Question

Research department tells you that an annual income of a city resident is normally distributed with mean (µ) $50,000 and standard deviation (σ) of $10,000. Use this information for questions 1-4. Use Table A in your book to lookup proportions for z-scores. 1. Using above information, what proportion of city residents earn below $35,700? Show work. 2. Using above information, what proportion of city resident earn above $68,400? Show work. 3. Using above information, what proportion of city resident earn above $65,500? Show work. 4. Using above information, what proportion of city resident earn above $35,700 and below $65,500? Show work.

Answer 1

Here we have given an annual income of a city resident is normally distributed with mean = $50000,

standard deviation = $10000.

Here let x be the an annual income of city resident.

1)

So here we have to find P(x<35700)

By using z score,

So by using z table,

Table value always gives left side (<) probability

See area value in the middle body of the table, having row heading -1.4 and column heading 0.03

so P(Z < -1.43) = 0.0764

So the proportion of city residents earn below $35,700 = 0.0764

2) P(x > 68400)

See like above example 1) row heading 1.8 and column heading 0.04 you will get 0.9671

proportion of city resident earn above $68,400 is 0.0329

3) P(x > 65500)

proportion of city resident earn above $65,500 is 0.0606

4) P(35700 < x < 65500) =  P(x< 65500) - P(x < 35700)

= 0.9394 - 0.0764

= 0.863  

proportion of city resident earn above $35,700 and below $65,500 is 0.863