Question

for certain workers, the median wage is $7.50/hr. with a standard deviation of $0.75. if a worker is chosen at random, what is the probability that the worker's wage is between $5.25-$9.75? Assume a normal distribution of wages.

Answer 1

We have the mean value and the standard deviation. The question wants to know the probability between a given range. To answer that, we can just calculate the area below the graph inside of this range.

The range is $5.25-$9.75, now let's check how far each value is apart from the mean value.

`\begin{gathered} 7.50-5.25=2.25=3*0.75 \\ 9.75-7.50=2.25=3*0.75 \end{gathered}`

From this, we know the range is 3 times the standard deviation to the negative side and 3 times the standard deviation to the positive side from the mean.

Since we're dealing with a normal distributions, we have some properties called the empirical rules that should help us to solve this problem.

Those rules are:

• 68% of the data falls within one standard deviation of the mean.

• 95% of the data falls within two standard deviations of the mean.

• 99.7% of the data falls within three standard deviations of the mean.

The third one, tells us exactly what we want to know. Our range fall exactly within three standard deviations of the mean, which means the probability that the worker's wage is inside this range is 99.7%!