The sample proportion ˆp can be defined as ˆp = X/n, where X B(n, p). For the fo

Question

The sample proportion ˆp can be defined as ˆp = X/n, where X B(n, p). For the four cases given as follows compute the probabilities P(ˆp 0.3) using normal approximations (i.e., ˆp N(p, p p(1 p)/n)). Intuitively (or based on what you have observed in (a)), which approximation(s) do you think may be the most accurate one? (Note: the sample proportion ˆp has no exact distribution available, so we have to use normal approximations.)

(i) choose n = 10 and p = 0.2, and then compute P(X 3) using both methods;

(ii) choose n = 10 and p = 0.4, and then compute P(X 3) using both methods;

(iii) choose n = 50 and p = 0.2, and then compute P(X 8) using both methods;

(iv) choose n = 50 and p = 0.4, and then compute P(X 8) using both methods;

Explanation / Answer

When n =10, p= 0.2, exact binomial probability nearly equal to first normal approximation but not in second approximation.

When n =10, p= 0.4, exact binomial probability nearly not equal to first normal approximation and also not in second approximation.

When n =50, p= 0.2, exact binomial probability nearly not equal to first normal approximation and also not in second approximation.

When n =50, p= 0.4, exact binomial probability nearly not equal to first normal approximation and also not in second approximation.

n p n*p sqrt(p*n*p*(1-p)) sqrt(p*p*(1-p)/n) Exact Binomial ProbP Prob using first Approximation Prob using secod Approximation 10 0.2 2 0.5657 0.0566 0.8791 0.9615 1 10 0.4 4 0.9798 0.098 0.3823 0.1537 1 50 0.2 10 1.2649 0.0253 0.3037 0.0569 1 50 0.4 20 2.1909 0.0438 0.0002 2.2E-08 1

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.