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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? (Round to four decimal places as needed.)

Ask by Ryan Ramos. in the United States
Jan 21,2025

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The probability that Company A will receive the discount is approximately 0.0001.

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The Deep Dive

To determine the probability that Company A, with 298 employees, will have at least 30 employees earning the prevailing minimum wage or less, we can apply the binomial probability formula. Given that 5% of workers earn the minimum wage or less, we will treat this as a binomial experiment where: - \( n = 298 \) (total number of employees) - \( p = 0.05 \) (probability of an employee earning minimum wage or less) - \( k \geq 30 \) (we want the probability of getting at least 30 employees earning minimum wage or less) To find the probability \( P(X \geq 30) \), it is often easier to compute the complementary probability \( P(X < 30) \), and then subtract this from 1: \[ P(X \geq 30) = 1 - P(X < 30) \] For computations of binomial probabilities, it can be useful to use a normal approximation since \( n \) is large. We calculate the mean and standard deviation of the binomial distribution: - Mean \( \mu = n \cdot p = 298 \cdot 0.05 = 14.9 \) - Standard Deviation \( \sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{298 \cdot 0.05 \cdot 0.95} \approx 3.87 \) Next, we approximate \( X \) with a normal distribution \( N(\mu, \sigma^2) \): 1. Compute the z-score for \( k = 29.5 \) (continuity correction): \[ z = \frac{29.5 - \mu}{\sigma} = \frac{29.5 - 14.9}{3.87} \approx 3.75 \] 2. Look up this z-score in the standard normal distribution table or use a calculator: The probability \( P(Z < 3.75) \approx 0.9999 \) Thus, \[ P(X < 30) \approx 0.9999 \] Finally, we find \( P(X \geq 30) \): \[ P(X \geq 30) = 1 - P(X < 30) = 1 - 0.9999 = 0.0001 \] Rounded to four decimal places, the probability that Company A will get the discount is approximately: **0.0001**.

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