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 $206\mathrm{mg}/\mathrm{dL}$ (milligrams per deciliter) and a standard deviation of $44.7\mathrm{mg}/\mathrm{dL}$. 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 standardized 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 $150\mathrm{mg}/\mathrm{dL}$ and $250\mathrm{mg}/\mathrm{dL}$ is equal to the area under the standard normal curve between-------- .

Step-by-Step Solution

Answer

a). The distribution of the variable $x$:

b). The standardization of the variable,

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

c). The distribution of the curve $z$ as,

d). The percentage of U.S. females $20$ years old or older who have a serum total cholesterol level between $150\mathrm{mg}/\mathrm{dL}$ and $250\mathrm{mg}/\mathrm{dL}$ is equal to the area under the standard normal curve between $-1.25$ and $0.98$.

Part (a) Step 1: Given Information

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

Part (a) Step 2: Explanation

The distribution of the variable $x$:

Part (b) Step 1: Given Information

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

Part (b) Step 2: Explanation

The standardized version can be obtained as

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

Part (c) Step 1: Given Information

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

Part (c) Step 2: Explanation

The distribution curve of $z$ is,

Part (d) Step 1: Given Information

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

Part (d) Step 2: Explanation

You can get the standardized version as

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

When $x=150$,

$z=\frac{150-206}{44.7}$

$z=-1.25$

When $x=250$,

$z=\frac{250-206}{44.7}$

$z=0.98$