State Income Taxes. All 50 U.S. states are given the ability to set their own st

Question

State Income Taxes. All 50 U.S. states are given the ability to set their own state income tax rates, which is measured as a proportion of an individuals gross income. In a recent study of state income taxes it was discovered that income tax rates for all 50 states follow a Normal distribution with mean 5.0 and a standard deviation 0.75. State income tax rates are reported as a (proportion × 100). Use this information to answer the following questions.

(a) What is the probability a state is charging an income tax rate greater than 3.9? Answer this question by completing parts (i) and (ii). i. Provide the z-score corresponding to the income tax rate of 3.9. ii. Based on your answer in (i), what is the probability a state is charging an income tax rate greater than 3.9?

(b) What is the probability a state is charging an income tax rate less than 5.3? Answer this question by completing parts (i) and (ii). i. Provide the z-score corresponding to the income tax rate of 5.3. ii. Based on your answer in (i), what is the probability a state is charging an income tax rate less than 5.3?

(c) Find Q1 (First Quartile). Answer this question by completing parts (i) and (ii). i. Provide the z-score corresponding to Q1. ii. Based on your answer in (i), provide the value of Q1.

(d) Find Q3 (Third Quartile). Answer this question by completing parts (i) and (ii). i. Provide the z-score corresponding to Q3. ii. Based on your answer in (i), provide the value of Q3.

(e) What is the value of the IQR for the distribution of income tax rates for all 50 states?

(f) What is the income tax rate such that only 12.1% of all other states charge a higher income tax rate? Answer this question by completing parts (i) and (ii). i. Provide the z-score corresponding to the above statement. ii. Based on your answer in (i), what is the income tax rate such that only 12.1% of all other states charge a higher income tax rate? 2 iii. What percentile does the income tax rate obtained in part (ii) correspond to? (Enter the value as a whole number, not a proportion.)

(g) What proportion of states charge an income tax rate within 1.25 standard deviations from the mean? Answer this question by completing parts (i) through (iii). i. Provide the SMALLER z-score corresponding to the above statement. ii. Provide the LARGER z-score corresponding to the above statement. iii. Based on your answers in (i) and (ii), what proportion of states charge an income tax rate within 1.25 standard deviations from the mean?

Explanation / Answer

a) i. ) z score =(X-mean)/std deviation =(3.9-5)/0.75=-1.4667

(ii)_probability a state is charging an income tax rate greater than 3.9=0.9288

b)i) z score =(X-mean)/std deviation =(5.3-5)/0.75 =0.4

probability a state is charging an income tax rate less than 5.3 =0.6554

c) for Q1 ; z =-0.6745

herefore Q1 =mean+z*std deviation =4.494

d)

for Q1 ; z =0.6745

herefore Q1 =mean+z*std deviation =5.506

e) value of the IQR =Q3-Q1 =1.012

f)(i) for above z score =1.17

(ii) tax rate =mean+z*std deviation =5.878

(iii) percentie =100-12.1=87.9

g) smaller z score =-1.25

(ii) larger score =1.25

(iii) proportion =0.8944-0.1056 =0.7887

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