Question

A sample of 500 part-time students revealed that their annual incomes were normally distributed with a mean income of $30,000 and a standard deviation of $3,000. The number of students that earned between $27,000 and $33,000 is _________.

Answer 1

341 students

Explanation:

Here, we are to calculate the number of students earning between a particular range of income.

Please check attachment for complete solution and step by step explanation

Answer 2

y = 340 students

Explanation:

Solution:-

- The sample of size, n = 500 part time students

- The random variable "X" is defined : annual incomes of part time students. The random variable "X" follows normal distribution with the following parameters (mean income, standard deviation):

X ~ N ( 30,000 , 3,000)

- The proportion of students p that earn between $27,000 and $33,000 are defined by:

p = P ( 27,000 < X < 33,000 )

- Standardize the test values:

P ( (27,000 - 30000) / 3000 < Z < (33,000 - 30000) / 3000 )

P ( - 1 < Z < 1 )

- The empirical rule for the 68 - 95 - 99.7, states:

p = P ( - 1 < Z < 1 ) = 0.68

- So the number of students "y" that get their income between $27,000 and $33,000 are 68% of the sample taken:

y = 0.68*500

y = 340 students