About 5​% of hourly paid workers in a region earn the prevailing minimum wage or less. A grocery chain offers discount rates to companies that have at least 30 employees who earn the prevailing minimum wage or less. Complete parts​ (a) through​ (c) below. ​

a. Company A has 285 employees. What is the probability that Company A will get the​ discount? ​(Round to four decimal places as​ needed.)
b. Company B has 502 employees. What is the probability that Company B will get the​ discount? ​(Round to four decimal places as​ needed.)
c. Company C has 1033 employees. What is the probability that Company C will get the​ discount? ​(Round to four decimal places as​ needed.)

Answer:

a. 0.0000

b. 0.1841

c. 0.9992

Explanation:

a. n = 285

p = 5% = 0.05

μ = np = 285 x 0.05

= 14.25

we fnd the standard deviation

sd = √np(1-p)

= [tex]\sqrt{285*0.05*0.95}[/tex]

= 3.6793

we find the z score

x = 30-0.5 = 29.5

[tex]z=\frac{29.5-14.25}{3.6793} \\= 4.14[/tex]

using the microsoft excel function

1-NORMSDIST(4.14)

probability = 1 -0.999982

= 0.0000

b.

n = 502

p = 0.05

np = 502x0.05

= 25.1

sd = [tex]\sqrt{np(1-p)}[/tex]

= [tex]\sqrt{502*0.05*0.95} \\= 4.8831[/tex]

x = 29.5

[tex]z = \frac{29.5-25.1}{4.8831} \\= 0.90[/tex]

1 - NORMSDIST(0.90)

= 1 - 0.815939875

PROB = 0.1841

c. n = 1033

p = 0.05

np = 1033*0.05

= 51.65

sd  [tex]\sqrt{np(1-p)} \\= \sqrt{1033*0.05*0.95}[/tex]

= 7.0048

x = 29.5

[tex]z=\frac{29.5-51.65}{7.0048} \\= -3.16[/tex]

probability =

1 - normsdist(-3.16)

= 1 - 0.000788846

= 0.9992