12.1 Sampling Techniques and Misuses of Statistics 765 Let’s examine two advertisements. “Four out of five dentists recommend sugarless gum for their patients who chew gum.” In this advertisement, we do not know the sample size and the number of times the experiment was performed to obtain the desired results. The advertisement does not mention that possibly only 1 out of 100 dentists recommended gum at all. In an advertisement for golf balls, a Driver golf ball is hit and another brand golf ball is hit in the same manner. We are told that the Driver golf ball travels farther and we are supposed to conclude that the Driver golf ball is the better golf ball. The advertisement does not mention the number of times the experiment was previously performed or the results of the earlier experiments. Possible sources of bias include (1) wind speed and direction, (2) that no two swings are identical, and (3) that the ball may land on a rough or smooth surface. Vague or ambiguous words also lead to statistical misuses or misinterpretations. The word average is one such culprit. There are many different averages used in the study of statistics. In Section 12.3 we discuss four of these averages. Each is calculated differently, and each may have a different value for the same sample. During contract negotiations, it is not uncommon for an employer to state publicly that the average salary of its employees is $45,000, whereas the employees’ union states that the average salary is $40,000. Who is lying? Actually, both sides may be telling the truth. Each side will use the average that best suits its needs to present its case. Advertisers also use the average that most enhances their products. Consumers often misinterpret this average as the one with which they are most familiar. Another vague word is largest . For example, Hardings claims that it is the largest department store in the United States. Does that mean largest profit, largest sales, largest building, largest staff, largest acreage, or largest number of outlets? Still another deceptive technique used in advertising is to state a claim from which the public may draw irrelevant conclusions. For example, a disinfectant manufacturer claims that its product killed 40,760 germs in a laboratory in 5 seconds. “To prevent colds, use disinfectant A.” It may well be that the germs killed in the laboratory were not related to any type of cold germ. In another example, company C claims that its paper towels are heavier than its competition’s towels. Therefore, they will hold more water. Is weight a measure of absorbency? A rock is heavier than a sponge, yet a sponge is more absorbent. An insurance advertisement claims that in Duluth, Minnesota, 212 people switched to insurance company Z. One may conclude that this company is offering something special to attract these people. What may have been omitted from the advertisement is that 415 people in Duluth, Minnesota, dropped insurance company Z during the same period. A car manufacturer claims that 9 of every 10 of a popular-model car it sold during the previous 10 years were still on the road. From this statement, the public is to conclude that this car is well manufactured and would last for many years. The commercial neglects to state that this model has been selling for only a few years. The manufacturer could just as well have stated that 9 of every 10 of these cars sold in the previous 100 years were still on the road. Charts and graphs can also be misleading or deceptive. In Fig. 12.2, the two graphs show the performance of two stocks over a 6-month period. Based on the MATHEMATICS TODAY Creative Displays Visual graphics are often used to “dress up” what might otherwise be considered boring statistics. Although visually appealing, such creative displays of numerical data can be misleading. The graph above shows the percentage of employers that offer “perks” such as stress reduction, massage therapy, or a nap during the workday to make workers happy. This graph is misleading because the lengths of the bars are not proportional to one another as they should be to accurately reflect the percent of employers offering each of the named perks. For example, the bar for massage therapy should be eight times as long as the bar for nap during workday instead of being approximately four times as long, as the graph shows. Why This Is Important In order to correctly interpret data from a graph, it is important to be aware that a graph may be misleading. Massage therapy Nap during workday How Employers Make Workers Happy Stress reduction 1% 21% 8% Price Price Month Month Stock A Stock B J J F M A M J J F M A M 40 30 30 0 25 35 20 10 0 Figure 12.2
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