Question

A private high school would like to offer free tuition to residents of Missouri whose income (as indicated by their most recent annual tax statement) place the family in the bottom 12% of the state's family income. Assume that the average family income in Missouri is $44,250, with a standard deviation of $13,600. At what income level would a family be qualified for the free tuition program?

Provide final answer rounded to 2 decimal places. The Z score to 4 decimal places if applicable

1) Associated with bottom % of income earners is:

2) Translated into annual income dollars, this would indicate the families who earn less than $_____ per year would qualify for the free tuition program.

Answer 1

mean income = $44,250

standard deviation of income = $13,600

Let X be the income of family, since data is large it follows normal

Let X_1 be the income of family where the family classifies for free tuition

P[ X < X_1 ] = 12% = 0.12

P[ ( X - mean ) / standard deviation < ( X_1 - mean ) / standard deviation ] = 0.12

P[ ( X - $44,250 ) / $13,600 < ( X_1 - $44,250 ) / $13,600 ] = 0.12

P[ Z < ( X_1 - $44,250 ) / $13,600 ] = 0.12

Also, from the table we know that , P[ Z < -1.1750 ] = 0.12

equating, ( X_1 - $44,250 ) / $13,600 = -1.1750

X_1 - $44,250 = -$15980

X_1 = $44,250 - $15980 = $28270

1) Z score associated with bottom % of income earners is : -1.1750

2)  Translated into annual income dollars, this would indicate the families who earn less than $28270 per year would qualify for the free tuition program.