Question

6. In this question, you are going to study the approximation to binomial probabilities using the nor mal distribution. The binomial distribution is discrete while the normal distribution is continuous Therefore, we would expect some issues with approximating the binomial with the normal. (a) (2 points) Suppose X ~ Bin (25,04). Calculate E (N) and Var . (b) (4 points) Use the central lit theorem along with (a) to approximate Pr (X 8). Compare this with your result in #4(a). Verify the sample size condition for the binomial case where p is known rather than estimated. What do you notice? with your result in # 4(b), what do you notice? Use this fact to improve the approximation obtained in (c) (c) (3 points) Use the central limit theorem along with (a) to approximate Pr (X<8). Compare this (d) (2 points) Explain why in the context of the binomial distribution Pr (X <8)-Pr(7.5)

Answer 1

(a)

E($\bar{X}$) = E(X) = np = 25 * 0.4 = 10

If $\bar{X}$ is mean of X, then Var($\bar{X}$) = p(1-p)/n = 0.4 * (1 - 0.4) / 25 = 0.0096

Var(X) = np(1-p) = 25 * 0.4 * (1 - 0.4) = 6

(b)

Pr(X $\le$ 8) = Pr(Z $\le$ (8 - 10)/6) = Pr(Z $\le$ -0.3333) = 0.3695

Sample size condition is np $\ge$ 5 and n(1-p) $\ge$ 5

np = 25 * 0.4 is greater than 5

n(1-p) = 25 * (1 - 0.4) = 15 which is greater than 5

Thus, the sample size condition is met.

(c)

Pr(X < 8) = Pr(Z < (8 - 10)/6) = Pr(Z < -0.3333) = 0.3695

(d)

Because the normal distribution can take all real numbers (is continuous) but the binomial distribution can only take integer values (is discrete), a normal approximation to the binomial should identify the binomial event "< 8" with the normal interval "< 7.5". To determine the approximate probability of observing at most 8 successes, we would find the area under the normal curve from X = 0 to X = 7.5 since, on a continuum, 7.5 is the upper boundary of X.