Q. 6.38 - Elementary Statistics

Question

Serum Cholesterol Levels. According to the National Health and Nutrition Examination Survey, published by the National Center for Health Statistics, the serum (noncellular portion of blood) total cholesterol level of U.S. females $20$ years old or older is normally distributed with a mean of $206$mg/dL (milligrams per deciliter) and a standard deviation of $44.7$ mg/dL. Let $x$ denote serum total cholesterol level for U.S. females $20$ years old or older.

a. Sketch the distribution of the variable $x$.

b. Obtain the standardization version,$z$ of $x$.

c. Identify and sketch the distribution of $z$.

d. The percentage of U.S. females $20$ years old or older who have a serum total cholesterol level between $150$ mg/dL and $250$ mg/dL is equal to the area under the standard normal curve between _____ .

Step-by-Step Solution

Answer

Part $\left(a\right)$ : The distribution of $x$ is as follows:

Part $\left(b\right)$: The standardized version of $x$ is, $z=\frac{x-206}{44.7}$.

Part $\left(c\right)$: The distribution of $z$ is as follows:

Part $\left(d\right)$: The percentage of females $20$ years old or older who have serum total cholesterol level between $150$ mg/dL and $250$  mg/dL is equal to the area under the standard normal curve between $-1.95$ and $0.98$.

Part $\left(e\right)$: The percentage of females $20$ years old or older who have serum total cholesterol level below $220$ mg/dL is equal to the area under the standard normal curve that lies to the left of $0.31$.

Part a Step 1 . Given information

Let $x$ denote serum total cholesterol level for females $20$ years old or older.

Part a Step 2 . The distribution of x is as follows:

Part b Step 1 . The standardized version of x is,

$z=\frac{x-206}{44.7}$.

Part c Step 1 . The distribution of z is as follows:

Part d Step 1 . Substitute x = 150 in the standardized version.

$$\begin{gathered} z=\frac{150-206}{44.7} \\ =-1.25 \end{gathered}$$

Substitute $x=250$ in the standardized version.

$$\begin{gathered} z=\frac{250-206}{44.7} \\ =0.98 \end{gathered}$$

Therefore, the percentage of females $20$ years old or older who have serum total cholesterol level between $150$mg/dL and $250$ mg/dL is equal to the area under the standard normal curve between $-1.25$ and $0.98$.

Part e Step 1 . Substitute x = 220 in the standardized version.

$$\begin{gathered} z=\frac{220-206}{44.7} \\ =0.31 \end{gathered}$$

Therefore, the percentage of females $20$ years old or older who have serum total cholesterol level below $220$ mg/dL is equal to the area under the standard normal curve that lies to the left of $0.31$.