Question

A particular telephone number is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages, and consider a sample of 20 incoming calls.

(a)

What is the expected number of calls among the 20 that involve a fax message?

E(X) =

(b)

What is the standard deviation of the number among the 20 calls that involve a fax message? (Round your answer to three decimal places.)

σ_X =

(c)

What is the probability that the number of calls among the 20 that involve a fax transmission exceeds the expected number by more than 2 standard deviations? (Round your answer to three decimal places.)

Answer 1

According to the given question ,a  particular telephone number is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages, and consider a sample of 20 incoming calls.

Therefore the probability that the incoming calls involve fax messages is

p=0.25

(a) Therefore the expected number of calls among the 20 that involve a fax message is 5 which is determined as:

E(X) np20 x 0.25 5

(b) Therefore the standard deviation of the number among the 20 calls that involve a fax message is 1.9365 which is determined as:

np(1-p) =20 x 0.25 x (1 0.25) = 1.9365 o

(Round to three decimal places.)

(c) Therefore  the probability that the number of calls among the 20 that involve a fax transmission exceeds the expected number by more than 2 standard deviations is 0.014  which is determined as:

P[(X - 20,

1-P (X-4x)20

1-PX 20

1-PX< 5+2 x 1.9365

1-PX8.8729
we assume
8.87299

1-PX<9

-Σ 20Ck(0.25)(1 0.25) 20- 1- k-0

$\Rightarrow 1-0.9861$

0.0139

0.014

(Round to three decimal places.)

Note:

1-PX8.8729
we assume $8.8729\rightarrow 8$

$\Rightarrow 1-P\left [ X \leq 8]$

$\Rightarrow 1-\sum_{k=0}^{8}^{20}C_{k}(0.25)^{k}(1-0.25)^{20-k}$

$\Rightarrow 1-0.9590$

$\Rightarrow 0.0409$

0.041

(Round to three decimal places.