# MATHEMATICS

25. A researcher was studying how the weights of horseshoe crabs along the east coast varied between northern and southern habitats. In spring 2006, during breeding season, the researcher randomly selected 50 horseshoe crabs in southern habitats. In fall 2006, before breeding season begins, the researcher randomly selected 50 horseshoe crabs from northern habitats. It was found that crabs in southern habitats weighed more and so the researcher claimed that location had an effect on weight of horseshoe crabs. The results cannot be trusted because the investigator should have used more than two locations. investigator was biased. She knew beforehand what the study would show. time of year during which the samples were taken is a confounding variable. study was not double-blind. 26. Researchers wish to determine if a new experimental medication will reduce the symptoms of allergy sufferers without the side effect of drowsiness. To investigate this question, the researchers give the new medication to 50 adult volunteers who suffer from allergies. Forty-four of these volunteers report a significant reduction in their allergy symptoms without any drowsiness. This study could be improved by including people who do not suffer from allergies in the study in order to represent a more diverse population. repeating the study using only the volunteers who reported a significant reduction in their allergy symptoms without any drowsiness and giving them a higher dosage this time. using a control group. All of the above 27. Will a fluoride mouthwash used after brushing reduce cavities? Twenty sets of twins were used to investigate this question. One member of each set of twins used the mouthwash after each brushing, the other did not. After six months, the difference in the number of cavities of those using the mouthwash was compared with the number of those who did not use the mouthwash. This experiment uses double replication. a matched pairs design. double-blinding. random placeboes. 28. A study examined the effect of garlic on blood cholesterol levels. A total of 192 subjects with starting LDL (“bad cholesterol”) levels between 130 and 190 mg/dl were randomly assigned to one of four treatments for six months: raw garlic eaten daily, a powdered garlic supplement taken daily (Garlicin brand), an aged-garlic extract taken daily (Kyolic brand), or a placebo pill taken daily. The study found no significant or clinical differences in blood cholesterol levels between the four groups at the end of the study. This study uses the principles of randomization. blinding. blocking. All of the above 29. A population has an equal proportion of males and females. That is, when randomly selecting one individual, the probability that the individual is male (M) is 1/2 and the probability that the individual is female (F) is 1/2. In the first 50 randomly selected individuals, 20 were male. In the next 50 randomly selected individuals, which of the following must happen? More than half of the individuals will be males to balance out the low number of males in the first 50 individuals. However, the order in which these males will be selected is unpredictable. More than 20 of the individuals will be males because the proportion of males after 100 individuals must be closer to 1/2 than the proportion after 50 individuals. The number of males will be very close to 30 in the next 50 individuals because the proportion must be close to 1/2 after 100 individuals. None of the above 30. The Centers for Disease Control and Prevention receives information about the causes for HIV/AIDS infection in the United States. Here are the causes of infection and their respective probability among American women ages 20 to 24 years old. Cause Probability Injection drug use ? Heterosexual sex 0.664 Perinatal infection 0.125 Other causes 0.095 The probability that an American woman age 20 to 24 contracted HIV/AIDS either via heterosexual sex or via perinatal infection is 0.789. 0.211. 0.500. 0.083. 31. Based on a recent Gallup survey, we define the following probability model for the number X of daily cups of coffee consumed by Americans. X 0 1 2 3 4 or more Probability 0.36 0.26 0.19 0.09 ? The probability that an American does not consume 2 cups of coffee daily is _______ . 32. A variable whose value is a numerical outcome of a random phenomenon is called a random variable. biased. a random sample. a parameter. 33. The frequency of color blindness (dyschromatopsia) in the Caucasian American male population is estimated to be about 8%. Let’s call X the number of color-blind individuals in a random sample of 8 Caucasian American males. The table below gives the probability distribution for X. X 0 1 2 3 4 5 6 7 8 P(X) 0.513 0.357 0.109 0.019 0.002 0.000 0.000 0.000 0.000 What is the probability of getting 1 or 2 color-blind individuals in the sample? 0.248 0.870 0.466 0.039 34. Using the standard Normal distribution tables, the area under the standard Normal curve corresponding to Z > –2.62 is 0.0047. 0.9956. 0.0044. 0.9953. 35. The pH measurements of water specimens from various locations along a given river basin are Normally distributed with mean 8 and standard deviation 0.3. What is, approximately, the probability that the pH measurement of a randomly selected water specimen is greater than 8.2? 0.2525 0.7475 0.7525 0.2475 36. A researcher is interested in the lengths of Salvelinus fontinalis (brook trout), which are known to be approximately Normally distributed with mean 80 centimeters and standard deviation 5 centimeters. To help preserve brook trout populations, some regulatory standards need to be set limiting the size of fish that can be caught. The probability of catching a brook trout less than 72 centimeters in length is 0.3745. 0.9452 0.0548. 0.6255.