Q 8.24. - Elementary Statistics

Question

Serum Cholesterol Levels. Information on serum total cholesterol levels is published by the Centers for Disease Control and 

Prevention in National Health and Nutrition Examination Survey. A simple random sample of $12$ U.S. females $20$ years old or older provided the following data on serum total cholesterol level, in milligrams per deciliter (mg/dL).

a. Obtain a normal probability plot of the data.

b. Based on your result from part (a), is it reasonable to presume that serum total cholesterol level of U.S. females $20$ years old or older is normally distributed? Explain your answer.

c. Find and interpret a $95\%$ confidence interval for the mean serum total cholesterol level of U.S. females 20 years old or older. The population standard deviation is $44.7\mathrm{mg}/\mathrm{dL}$

d. In Exercise $6.98$, we noted that the mean serum total cholesterol level of U.S. females $20$ years old or older is $206\mathrm{mg}/\mathrm{dL}$ Does your confidence interval in part (c) contain the population means? Would it necessarily have to? Explain your answers.

Step-by-Step Solution

Answer

Part (b) Cholesterol levels are distributed in a rather regular manner.

Part (c) The $95\%$ confidence interval for the mean serum total cholesterol level is $(174.11,214.09)$

Part (d) Because the mean serum total cholesterol level $206$ lies between $174.11$ and $214.09$

Part (a) 

Part (a) Step 1: Given information

260	289	190	214	110	241
169	173	191	178	129	185

Part (a) Step 2: Concept

The formula used: $\overline{x}\pm 2\frac{\sigma }{\sqrt{n}}$

Part (a) Step 3: Explanation

To create a normal probability plot for cholesterol levels, use MINITAB.

Procedure for MINITAB:

Step 1: Select Probability Plot from the Graph menu.

Step 2: Click OK after selecting Single.

Step 3: In the Graph variables section, type LEVEL in the column.

Step 4: Click the OK button.

OUTPUT FROM MINITAB: 

Part (b) Step 1: Explanation

Yes, serum total cholesterol levels should be expected to be regularly distributed. It is obvious from the standard probability plot that all of the points are plotted in a straight line. As a result, cholesterol levels are distributed in a rather regular manner.

Part (c) Step 1: Calculation

Calculate the mean serum total cholesterol level's $95\%$ confidence interval.

$$\begin{gathered} \overline{x}\pm 2\frac{\sigma }{\sqrt{n}}=194.1\pm 2\left(\frac{44.7}{\sqrt{20}}\right) \\ =194.1\pm 19.99 \\ =(194.1-19.99,194.1+19.99) \\ =(174.11,214.09) \end{gathered}$$

As a result, for the mean blood total cholesterol level, the $95\%$ confidence interval is $(174.11,214.09)$

Part (d) Step 1: Explanation

The mean serum total cholesterol level $206$ is included in the confidence interval in component (c), since the mean serum total cholesterol level $206$ is between $174.11$ and $214.09$