Question

Relationships between means, standard deviations, X values,and z-scores

The International Average Salary Income Database provides a comparison of average salaries for various professions. The data are gathered from publications and reports obtained directly from government agencies (such as the U.S. Bureau of Labor Statistics) or from the International Labour Organization.
1. Suppose you are told that a firefighter in Portugal who makes $510.10 has a z-score of -0.50 relative to other firefighters in Portugal and that the standard deviation of the salary of firefighters in Portugal is $145.80. The average salary, then, for firefighters in Portugal is __________.
2. Suppose you are also told that a firefighter in Finland who makes $2,291.90 has a z-score of +0.50 relative to other firefighters in Finland and that the mean of the salary of firefighters in Finland is $2,132. The standard deviation of the salary, then, for firefighters in Finland is ___________ .

Answer 1

a) The average salary of firefighters in Portugal is $583.

b) The standard deviation of the salary of firefighters in Portugal is $319.8

Explanation:

We are given the following in the question:

a) Standard Deviation, σ = $145.80

We are given that the distribution of salary of professor is a bell shaped distribution that is a normal distribution.

Formula:

`z_(score) = \displaystyle(x-\mu)/(\sigma)`

A salary of $510.10 has a z-score of -0.50.

Putting values, we get,

`-0.50 = \displaystyle(510.10-\mu)/(145.80)\\\\\mu = 0.50(145.80) + 510.10\\\Rightarrow \mu = 583`

Thus, the average salary of firefighters in Portugal is $583.

b) Mean, μ = $2,132

We are given that the distribution of salary of computer programmers is a bell shaped distribution that is a normal distribution.

Formula:

`z_(score) = \displaystyle(x-\mu)/(\sigma)`

The salary of $2,291.90 has a z-score of +0.50

Putting values, we get,

`0.50 = \displaystyle(2291.90-2132)/(\sigma)\\\\\sigma= (2291.90-2132)/(0.50)\\\\\sigma = 319.8`

Thus, the standard deviation of the salary of firefighters in Portugal is $319.8