Question

the per capita energy consumption level (in kilowatt - hours) in a certain country for a recent year can be approximated by a normal distribution, as shown in the figure. (a) what consumption level represents the 4th percentile? (b) what consumption level represents the 21st percentile? (c) what consumption level represents the third quartile? (a) the consumption level that represents the 4th percentile is kilowatt - hours. (round to the nearest integer as needed.)

Explanation:

Step1: Recall z - score formula

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the mean and $\sigma$ is the standard deviation. We need to find the z - score corresponding to the given percentiles and then solve for $x$.

Step2: Find z - score for 4th percentile

Looking up the 4th percentile (0.04) in the standard normal distribution table (z - table), the z - score $z_{1}\approx - 1.75$.

Step3: Solve for $x$ for 4th percentile

We know that $\mu = 2328$ and $\sigma=593.9$. Substitute into the z - score formula: $-1.75=\frac{x - 2328}{593.9}$. Cross - multiply: $x-2328=-1.75\times593.9$. Then $x = 2328-1.75\times593.9=2328 - 1039.325\approx1289$.

Step4: Find z - score for 21st percentile

Looking up the 21st percentile (0.21) in the z - table, the z - score $z_{2}\approx - 0.81$.

Step5: Solve for $x$ for 21st percentile

Substitute into the z - score formula: $-0.81=\frac{x - 2328}{593.9}$. Cross - multiply: $x-2328=-0.81\times593.9$. Then $x = 2328-0.81\times593.9=2328 - 481.059\approx1847$.

Step6: Find z - score for third quartile

The third quartile is the 75th percentile. Looking up 0.75 in the z - table, the z - score $z_{3}\approx0.67$.

Step7: Solve for $x$ for third quartile

Substitute into the z - score formula: $0.67=\frac{x - 2328}{593.9}$. Cross - multiply: $x-2328 = 0.67\times593.9$. Then $x=2328+0.67\times593.9=2328 + 397.913\approx2726$.

Answer:

(a) 1289
(b) 1847
(c) 2726