Question

Salaries for teachers in a particular elementary school district have a mean of $44,000 and a standard deviation of $6,500. We randomly survey 36 teachers from that district.

1. Why can we say the sampling distribution of mean salaries for teachers in this district is approximately normal?
2. Find the probability that the mean salary is less than $43,000.
3. Find the probability that the mean salary is between $45,000 and $47,000.

Answer 1

Solution :

Given that ,

mean = = 44000

standard deviation = = 6500

n = 36

= 44000

= / n = 6500 / 36  = 1083.3333

The sampling distribution of mean = 44000

The standard deviation of the sampling mean = 1083.3333

P( < 43000 ) = P(( - ) / < ( 43000 - 44000 ) / 1083.3333 )

= P(z < -0.92)

Using z table

= 0.1788

Probability = 0.1788

P( 45000 < < 47000 )  

= P[( 45000 - 44000) / < ( - ) / < ( 47000 - 44000) / 1083.3333 )]

= P(0.92 < Z < 2.77 )

= P(Z < 2.77 ) - P(Z < 0.92 )

Using z table,  

= 0.9972 - 0.8212

= 0.1760

Probability = 0.1760