32. A government agency has specialists who analyze the frequencies of letters o

Question

32. A government agency has specialists who analyze the frequencies of letters of the alphabet n an attempt to decipher intercepted messages. In standard English text, a particular letter is used at a rate of 7.9% a. Find the mean and standard deviation for the number of times this letter will be found on a typical page of 3200 characters (Round to one decimal place as needed.) b. In an intercepted message, a page of 3200 characters is found to have the letter occurring 301 times. Is this unusual? O A. Yes, because 301 is below the minimum usual value. O B. Yes, because 301 is within the range of usual values. ° C. No, because 301 is within the range of usual values. D. Yes, because 301 is greater than the maximum usual value

Explanation / Answer

a. There are two possible outcomes-success (leter is found in a typical page) and failure (letter is not found in a typical page), and the probability of success, p=0.079. There are n=3200 independent, random trials, and probability of success is constant throughout the trials. This accounts for binomial distribution. Now, note that n is too large and p is comparitively too small, which makes the task of applying binomial distribution to Bernoulli trials a difficult one. Since, both np=3200*0.079=252.8 and n(1-p)=3200*(1-0.079)=2947.2 are atleast 10, use normal approximation to binomial distribution.

Mean, mu=np=3200*0.079=252.8

Standard deviation, sigma=sqrt np(1-p)=sqrt 3200*0.079*(1-0.079)=15.3

b. The proportion of the letter occuring in an intercepted message is: 301/3200=0.09. An event is called unusual, if the proportion of occurrence of the event is less than 5%. Since, 0.09>0.05, the event is not unusual. Ans>C.

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