## Solution for problem 16BSC Chapter 5.4

# Solved: Deciphering Messages The Central Intelligence

Elementary Statistics | 12th Edition

Problem 16BSC

Deciphering Messages The Central Intelligence Agency has specialists who analyze the frequencies of letters of the alphabet in an attempt to decipher intercepted messages that are sent as ciphered text. In standard English text, the letter e is used at a rate of 12.7%.

a. Find the mean and standard deviation for the number of times the letter e will be found on a typical page of 2600 characters.

b. In an intercepted ciphered message sent to France, a page of 2600 characters is found to have the letter e occurring 290 times. Is 290 unusually low or unusually high?

**Accepted Solution**

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Problem 16BSC

Answer:

Step1 of 2:

We have The Central Intelligence Agency has specialists who analyze the frequencies of letters of the alphabet in an attempt to decipher intercepted messages that are sent as ciphered text. In standard English text, the letter e is used at a rate of 12.7%.

That is p = 0.127

a).

The mean and standard deviation for the number of times the letter e will be found on a typical page of 2600 characters is given by

Consider a random variable “x” follows binomial distribution with sample size “n” and proportion “p”.

i,e.X B(n, p)

Where,

n = 2600 and p = 0.127.

q = 1 - p

= 1 - 0.127.

= 0.873

Probability mass function of binomial distribution is given by

P(x) = , x = 0,1,2,...,n.

With mean E(x) = np and variance var(x) = npq

Consider,

E(x) = np

= 26000.127

= 330.2

Therefore,E(x) = 330.2

var(x) = npq

= 26000.1270.873

= 288.2646

Standard deviation is given by =

=

= 16.9783.

17 letters.

Step2 of 2:

b).

In an intercepted ciphered message sent to France, a page of 2600 characters is found to have the letter e occurring 290 times.

Consider,A range rule It is especially important to interpret results. The range rule of thumb suggests that values are unusual if they lie outside of these limits:

Minimum = and Maximum =

1).Minimum =

= 330.2 - 2(17)

= 296.

2).Maximum =

= 330.2 + 2(17)

= 364.2

Therefore, 290 lies between intervals that is (296, 364.2) hence unusually effective.

###### Chapter 5.4, Problem 16BSC is Solved

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Solved: Deciphering Messages The Central Intelligence