Suppose the hourly wage of a detasseler is normally distributed with a mean of $15$ dollars and a standard deviation of $2$ dollars. What is the approximate probability that a randomly selected detasseler makes between $13$ and $17$ dollars an hour?

• A. $0.6826$
• B. $0.3413$
• C. $0.9544$
• D. $0.5000$

Answer: (A)

Verified step by step guidance

1. Identify the given parameters: the mean hourly wage \( \mu = 15 \) dollars and the standard deviation \( \sigma = 2 \) dollars. The wage is normally distributed.
2. Convert the raw scores (13 and 17 dollars) to their corresponding z-scores using the formula: \[ z = \frac{X - \mu}{\sigma} \] where \( X \) is the value of interest.
3. Calculate the z-scores for 13 and 17 dollars: \[ z_{13} = \frac{13 - 15}{2} \] \[ z_{17} = \frac{17 - 15}{2} \]
4. Use the standard normal distribution table (z-table) or a calculator to find the probabilities corresponding to these z-scores. These probabilities represent the area under the normal curve to the left of each z-score.
5. Find the probability that the wage is between 13 and 17 dollars by subtracting the smaller cumulative probability from the larger one: \[ P(13 < X < 17) = P(Z < z_{17}) - P(Z < z_{13}) \]