Question

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 300 employees. What is the probability that Company A will get the discount? (Round to four decimal places as needed.) (b) Company B has 516 employees. What is the probability that Company B will get the discount? (Round to four decimal places as needed.) (c) Company C has 1015 employees. What is the probability that Company C will get the discount? (Round to four decimal places as needed.)

Answer 1

SOLUTION:

From given data,

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.

Probability (p) = 5% = 5/100 = 0.05

Where ,

Use Binomial distribution mean ($\mu$) and standard deviation ($\sigma$)

​(a) Company A has 300 employees. What is the probability that Company A will get the​ discount?

Sample size = n = 300

$\mu$ = n*p = 300*0.05 = 15

$\sigma$ = sqrt(np (1-p)) = sqrt(300*0.05(1-0.05)) = sqrt(300*0.05*0.95) = 3.7749

Now we have to find

Use continuity correction factor if, P(X > x) then use P(X > x-0.5)

P(X > 30) = P(X > 30-0.5) = P(X > 29.5)

P(X > 30) = P(X-$\mu$ / $\sigma$   >   29.5-15 / 3.7749)

P(X > 30) = P(Z  >   14.5 / 3.7749)

P(X > 30) = P(Z  >   3.84)

P(X > 30) = 1- P(Z  <   3.84)

P(X > 30) = 1- 0.99994

P(X > 30) = 0.00006

P(X > 30) = 0.0001

The probability that Company A will get the​ discount is 0.0001

​(b) Company B has 516 employees. What is the probability that Company B will get the​ discount?

Sample size = n = 516

$\mu$ = n*p = 516*0.05 = 25.8

$\sigma$ = sqrt(np (1-p)) = sqrt(516*0.05(1-0.05)) = sqrt(516*0.05*0.95) = 4.9508

Now we have to find

Use continuity correction factor if, P(X > x) then use P(X > x-0.5)

P(X > 30) = P(X > 30-0.5) = P(X > 29.5)

P(X > 30) = P(X-$\mu$ / $\sigma$   >   29.5-25.8/ 4.9508)

P(X > 30) = P(Z  >   3.7 / 4.9508)

P(X > 30) = P(Z  >   0.75)

P(X > 30) = 1- P(Z  <    0.75)

P(X > 30) = 1- 0.77337

P(X > 30) = 0.22663

P(X > 30) = 0.2266

The probability that Company B will get the​ discount is 0.2266

​(c) Company C has 1015 employees. What is the probability that Company C will get the​ discount?

Sample size = n = 1015

$\mu$ = n*p = 1015*0.05 = 50.75

$\sigma$ = sqrt(np (1-p)) = sqrt(1015*0.05(1-0.05)) = sqrt(1015*0.05*0.95) = 6.9435

Now we have to find

Use continuity correction factor if, P(X > x) then use P(X > x-0.5)

P(X > 30) = P(X > 30-0.5) = P(X > 29.5)

P(X > 30) = P(X-$\mu$ / $\sigma$   >   29.5-50.75/ 6.9435)

P(X > 30) = P(Z  >   -21.25 / 6.9435)

P(X > 30) = P(Z  >  -3.06)

P(X > 30) = 1- P(Z  <   -3.06)

P(X > 30) = 1- 0.00111

P(X > 30) = 0.99889

P(X > 30) = 0.9989

The probability that Company C will get the​ discount is 0.9989

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