Question

A survey found that 25% of pet owners had their pets bathed professionally rather than do it themselves. 18 pet owners are randomly selected.

(Please with excel function)

a) What is n?

b) What is p?

c) What is 1 – p?

Find the probability that:

d) Exactly 12 pet owners had their pets bathed professionally

e) More than 12 pet owners had their pets bathed professionally.

f) No more than 12 pet owners had their pets bathed professionally.         

g) At least 12 pet owners had their pets bathed professionally.

h) Less than 12 pet owners had their pets bathed professionally.

i) Between 10 and 15 (inclusive) pet owners had their pets bathed professionally.

Thank you . I will rate answer right away.

Answer 1

Let us consider X be the random variable that denotes the number of pet owners have their pets bathed professionally.

Clearly,

$X\sim bin (18, 25)$

Therefore, probability mass function (p.m.f) is,

$P(X) = C_{x}^{n} p^{x} q^{n-x}$

$P(X) = C_{x}^{18}*0.25^{x} (1-0.25)^{18-x}$

a) n = 18

b) p = 0.25

c) 1 - p = 1 - 0.25 = 0.75

d) Exactly 12 pet owners had their pets bathed professionally

i.e, X = 12

$P(X) = C_{x}^{18}*0.25^{x} (1-0.25)^{18-x}$

$P(X=12) = C_{12}^{18}*0.25^{12} (1-0.25)^{18-12}$

P(X = 12) = 0.0002

e) More than 12 pet owners had their pets bathed professionally. i.e Find P(X >12)

P(X > 12) = P(X = 13) + P(X = 14) + P(X = 15) + P(X = 16) + P(X = 17) + P(X = 18)

$P(X>12) = C_{13}^{18} *0.25^{13} (1-0.25)^{18-13} + ................+ C_{18}^{18} *0.25^{18} (1-0.25)^{18-18}$

P(X > 12) = 0.000030 + 0.000 + 0.00 + 0.00 + 0.00 + 0.00

P(X > 12) = 0.000

f) No more than 12 pet owners had their pets bathed professionally. i.e Find: $P(X\leq 12)$

$P(X\leq 12) = P(X = 0) + P(X = 1) + ...................+ P(X = 12)$

$P(X\leq 12) = C_{0}^{18} *0.25^{0} (1-0.25)^{18-0} + ................+ C_{12}^{18} *0.25^{12} (1-0.25)^{18-12}$

$P(X\leq 12) = 0.0056 + 0.0338 + 0.0958 + 0.1704 + 0.2130 + 0.1988 + 0.1436 + 0.0820 + 0.0376 + 0.0139 + 0.0042 + 0.0010 + 0.0002$

  

P(X $\leq$ 12) = 1.000

g) At least 12 pet owners had their pets bathed professionally.

$P(X\geq 12) = P(X=12) + P(X=13) + P(X = 14) + P(X = 15) + P(X = 16) + P(X = 17) + P(X = 18)$

$P(X\geq 12) = 0.000$

h) Less than 12 pet owners had their pets bathed professionally.

P(X < 12) = P(X = 0) + .............P(X = 11)

P(X < 12) = 1.000

i) Between 10 and 15 (inclusive) pet owners had their pets bathed professionally.

$P(10 \leq X\leq 15) = P(X = 10) &plus; ..........&plus; P(X = 15)$

$P(10 \leq X\leq 15)$ = 0.004 + 0.001 + 0.000 + 0.000 + 0.000

$P(10 \leq X\leq 15) = 0.005$