Question

5.54 A survey by Frank N.Magid Associates revealed that 3% of Americans are not connected to the Internet at home. Another researcher randomly selects 70 Americans. a. What is the expected number of these who would not be connected to the Internet at home?
b. What is the probability that eight or more are not connected to the Internet at home? c. What is the probability that between three and six (inclusive) are not connected to the Internet at home?

5.51 An office in Albuquerque has 24 workers including management. Eight of the workers commute to work from the west side of the Rio Grande River.Suppose six of the office workers are randomly selected. a. What is the probability that all six workers commute from the west side of the Rio Grande?

b. What is the probability that none of the workers commute from the west side of the Rio Grande?

c. Which probability from parts (a) and (b) was greatest? Why do you think this is?

d. What is the probability that half of the workers do not commute from the west side of the Rio Grande?

Answer 1

Question 5.54

P(Americans not connected to the internet at home) = 0.03

sample size = n = 70

(a) Expected number of those not connected to the internet at home = 70 * 0.03 = 2.1

(b) Here if x is the number of people not connected at home out of 70 Americans,

P(x 8) =1 - P(X < 8)

=1 - BINOMDIST(8; 70 ; 0.03; true)

=1 - 0.9988 = 0.0012

(c) then

P(3 x 6) = P(x 6) - P(x < 3)

Now as x ~ BINOMIAL(n = 70; p = 0.03)

we can't use normal approximation as np < 5

so we will use binomial table to calculate the probability.

P(3 x 6) = P(x 6) - P(x < 3)

= BINOMDIST(6; 70 ; 0.03; True) - P(2; 70 ; 0.03; true)

= 0.9950 - 0.6492 = 0.3457

Question 5.51

Here total workers = N = 24

Workers commute to work from the west side of the Rio Grande River= K = 8

Selected workers = n = 6

(a) Here if x is the number of workers out of 6 that comes from west side.

then x ~ HYPERGEOMETRIC(N = 24; K = 8 ; n = 6)

p(x) = ⁸C_x¹⁶C_(6-x)/ ²⁴C₆

now we have to find all seix workers from the west side of the Rio Grande

P(x = 6) = ⁸C₆¹⁶C₀/ ²⁴C₆ = 0.0002

(b) P(x = 0) = ⁸C₀¹⁶C₆/ ²⁴C₆ = 0.0595

(c) Here probability b is greater than probability as it is higher chance to have all person from the east side of the river instead of west side as there are 8 workers are from west side and 16 from east side.

(d) P(Half of the workers do not cummute from the west side)

= P(x = 3) = ⁸C₃¹⁶C₃/ ²⁴C₆ = 0.2330