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study began, there was a lower incidence of heart attack for those who took aspirin than for those who took a placebo. Suppose that some medical researchers want to study this phenomenon more closely. They recruit 2000 healthy women aged 65 years and older, and randomly divide them into two groups. One group takes 100 mg of aspirin every other day, and the other group takes a placebo. The women do not know to which group they belong, but the doctors who are conducting the study have access to this information. a. Is this an observational study or a designed experiment? Explain. b. Is this a double-blind study? Explain. 1.32 In March 2005, The New England Journal of Medicine pub- lished the results of a 10-year clinical trial of low-dose aspirin therapy for the cardiovascular health of women (Time, March 21, 2005). The study was based on 40,000 healthy women, most of whom were in their 40s and 50s when the trial began. Half of these women were administered 100 mg of aspirin every other day, and the others were given a placebo. The study noted that among women who were at least 65 years old when the study began, there was a lower incidence of heart attack for those who took aspirin than for those who took pla- cebo. Some medical researchers want to study this phenomenon more closely. They recruit 2000 healthy women aged 65 years and older, and randomly divide them into two groups. One group takes 100 mg of aspirin every other day, and the other group takes a placebo. Neither patients nor doctors know which group patients belong to. a. Is this an observational study or a designed experiment? Explain. b. Is this study a double-blind study? Explain. 1.33 A federal government think tank wanted to investigate whether a job training program helps the families who are on welfare to get off the welfare program. The researchers at this agency selected 5000 volunteer families who were on welfare and offered the adults in those families free job training. The researchers selected another group of 5000 volunteer families who were on welfare and did not offer them such job training. After 3 years the two groups were compared in regard to the percentage of families who got off welfare. Is this an observational study or a designed experiment? Explain. 1.34 A federal government think tank wanted to investigate whether a job training program helps the families who are on welfare to get off the welfare program. The researchers at this agency selected 10,000 families at random from the list of all families that were on welfare. Of these 10,000 families, the agency randomly selected 5000 families and offered them free job training. The remaining 5000 families were not offered such job training. After 3 years the two groups were com- pared in regard to the percentage of families who got off welfare. Is this an observational study or a designed experiment? Explain. 1.35 A federal government think tank wanted to investigate whether a job training program helps the families who are on welfare to get off the welfare program. The researchers at this agency selected 5000 volunteer families who were on welfare and offered the adults in those families free job training. The researchers selected another group of 5000 volunteer families who were on welfare and did not offer them such job training. After three years the two groups were compared in regard to the percentage of families who got off welfare. Based on that study, the researchers concluded that the job training program causes (helps) families who are on welfare to get off the welfare program. Do you agree with this conclusion? Explain. 1.36 A federal government think tank wanted to investigate whether a job training program helps the families who are on welfare to get off the welfare program. The researchers at this agency selected 10,000 families at random from the list of all families that were on welfare. Of these 10,000 families, the researchers randomly selected 5000 families and offered the adults in those families free job training. The remaining 5000 families were not offered such job training. After three years the two groups were compared in regard to the percentage of families who got off welfare. Based on that study, the researchers concluded that the job training program causes (helps) families who are on welfare to get off the welfare program. Do you agree with this conclusion? Explain. Sometimes mathematical notation helps express a mathematical relationship concisely. This section describes the summation notation that is used to denote the sum of values. Suppose a sample consists of five books, and the prices of these five books are $175, $80, $165, $97, and $88, respectively. The variable, price of a book, can be denoted by x. The prices of the five books can be written as follows: Price of the first book = x1 = $175 1.7 Summation Notation Subscript of x denotes the number of the book 1.7 Summation Notation 23 Similarly, Price of the second book = x2 = $80 Price of the third book = x3 = $165 Price of the fourth book = x4 = $97 Price of the fifth book = x5 = $88 In this notation, x represents the price, and the subscript denotes a particular book. Now, suppose we want to add the prices of all five books. We obtain x1 + x2 + x3 + x4 + x5 = 175 + 80 + 165 + 97 + 88 = $605 The uppercase Greek letter ∑ (pronounced sigma) is used to denote the sum of all values. Using ∑ notation, we can write the foregoing sum as follows: ∑x = x1 + x2 + x3 + x4 + x5 = $605 The notation ∑x in this expression represents the sum of all values of x and is read as “sigma x” or “sum of all values of x.” Using summation notation: one variable. © Troels Graugaard/iStockphoto EXAMPLE 1–4 The following table lists four pairs of m and f values: m 12 15 20 30 f 5 9 10 16 Compute the following: (a) ∑m (b) ∑ f 2 (c) ∑mf (d) ∑m2f Solution We can write m1 = 12 m2 = 15 m3 = 20 m4 = 30 f1 = 5 f2 = 9 f3 = 10 f4 = 16 (a) ∑m = 12 + 15 + 20 + 30 = 77 (b) ∑ f 2 = (5)2 + (9)2 + (10)2 + (16)2 = 25 + 81 + 100 + 256 = 462 Using summation notation: two variables. EXAMPLE 1–3 Annual Salaries of Workers Annual salaries (in thousands of dollars) of four workers are 75, 90, 125, and 61, respectively. Find (a) ∑x (b) (∑x)2 (c) ∑x2 Solution Let x1, x2, x3, and x4 be the annual salaries (in thousands of dollars) of the first, second, third, and fourth worker, respectively. Then, x1 = 75, x2 = 90, x3 = 125, and x4 = 61 (a) ∑x = x1 + x2 + x3 + x4 = 75 + 90 + 125 + 61 = 351 = $351,000 (b) Note that (∑x)2 is the square of the sum of all x values. Thus, (∑x)2 = (351)2 = 123,201 (c) The expression ∑x2 is the sum of the squares of x values. To calculate ∑x2, we first square each of the x values and then sum these squared values. Thus, ∑x2 = (75)2 + (90)2 + (125)2 + (61)2 = 5625 + 8100 + 15,625 + 3721 = 33,071 ◼ 24 Chapter 1 Introduction (c) To compute ∑mf, we multiply the corresponding values of m and f and then add the products as follows: ∑mf = m1 f1 + m2 f2 + m3 f3 + m4 f4 = 12(5) + 15(9) + 20(10) + 30(16) = 875 (d) To calculate ∑m2f, we square each m value, then multiply the corresponding m2 and f values, and add the products. Thus, ∑m2f = (m1) 2f1 + (m2) 2f2 + (m3) 2f3 + (m4) 2f4 = (12)2(5) + (15)2(9) + (20)2(10) + (30)2(16) = 21,145 The calculations done in parts (a) through (d) to find the values of ∑m, ∑ f 2, ∑mf, and ∑m2f can be performed in tabular form, as shown in Table 1.4. Table 1.4 m f f 2 mf m2f 12 5 5 × 5 = 25 12 × 5 = 60 12 × 12 × 5 = 720 15 9 9 × 9 = 81 15 × 9 = 135 15 × 15 × 9 = 2025 20 10 10 × 10 = 100 20 × 10 = 200 20 × 20 × 10 = 4000 30 16 16 × 16 = 256 30 × 16 = 480 30 × 30 × 16 = 14,400