Question

A survey collects demographic, socioeconomic, dietary, and health-related information on an annual basis. Here is a sample of 20 observations on HDL cholesterol level (mg/dl) obtained from the survey (HDL is "good" cholesterol; the higher its value, the lower the risk for heart disease):
35 49 51 54 65 51 52
47 87 37 46 33 39 44
39 64 94 34 30 48

Requried:
a. Calculate a point estimate of the population mean HDL cholesterol level.
b. Making no assumptions about the shape of the population distribution, calculate a point estimate of the value that separates the largest 50% of HDL levels from the smallest 50%.
c. Calculate a point estimate of the population standard deviation.

Answer 1

Answer:

(a) The point estimate of the population mean HDL cholesterol level is 49.95.

(b) The point estimate of the value that separates the largest 50% of HDL levels from the smallest 50% is 47.5.

(c) The point estimate of the population standard deviation is 16.85.

Step-by-step explanation:

We are given a sample of 20 observations on HDL cholesterol level (mg/dl) obtained from the survey below;

35, 49, 51, 54, 65, 51, 52,  47, 87, 37, 46, 33, 39, 44,  39, 64, 94, 34, 30, 48.

(a) The point estimate of the population mean HDL cholesterol level is given by the sample mean of the above data, i.e;

        Sample Mean, $\bar X$  =  $\frac{\sum X}{n}$

            =  $\frac{35+ 49+ 51+ 54 +65 +51+ 52+ 47+ 87+ 37+ 46+ 33+ 39+ 44+ 39+ 64+ 94+ 34+ 30+ 48}{20}$  

            =  $\frac{999}{20}$  =  49.95

So, the point estimate of the population mean HDL cholesterol level is 49.95.

(b) The point estimate of the value that separates the largest 50% of HDL levels from the smallest 50% is given by the Median of the above data.

Firstly, arranging the given data in ascending order we get;

30, 33, 34, 35, 37, 39, 39, 44, 46, 47, 48, 49, 51, 51, 52, 54, 64, 65, 87,  94.

Now, for calculating median we have to first observe that the number of observations (n) in our data is even or odd, i.e;

• If n is odd, then the formula for calculating median is given by;

                      Median  =  $(\frac{n+1}{2})^{th} \text{ obs.}$

• If n is even, then the formula for calculating median is given by;

                      Median  =  $\frac{(\frac{n}{2})^{th} \text{ obs.}+(\frac{n}{2}+1)^{th} \text{ obs.} }{2}$

Here, the number of observations is even, i.e. n = 20.

So,   Median  =  $\frac{(\frac{n}{2})^{th} \text{ obs.}+(\frac{n}{2}+1)^{th} \text{ obs.} }{2}$  

                      =  $\frac{(\frac{20}{2})^{th} \text{ obs.}+(\frac{20}{2}+1)^{th} \text{ obs.} }{2}$

                      =  $\frac{(10)^{th} \text{ obs.}+(11)^{th} \text{ obs.} }{2}$

                      =  $\frac{47+48 }{2}$

        Median  =  47.5

Hence, the point estimate of the value that separates the largest 50% of HDL levels from the smallest 50% is 47.5.

(c) The point estimate of the population standard deviation is given by the following formula;

      Standard deviation, s  =  $\sqrt{\frac{\sum(X-\bar X)^{2} }{n-1} }$

       =  $\sqrt{\frac{ (30-49.95)^{2}+(33-49.95)^{2}+(34-49.95)^{2}+........+(94-49.95)^{2}}{{20-1}} }} }$

       =  16.85