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 less than $11.00? Round your answer to the nearest worker.

Answer 1

To determine the approximate number of workers who earned less than $11.00 per hour, we can use the concept of z-scores and the properties of the normal distribution.

First, we need to calculate the z-score for the given value of $11.00. The z-score formula is given by:

z = (x - μ) / σ

Where:

x = the given value ($11.00)

μ = mean of the distribution ($13.50)

σ = standard deviation of the distribution ($2.50)

Calculating the z-score:

z = (11.00 - 13.50) / 2.50

z ≈ -1.00

Now, we can use a z-table (or software) to find the proportion (or probability) associated with a z-score of -1.00. The z-table provides the area under the normal distribution curve up to a particular z-score.

Looking up the z-table (or using software), we find that the area to the left of a z-score of -1.00 is approximately 0.1587. This means that approximately 15.87% of the distribution is below $11.00.

Lastly, to estimate the number of workers who earned less than $11.00, we can multiply the proportion (0.1587) by the total number of workers in the survey (1000):

Number of workers = 0.1587 * 1000 ≈ 158.7

Rounding to the nearest worker, approximately 159 workers earned less than $11.00 per hour.