b. Use this result and Corollary B of Section 7.3.2 to conclude that the quantity
\(\frac{1}{2} \sqrt{\frac{N-n}{N(n-1)}}\)
is a conservative estimate of the standard error of \(\hat{p}\) no matter what the value of p may be.
c. Use the central limit theorem to conclude that the interval
\(\hat{p} \pm \sqrt{\frac{N-n}{N(n-1)}}\)
contains p with probability at least .95.
Questions & Answers
QUESTION:
a. Show that the standard error of an estimated proportion is largest when p = 1/2.
b. Use this result and Corollary B of Section 7.3.2 to conclude that the quantity
\(\frac{1}{2} \sqrt{\frac{N-n}{N(n-1)}}\)
is a conservative estimate of the standard error of \(\hat{p}\) no matter what the value of p may be.
c. Use the central limit theorem to conclude that the interval
\(\hat{p} \pm \sqrt{\frac{N-n}{N(n-1)}}\)
contains p with probability at least .95.
ANSWER:Step 1 of 3
a.
The standard error (S.E) of an estimated proportion without using the finite population correction (FPC) method is given as,
To maximize the standard error, differentiating p(1-p) with respect to p, the value of p is given as,
Again, differentiating with respect to p,
Since the double differentiation is negative, this implies that maximum appears at 