1. TanFin 12 8.6.013.CMI Use the appropriate normal distribution to approximate

Question

1. TanFin 12 8.6.013.CMI Use the appropriate normal distribution to approximate the resulting binomial distribution. A coin is weighted so that the probability of obtaining a head in a single toss is 0.6. If the coin is tossed 25 times, find the following probabilities. (Round your answers to four decimal places.) (a) fewer than 15 heads 0.4207 (b) between 15 and 19 heads, inclusive x 0.5464 (c) more than 21 heads 0.0040 You may need to use the appropriate table in the Appendix of Tables to answer this question. Grade This Hide Answer Try Again

Explanation / Answer

here, n = number of trials = 25

p = probability of success = 0.6

q = probability of failure = 1 - 0.6 = 0.4

the estimates for mean and variance are :

= np = 25 * 0.6 = 15 and 2 = npq = 25*0.6*0.4 = 6 respectively.

let X be the random variable which denotes the number of heads out of 25(n) tosses of the coin.

hence, (a) P(X<15) = P( (X-)/ < (15-)/ ) = P( (X-)/ < (15-15)/6 ) = P( (X-)/ < 0 ) = (0) = 0.5

(b) P(15<X<19) = P( (15-)/ < (X-)/ < (19-)/ ) = P( 15-15)/6 < (X-)/ < (19-15)/6 ) = P( 0 < (X-)/ < 1.63 ) = (1.63) - (0) = 0.9484 - 0.5 = 0.4484

(c) P(X>21) = P( (X-)/ > (21-)/ ) = P( (X-)/ > (21-15)/6 ) = P( (X-)/ > 6 ) = 1 - (6 ) = 1 - (2.45 ) = 1 - 0.9929 = 0.0071

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