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Chapter 5: Q15E (page 314)
Will the sampling distribution of always be approximately normally distributed? Explain
Answer
A sampling distribution is statistics derived by continuous sampling from a greater populace.
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Motivation of drug dealers. Refer to the Applied Psychology in Criminal Justice (September 2009) investigation of the personality characteristics of drug dealers, Exercise 2.80 (p. 111). Convicted drug dealers were scored on the Wanting Recognition (WR) Scale. This scale provides a quantitative measure of a person’s level of need for approval and sensitivity to social situations. (Higher scores indicate a greater need for approval.) Based on the study results, we can assume that the WR scores for the population of convicted drug dealers have a mean of 40 and a standard deviation of 5. Suppose that in a sample of 100 people, the mean WR scale score is x = 42. Is this sample likely selected from the population of convicted drug dealers? Explain.
Fecal pollution at Huntington Beach. California mandates fecal indicator bacteria monitoring at all public beaches. When the concentration of fecal bacteria in the water exceeds a certain limit (400 colony-forming units of fecal coliform per 100 millilitres), local health officials must post a sign (called surf zone posting) warning beachgoers of potential health risks. For fecal bacteria, the state uses a single-sample standard; if the fecal limit is exceeded in a single sample of water, surf zone posting is mandatory. This single-sample standard policy has led to a recent rash of beach closures in California. A study of the surf water quality at Huntington Beach in California was published in Environmental Science & Technology (September 2004). The researchers found that beach closings were occurring despite low pollution levels in some instances, while in others, signs were not posted when the fecal limit was exceeded. They attributed these "surf zone posting errors" to the variable nature of water quality in the surf zone (for example, fecal bacteria concentration tends to be higher during ebb tide and at night) and the inherent time delay between when a water sample is collected and when a sign is posted or removed. To prevent posting errors, the researchers recommend using an averaging method rather than a single sample to determine unsafe water quality. (For example, one simple averaging method is to take a random sample of multiple water specimens and compare the average fecal bacteria level of the sample with the limit of 400 CFU/100 mL to determine whether the water is safe.) Discuss the pros and cons of using the single sample standard versus the averaging method. Part of your discussion should address the probability of posting a sign when the water is safe and the probability of posting a sign when the water is unsafe. (Assume that the fecal bacteria concentrations of water specimens at Huntington Beach follow an approximately normal distribution.
A random sample of n = 80 measurements is drawn from a binomial population with a probability of success .3.
Study of why EMS workers leave the job. A study of fulltimeemergency medical service (EMS) workers publishedin the Journal of Allied Health(Fall 2011) found that onlyabout 3% leave their job in order to retire. (See Exercise3.45, p. 182.) Assume that the true proportion of all fulltime
EMS workers who leave their job in order to retire is p= .03. In a random sample of 1,000 full-time EMS workers, let represent the proportion who leave their job inorder to retire.
Improving SAT scores. Refer to the Chance(Winter2001) examination of Scholastic Assessment Test (SAT)scores of students who pay a private tutor to help them improve their results, Exercise 2.88 (p. 113). On the SAT—Mathematics test, these students had a mean score change of +19 points, with a standard deviation of 65 points. In a random sample of 100 students who pay a private tutor to help them improve their results, what is the likelihood that the sample mean score change is less than 10 points?
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