Question

Adjusted Gross Incomes. Refer to Problem 30. A federal individual income tax return is selected at random. Let

$H=$event the return shows an AGI between \(20K and \)100K,

$I=$event the return shows an AGI of less than \(50K.

$J=$event the return shows an AGI of less than \)100K, and

$K=$event the return shows an AGI of at least $50K.

Describe each of the following events in words and determine the number of outcomes ( returns ) that constitute each event.

(a) (not J) (b) (H& I)

(c) (Hor K) (d) (H&K)

Short answer

Part (a) 18,227 thousand.

Part (b) 44,251 thousand.

Part (c) 93,404 thousand.

Part (d) 30,926 thousand.

Step-by-step solution

Part (a) Step 1. Given information.

The frequency distribution (number of returns) for adjusted gross income (AGI) from federal individual income tax returns is shown in the first two columns of Table 5.21, where K equals thousand.

The given table is:

Adjusted gross income	Frequency (1000s)	Event
Under $10K	26,268	A
$10K-under $20K	22,778	B
$20K-under $30K	18,610	C
$30K-under $40K	14,554	D
$40K-under $50K	11,087	E
$50K-under $100K	30,926	F
$100K & over	18,227	G
	142,450

Part (a) Step 2. Determine the number of possible outcomes for an event (not J).

The return shows an AGI of less than $100K in event "J".

Event "not J" will represent the return of at least $100K.

From the table, we know that there are only 18,227 thousand such returns that represent $100k and above.

Part (b) Step 1. Determine the number of possible outcomes for an event (H & I)

The event (H & I) indicates that the return has an AGI of $20,000 to $50,000.

The number of returns with an AGI of $20,000 to $50,000 is:

$$\begin{gathered} \left(H\&I\right)=C+D+E \\ =18,610+14,554+11,087 \\ =44,251 \end{gathered}$$

(H & I) includes 44,251 thousand returns.

Part (c) Step 1. Determine the number of possible outcomes for an event (H or K).

The event (H or K) indicates that the return has a minimum AGI of $20K.

The percentage of returns with an AGI of at least $20,000 is:

$$\begin{gathered} (HorK)=C+D+E+F+G \\ =18,610+14,554+11,087+30,926+18,227 \\ =93,404 \end{gathered}$$

Therefore, (Hor I) includes 93,404 thousand returns.

Part (d) Step 1. Determine the number of possible outcomes for an event (H & K).

The event (H& K) indicates that the return must lie between AGI of $50K and $100K.

The percentage of returns with an AGI that lies between $50,000 and $100K is:

$$\begin{gathered} (H\&K)=F \\ =30,926 \end{gathered}$$

Therefore, (H& I) includes 30,926 thousand returns.