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Example. A sample of 100 people is to be chosen from the population of a town to investigate the attitude of the adult townsfolk toward smoking. Which of the following sampling methods are biased?
1. Every tenth person leaving a supermarket in the center of the town is surveyed.
More women than men use a supermarket, so this survey is biased toward women.
2. People are asked to respond to a newspaper advertisement requesting views on smoking, and the first 100 people who respond are surveyed.
The people who respond will be readers of that particular newspaper and therefore not a random selection. This also is biased.
3. Every 50th adult person on the Electoral Roll is chosen until 100 names are obtained. Each one is visited and surveyed.
There is no bias in this method of sampling.
4. The largest factory in the town is chosen and 100 workers are randomly selected from the work force.
This selection may not accurately represent the population of the town for a variety of reasons: The work force of this factory may be predominantly one sex, for example, and so the selection is biased toward that sex. Another reason is that the largest factory could manufacture cigarettes! This sampling method is biased.
References: Population, Random Sample, Sample, Statistics, Survey.
Common usage, especially in the United States, puts this number at 1000 million, which is 1,000,000,000, or 109. In some countries, such as the United Kingdom and Germany, it is regarded as 1012, which is one million million.
When a set of data, or a frequency distribution, has two separate modes, then the data are said to be bimodal, and the frequency curve of the distribution has two “humps.”
56 BINARY NUMBERS
Example. Forty students took a short mathematics test that was graded out of 10. The frequency distribution of the results was recorded in a table. Draw a suitable bar graph of the results and identify the mode.
Mark out of 10 (x) 01 23456789 10
Frequency (f) 0135764751 1
Solution. The mode is the mark with the highest frequency. The highest frequency is 7, and occurs twice. The modes of this distribution are x = 4 and x = 7. Since there are two modes in this distribution, we say the distribution is bimodal. A suitable graph is a bar graph (see figure), because the data are discrete.
Bar Graph of Test Marks
References: Bar Graph, Data, Discrete, Frequency, Frequency Distribution, Mode. BINARY DIGIT
A binary digit is either a 0 or a 1. Often abbreviated “bit.”
Reference: Binary numbers.
These are numbers which are written in base 2, whereas the numbers we usually calculate with are in base 10, which are called denary numbers. Suppose you have a set of weights in grams (abbreviated g) and use them to weigh out certain quantities. The weights are 1, 2, 4, 8, and 16 g. The table shows the weights needed to weigh certain quantities.
BINARY NUMBERS 57
Number of Each Weight Needed Quantity to be Weighed
16 g 8 g 4g 2 g 1 g
1 0 2g
1 1 зд
1 0 0 0 8g
1 1 0 0 12 g
1 1 1 1 15 g
1 0 0 0 1 17 g
1 0 0 1 1 19 g
The numbers 1,10,11,1000,1100,1111,10001, and 10011, stating the numbers of weights needed to weigh certain quantities, are examples of binary numbers. Binary numbers can be added together as demonstrated in the following example.
Example. Find the weights needed to weigh 27 g.
Solution. The answer can be found by adding together the weights that are needed to weigh 12 g and 15 g, which together make 27 g. The weights required for 12 g are 1100, and the weights required for 15 g are 1111, which can be added using columns, as shown in the table.
16 g 8g 4g 2g 1 g
1 1 0 0
1 ii 1 1 1
1 1 0 1 1
The process known as “carrying” to the next column is used. Since we are working in base 2, we carry over to the next column when the total in a column is 2 or greater. From the columns we can see that 1100+ 1111 = 11011.
Example. Luke counted 22 sheep as they went into the paddock. How would you write 22 as a base 2 number?
Solution. The column headings needed are 1, 2,4, 8, and 16. It is unnecessary to go to the next column of 32, because there are only 22 sheep. Starting with the greatest column heading of 16, we subtract 16 from 22, leaving 6. The next column heading is 8 and no subtraction is necessary, because 8 is greater than 6. The next column heading is 4, which we subtract from 6, leaving 2. The next column heading is 2, which we subtract from 2, leaving 0. There is zero left over for the last column of units. We express this in numbers as follows:
22 = (16 x 1) + (8 x 0) + (4 x 1) + (2 x 1) + (1 x 0)
This is abbreviated to 101102 with the subscript 2, indicating that we are working in base 2. Using columns, we write
16 8 4 2 Unit
10 11 0
Base 2 numbers can be added, subtracted, and multiplied using methods similar to those for base 10 numbers. It is important to remember the following rules for binary numbers.