Chapter 2.1, Problem 13E

(a)

To determine

Interpretation of z − score for Montana.

Answer to Problem 13E

The percent of foreign − born residents in Montana is 1.12 standard deviations below the mean percent of foreign − born residents of all 50 states.

Explanation of Solution

Given information:

Value, $x=1.9$

Mean, $\overline{x}=8.73$

Standard deviation, $\text{SD}=6.12$

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:

  $z=\frac{x-\overline{x}}{\text{SD}}=\frac{1.9-8.73}{6.12}=\frac{-6.83}{6.12}\approx -1.12$

Therefore,

In Montana, the percent of foreign − born residents is 1.12 standard deviations below the mean percent of foreign − born residents of all 50 states.

(b)

To determine

Percentage of foreign − born residents in New York.

Answer to Problem 13E

New York has approx. 21.582% of foreign − born residents.

Explanation of Solution

Given information:

Mean, $\overline{x}=8.73$

Standard deviation, $\text{SD}=6.12$

Standardized score, $z=2.10$

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:

  $2.10=\frac{x-8.73}{6.12}$

Multiply both sides by 6.12:

  $12.852=x-8.73$

Add 8.73 to both sides:

  $12.852+8.73=x-8.73+8.73$

That becomes

  $x=21.582$

Therefore,

New York hasapprox. 21.582% of foreign − based residents.