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 297 employees. what is the probability that company a will get the discount? 0.0000 (round to four decimal places as needed.) (b) company b has 506 employees. what is the probability that company b will get the discount? 0.1949 (round to four decimal places as needed.) (c) company c has 1043 employees. what is the probability that company c will get the discount? (round to four decimal places as needed.)

Explanation:

Step1: Identify the distribution

This is a binomial - like problem that can be approximated by a normal distribution. Let $X$ be the number of employees who earn the prevailing minimum wage or less. The mean of the binomial distribution is $\mu = np$ and the standard deviation is $\sigma=\sqrt{np(1 - p)}$, where $n$ is the number of employees and $p = 0.05$.

Step2: Calculate mean and standard deviation for Company C

For Company C, $n = 1043$ and $p=0.05$.
$\mu=np=1043\times0.05 = 52.15$
$\sigma=\sqrt{np(1 - p)}=\sqrt{1043\times0.05\times(1 - 0.05)}=\sqrt{1043\times0.05\times0.95}=\sqrt{49.5425}\approx7.04$

Step3: Standardize the value

We want to find $P(X\geq30)$. Using the continuity - correction for the normal approximation to the binomial, we find $P(X\geq29.5)$. The z - score is $z=\frac{x-\mu}{\sigma}$, where $x = 29.5$.
$z=\frac{29.5 - 52.15}{7.04}=\frac{-22.65}{7.04}\approx - 3.22$

Step4: Find the probability

$P(X\geq29.5)=1 - P(X\lt29.5)=1 - P(Z\lt - 3.22)$. From the standard normal table, $P(Z\lt - 3.22)=0.0006$. So $P(X\geq29.5)=1 - 0.0006 = 0.9994$

Answer:

$0.9994$