Answer:
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.