Question

Consider a binomial experiment with 15 trials and probability 0.65 of success on a single trial.

(a) Use the binomial distribution to find the probability of exactly 10 successes. (Round your answer to three decimal places.)


(b) Use the normal distribution to approximate the probability of exactly 10 successes. (Round your answer to three decimal places.)


(c) Compare the results of parts (a) and (b).

These results are almost exactly the same.These results are fairly different.

Answer 1

a)

X ~ Bin( n,p)

Where n = 15 , p = 0.65

P(X) = ⁿC_x p^x ( 1 - p)^n-x

P(X = 10) = ¹⁵C₁₀ * 0.65¹⁰ * 0.35⁵

= 0.2123

b)

Mean = np = 15 * 0.65 = 9.75 ,

Standard deviation = sqrt (np(1-p) ) = sqrt ( 15 * 0.65 * 0.35) = 1.8473

Using normal approximation,

P(X < x) = P(Z < x - Mean / SD)

With continuity correction,

P(X = 10) = P(9.5 < X < 10.5)

= p(X < 10.5) - P(X < 9.5)

= P(Z < (10.5 - 9.75) / 1.8473) - P(Z < (9.5 - 9.75) / 1.8473)

= P(Z < 0.41) - P(Z < -0.14)

= 0.6591 - 0.4443

= 0.2148

c)

Difference between these two probabilities = 0.2148 - 0.2123 = 0.0025

Since this probability difference is less than 0.5, these results are almost exactly the same.