**Sampling and Estimation**

Here are some problems involving data sampling and estimation. Recall the steps that must be followed when conducting a statistical experiment:

- Formulate the question(s) we want to answer.
- Gather sufficient data from the required sources.
- Organize and analyze the data.
- Interpret the information that has been gathered and organized.

**Practice 1**

Most of us have seen charts that tell us how much mass (or weight) our bodies ought to have, in the opinions of medical experts. The mass values are based on height and "frame type" (small, medium, or large), and may also vary depending on your age. Charts made up this year may differ from those made up 10 years ago, or 30 years ago, or 60 years ago.

Let's consider hypothetical distributions of human body mass as functions of body height, based on observations of real people rather than on idealized theories. Imagine that such distributions are compiled for the whole world, and are based solely on how tall people are; age, ethnicity, gender, and all other factors are ignored. Imagine that we end up with a set of many graphs, one plot for each height in centimeters, ranging from the shortest person in the world to the tallest, showing how massive people of various heights actually are.

Suppose we obtain a graph (Fig. 8-12, hypothetically) showing the mass distribution of people in the world who are 170 centimeters tall, rounded to the nearest centimeter. Suppose we round each individual person's mass to the nearest kilogram. Further suppose that this is a normal distribution. Therefore, the mean, median, and mode are all equal to 55 kilograms. We obviously can't put every 170-centimeter-tall person in the world on a scale, measure his or her mass, and record it! Suggest a sampling frame that will ensure that Fig. 8-12 is a good representation of the actual distribution of Masses for people throughout the world who are 170 centimeters tall. Suggest another sampling frame that might at first seem good, but that in fact is not.

**Fig. 8-12. **Illustration for Practice 1 and 2.

**Solution 1**

Recall the definition of a sampling frame: a set of elements from within a population, from which a sample is chosen. A sampling frame is a representative cross-section of a population.

Let's deal with the not-so-good idea first. Suppose we go to every recognized country in the world, randomly select 100 males and 100 females, all 170 centimeters tall, and then record their masses. Then, as our samples, we randomly select 10 of these males and 10 of these females. This won't work well, because some countries have much larger populations than others. It will skew the data so it over-represents countries with small populations and under-represents countries with large populations.

If we modify the preceding scheme, we can get a fairly good sampling frame. We might go to every recognized country in the world, and select a number of people in that country based on its population in thousands. So if a country has, say, 10 million people, we would select, as nearly at random as possible, 10,000 males and 10,000 females, all 170 centimeters tall, as our sampling frame. If a country has 100 million people, we would select 100,000 males and 100,000 females. If a country has only 270,000 people, we would select 270 males and 270 females, each 170 centimeters tall. Then we put every tenth person on a scale and record the mass. If a country has so few people that we can't find at least 10 males and 10 females 170 centimeters tall, we lump that country together with one or more other small countries, and consider them as a single country. From this, we obtain the distribution by using a computer to plot all the individual points and run a curve-fitting program to generate a smooth graph.

**Practice 2**

Suppose Jorge and Julie are both 170 centimeters tall. Jorge has a mass of 61 kilograms, and Julie has a mass of 47 kilograms. Where are they in the distribution?

**Solution 2**

Figure 8-12 shows the locations of Jorge and Julie in the distribution, relative to other characteristics and the curve as a whole.

**Practice 3**

Suppose that we define the "typical human mass" for a person 170 centimeters tall to be anything within 3 kilograms either side of 55 kilograms. Also suppose that we define ranges called "low typical" and "high typical," representing masses that are more than 3 kilograms but less than 5 kilograms either side of 55 kilograms. Illustrate these ranges in the distribution.

**Solution 3**

Figure 8-13 shows these ranges relative to the whole distribution.

**Fig. 8-13. **Illustration for Practice 3.

More practice problems for these concepts can be found at:

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