Question

Household incomes Refer to Exercise $10$Here are summary statistics for the state median household incomes:

n	Mean	SD	Min	Q1	Med	Q3	Max
50	51742.44	8210.64	36641	46071	50009	57020	71836

a. Find and interpret the $z-$score for North Carolina, with a median household income of $\$41,553$

b. New Jersey had a standardized score of $1.82$Find New Jersey’s median household income for that year.

Short answer

Part (a) North California's median household income is $1.24$standard deviations below the mean of all $50$states' median household incomes.

Part (b) Median household income in New Jersey is approx. $\$66,685.80$

Step-by-step solution

Part (a) Step 1: Given information

The table is

n	Mean	SD	Min	$Q_1$	Med	$Q_3$	Max
50	51742.44	8210.64	36641	46071	50009	57020	71836

Value, $x=41,553$

Mean, $\overline{x}=51,742.44$

Standard deviation, $SD=8210.64$

Part (a) Step 2: Concept

The formula used:$z=\frac{(value-mean)}{(s\tan darddeviation)}$

Part (a) Step 3: Calculation

The $z-$score is the number of standard deviations a value deviates from the mean.

The value above the mean is indicated by a positive $z-$score.

Whereas,

The value below the mean is indicated by a negative $z-$score.

Calculate the $z-$ score:

$z=\frac{x-x}{SD}=\frac{41,553-51,742.44}{8210.64}=\frac{-10,189.44}{8210.64}\approx -1.24$

Therefore,

North California's median household income is $1.24$ standard deviations lower than the mean of all $50$ states' median household incomes.

Part (b) Step 1: Calculation

The $z-$score is the number of standard deviations a value deviates from the mean.

The value above the mean is indicated by a positive $z-$score.

Whereas,

A value below the mean is indicated by a negative $z-$score.

Calculate the $z-$ score:$z=\frac{x-\overline{x}}{SD}$

Substitute the values:

$1.82=\frac{x-51,742.44}{8210.64}$

Multiply both sides by $8210.64$:

$14,943.36=x-51,742.44$

Add $51,742.44$ to both sides:

$14,943.36+51,742.44=x-51,742.44+51,742.44$

That becomes

$x=66,685.80$

Therefore,

For New Jersey, the median household income is approx. $\$66,685.80$