Chapter 2.1, Problem 14E

(a)

To determine

Interpretation of z − score for North California.

Answer to Problem 14E

The median household income for North California is 1.24 standard deviation below the mean of all median household incomes of the 50 states.

Explanation of Solution

Given information:

Value, $x=41,553$

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

Standard deviation, $\text{SD}=8210.64$

Calculations:

The number of standard deviations that a value is from the mean represents the z − score.

Positive z − score indicates the value above the mean.

Whereas,

Negative z − score indicates the value below the mean.

Calculate the z − score:

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

Therefore,

For North California, the median household income is 1.24 standard deviation below the mean of all median household incomes of the 50 states.

(b)

To determine

Median household income of New Jersey.

Answer to Problem 14E

Median household income of New Jersey is approx. $66,685.80.

Explanation of Solution

Given information:

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

Standard deviation, $\text{SD}=8210.64$

Standardized score, $z=1.82$

Calculations:

The number of standard deviations that a value is from the mean represents the z − score.

Positive z − score indicates the value above the mean.

Whereas,

Negative z − score indicates the value below the mean.

Calculate the z − score:

  $z=\frac{x-\overline{x}}{\text{SD}}$

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.