Question

The average hourly wage of production workers in manufacturing is $13.50 and a standard deviation of $2.50. The wages are normally distributed. One thousand workers were chosen to participate in the survey Approximately how many workers earned between $8.50 and $16.00? Round your answer to the nearest worker.

Answer 1

To find approximately how many workers earned between $8.50 and $16.00, we can use the standard normal distribution and z-scores.

Given:

- Mean \( \mu = 13.50 \)

- Standard deviation \( \sigma = 2.50 \)

- Lower value \( x_1 = 8.50 \)

- Upper value \( x_2 = 16.00 \)

Calculate the z-scores for both values using the formula:

\[ z = \frac{x - \mu}{\sigma} \]

For \( x_1 = 8.50 \):

\[ z_1 = \frac{8.50 - 13.50}{2.50} = -2 \]

For \( x_2 = 16.00 \):

\[ z_2 = \frac{16.00 - 13.50}{2.50} = 1 \]

Next, use the z-table to find the areas corresponding to \( z_1 \) and \( z_2 \). The area for \( z_1 \) (approximately -2) is very close to 0, and the area for \( z_2 \) (approximately 1) is approximately 0.8413.

To find the proportion of workers earning between $8.50 and $16.00, subtract the two areas:

\[ \text{Proportion} = 0.8413 - 0 = 0.8413 \]

Finally, multiply the proportion by the total number of workers (1000) to get the approximate number of workers who earned between $8.50 and $16.00:

\[ \text{Number of workers} = 0.8413 \times 1000 \approx 841.3 \]

Rounded to the nearest worker, approximately 841 workers earned between $8.50 and $16.00.