Solved: Deciphering Messages The Central Intelligence

Problem 1BSC Notation In a clinical trial of the cholesterol drug Lipitor, 270 subjects were given a placebo, and 7% of them developed headaches. For such randomly selected groups of 270 subjects given a placebo, identify the values of n, p, and q that would be used for finding the mean and standard deviation for the number of subjects who develop headaches.

Standard Deviation For the binomial distribution described by the conditions in Exercise 1, find the standard deviation by evaluating n p q . Some books use the expression \(n p(1-p)\) for the standard deviation. Will that expression always yield the same result? Equation Transcription: Text Transcription: n p (1-p)

Problem 3BSC Variance An Office Team survey of 150 executives found that 93.3% of them said that they would be concerned about gaps in a résumé of a job applicant. Based on these results, such randomly selected groups of 150 executives have a mean of 140.0 executives and a standard deviation of 3.1 executives. Find the variance, and express it with the appropriate units.

Problem 5BSC Finding ?, ?, and Unusual Values. In Exercises, assume that a procedure yields a binomial distribution with n trials and the probability of success for one trial is p. Use the given values of n and p to find the mean ? and standard deviation ? . Also, use the range ruh of thumb to find the minimum usual value ? – 2? and the maximum usual value ? – 2 ?. Guessing on ACT Random guesses are made for the 60 multiple-choice questions on the math portion of the ACT test, so n = 60 and p =1/5 (because each question has possible answers of a, b, c, d, e, and only one of them is correct).

Why a Parameter? Given the results described in Exercise 3, is the mean of 140.0 executives expressed with \(\mu \text { or } x^{-}\) ? Explain why the mean of 140.0 executives is a parameter instead of a statistic. Equation Transcription: Text Transcription: \mu or x^-

Problem 6BSC Finding ?, ?, and Unusual Values. In Exercises, assume that a procedure yields a binomial distribution with n trials and the probability of success for one trial is p. Use the given values of n and p to find the mean ? and standard deviation ? . Also, use the range ruh of thumb to find the minimum usual value ? – 2? and the maximum usual value ? – 2 ?. Gender Selection In an analysis of preliminary test results from the XSORT gender-selection method, 14 babies are born and it is assumed that 50% of babies are girls, so n = 14 and p = 0.5.

Problem 7BSC Finding ?, ?, and Unusual Values. In Exercises, assume that a procedure yields a binomial distribution with n trials and the probability of success for one trial is p. Use the given values of n and p to find the mean ? and standard deviation ? . Also, use the range ruh of thumb to find the minimum usual value ? – 2? and the maximum usual value ? – 2 ?. Identity Theft In a Gallup poll of 1013 randomly selected adults, 66% said that they worry about identity theft, so n = 1013 and p = 0.66.

Problem 9BSC Gender Selection In a test of the YSORT method of gender selection, 291 babies are born to couples trying to have baby boys, and 239 of those babies are boys (based on data from the Genetics&IVF Institute). a. If the gender-selection method has no effect and boys and girls are equally likely, find the mean and standard deviation for the numbers of boys born in groups of 291. ________________ b. Is the result of 239 boys unusually high? Does it suggest that the YSORT gender-selection method appears to be effective?

Problem 10BSC Mendelian Genetics When Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 580 peas, and Mendel theorized that 25% of them would be yellow peas. a. If Mendel’s theory is correct, find the mean and standard deviation for the numbers of yellow peas in such groups of 580 offspring peas. ________________ b. The actual results consisted of 152 yellow peas. Is that result unusually high? What does this result suggest about Mendel’s theory?

Problem 13BSC Cell Phones and Brain Cancer In a study of 420,095 cell phone users in Denmark, it was found that 135 developed cancer of the brain or nervous system. If we assume that the use of cell phones has no effect on developing such cancer, then the probability of a person having such a cancer is 0.000340. a. Assuming that cell phones have no effect on developing cancer, find the mean and standard deviation for the numbers of people in groups of 420,095 that can be expected to have cancer of the brain or nervous system. ________________ b. Based on the results from part (a), is 135 cases of cancer of the brain or nervous system unusually low or high? ________________ c. What do these results suggest about the publicized concern that cell phones are a health danger because they increase the risk of cancer of the brain or nervous system?

Are 20% of M&M Candies Orange? Mars, Inc. claims that 20% of its M&M plain candies are orange, and a sample of 100 such candies is randomly selected. a. Find the mean and standard deviation for the number of orange candies in such groups of 100. b. Data Set 20 in Appendix B consists of a random sample of 100 M&Ms, including 25 that are orange. Is this result unusually high? Does it seem that the claimed rate of 20% is wrong?

Are 14% of M&M Candies Yellow? Mars, Inc. claims that 14% of its M&M plain candies are yellow, and a sample of 100 such candies is randomly selected. a. Find the mean and standard deviation for the number of yellow candies in such groups of 100. b. Data Set 20 in Appendix B consists of a random sample of 100 M&Ms, including 8 that are yellow. Is this result unusually low? Does it seem that the claimed rate of 14% is wrong?

