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 298 employees. What is the probability that Company A will get the discount? 0.0001 (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 1017 employees. What is the probability that Company C will get the discount? $$ \square $$ (Round to four decimal places as needed.)

Question

UpStudy AI · Accepted Answer

To determine the probability that **Company C** with **1,017 employees** will qualify for the discount (i.e., have at least **30 employees** earning the prevailing minimum wage or less), we can use the **normal approximation** to the binomial distribution. Here's a step-by-step breakdown:

### Given:
- **Total Employees (n)**: 1,017
- **Probability of Success (p)**: 5% or 0.05
- **Threshold for Discount**: At least 30 employees earning minimum wage or less

### Step 1: Calculate the Mean and Standard Deviation
- **Mean (μ)**: $$ \mu = n \times p = 1,017 \times 0.05 = 50.85 $$
- **Standard Deviation (σ)**: $$ \sigma = \sqrt{n \times p \times (1 - p)} = \sqrt{1,017 \times 0.05 \times 0.95} \approx 6.947 $$

### Step 2: Apply Continuity Correction
Since we're dealing with a discrete distribution, we'll use a continuity correction for the normal approximation:
- **Continuity Correction Point**: 29.5 (since we're looking for $$ X \geq 30 $$)

### Step 3: Calculate the Z-Score
$$ Z = \frac{X - \mu}{\sigma} = \frac{29.5 - 50.85}{6.947} \approx -3.072 $$

### Step 4: Find the Probability Using the Z-Score
Using standard normal distribution tables or a calculator:
- $$ P(Z \geq -3.072) $$
- The Z-score of -3.072 corresponds to a very low probability in the lower tail (approximately 0.00105).
- Therefore, $$ P(Z \geq -3.072) = 1 - P(Z < -3.072) \approx 1 - 0.00105 = 0.99895 $$

### Step 5: Round to Four Decimal Places
$$ P(X \geq 30) \approx 0.99895 $$
Rounded to four decimal places:
$$ P(X \geq 30) = \boxed{0.9989} $$
The probability that Company C will get the discount is **0.9989**.

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

(a) Company A has 298 employees.

(b) Company B has 506 employees.

© Company C has 1017 employees.

Final Answers:

preguntas relacionadas

Latest Probability Questions