Problem 14BSC Test of Touch Therapy Nine-year-old Emily Rosa conducted this test: A professional touch therapist put both hands through a cardboard partition and Emily would use a coin flip to randomly select one of the hands. Emily would place her hand just above the hand of the therapist, who was then asked to identify the hand that Emily had selected. Touch therapists believed that they could sense the energy field and identify the hand that Emily had selected. The trial was repeated 280 times. (Based on data from “A Close Look at Therapeutic Touch,” by Rosa et al Journal of the American Medical Association, Vol. 279, No. 13.) a. Assuming that the touch therapists have no special powers and made random guesses, find the mean and standard deviation for the numbers of correct responses in groups of 280 trials. b. The professional touch therapists identified the correct hand 123 times in the 280 trials. Is that result unusually low or high? What does the result suggest about the ability of touch therapists to select the correct hand by sensing an energy field?

Problem 15BSC Deciphering Messages The Central Intelligence Agency has specialists who analyze the frequencies of letters of the alphabet in an attempt to decipher intercepted messages that are sent as ciphered text. In standard English text, the letter r is used at a rate of 6%. a. Find the mean and standard deviation for the number of times the letter r will be found on a typical page of 2600 characters. ________________ b. In an intercepted ciphered message sent to Iran, a page of 2600 characters is found to have the letter r occurring 178 times. Is this unusually low or high?

Problem 16BSC Deciphering Messages The Central Intelligence Agency has specialists who analyze the frequencies of letters of the alphabet in an attempt to decipher intercepted messages that are sent as ciphered text. In standard English text, the letter e is used at a rate of 12.7%. a. Find the mean and standard deviation for the number of times the letter e will be found on a typical page of 2600 characters. ________________ b. In an intercepted ciphered message sent to France, a page of 2600 characters is found to have the letter e occurring 290 times. Is 290 unusually low or unusually high?

Problem 17BSC Too Young to Tat Based on a Harris poll of 370 adults who regret getting tattoos, 20% say that they were too young when they got their tattoos. a. For randomly selected groups of 370 adults who regret getting tattoos, find the mean and standard deviation for the number who say that they were too young when they got their tattoos. ________________ b. For a randomly selected group of 370 adults who regret getting tattoos, would 90 be an unusually low or high number who say that they were too young when they got their tattoos?

Problem 19BSC Born on the 4th of July For the following questions, ignore leap years. a. For classes of 30 students, find the mean and standard deviation for the number born on the 4th of July. Express results using seven decimal places. ________________ b. For a class of 30 students, would 2 be an unusually high number who were born on the 4th of July?

Problem 18BSC Roulette If you place a bet on the number 7 in roulette, you have a 1/38 probability of winning. a. Find the mean and standard deviation for the number of wins for people who bet on the number 7 fifty times. ________________ b. Would 0 wins in 50 bets be an unusually low number of wins?

Problem 20BSC Powerball Lottery As of this writing, the Powerball lottery is run in 42 states. If you buy one ticket, the probability of winning is 1/195,249,054. If you buy one ticket each week for 50 years, you play this lottery 2600 times. a. Find the mean and standard deviation for the number of wins for people who buy a ticket each week for 50 years. Express the results using six decimal places. ________________ b. Would it be unusual for someone to win this lottery at least once if they buy a ticket each week for 50 years?

Problem 21BB Finding n, p, q In a survey of randomly selected adults, subjects were asked if they could identify at least one current member of the Supreme Court. After obtaining the results, the range rule of thumb was used to find that for randomly selected groups of the same size, the minimum usual number who could identify at least one member of the Supreme Court is 48.0 and the maximum number is 72.0. Find the sample size n, the percentage of surveyed subjects who could identify at least one member of the Supreme Court, and the percentage of subjects who could not identify at least one member of the Supreme Court.

Problem 22BB Acceptable/Defective Products A new integrated circuit board is being developed for use in computers. In the early stages of development, a lack of quality control results in a 0.2 probability that a manufactured integrated circuit board has no defects. Engineers need 24 integrated circuit boards for further testing. What is the minimum number of integrated circuit boards that must be manufactured in order to be at least 98% sure that there are at least 24 that have no defects?

Problem 23BB Hypergeometric Distribution A statistics class consists of 10 females and 30 males, and each day, 12 of the students are randomly selected without replacement. Because the sampling is from a small finite population without replacement, the hypergeometric distribution applies. Using the hypergeometric distribution, find the mean and standard deviation for the numbers of females that are selected on the different days.

Problem 8BSC Finding ?, ?, and Unusual Values. In Exercises, assume that a procedure yields a binomial distribution with n trials and the probability of success for one trial is p. Use the given values of n and p to find the mean ? and standard deviation ? . Also, use the range ruh of thumb to find the minimum usual value ? – 2? and the maximum usual value ? – 2 ?. Clinical Trial In a clinical trial of the cholesterol drug Lipitor, 94 subjects were treated with 80 mg of Lipitor, and 6.4% of them developed headaches, so n = 94 and p = 0.064.