Find . d. Find . Lyman Ott Michael Longnecker

Bus. 4.41 The number of daily requests for emergency assistance at a fire station in a medium-sized city has the probability distribution shown here. y Py .06 1 .14 2 .16 3 .14 4 .12 5 .10 6 .08 7 .07 8 .06 9 .04 10 .03 a. What is the probability that four or more requests will be made in a particular day? b. What is the probability that the requests for assistance will be at least four but no more than six?

c. Suppose the fire station must call for additional equipment from a neighboring city

whenever the number of requests for assistance exceeds eight in a given day. The neighboring city then charges for its equipment. What is the probability the city will call for additional equipment on a given day? 4.8 Two Discrete Random Variables: The Binomial and The Poisson Bio.

4.42 A biologist randomly selects 10 portions of water, each equal to .1 cm

3 in volume, from the local reservoir and counts the number of bacteria present in each portion. The biologist then totals the number of bacteria for the 10 portions to obtain an estimate of the number of bacteria per cubic centimeter present in the reservoir water. Is this a binomial experiment? Pol. Sci. 4.43 Examine the accompanying newspaper clipping. Does this sampling appear to satisfy the characteristics of a binomial experiment? Poll Finds Opposition to Phone Taps New York—People surveyed in a recent poll indicated they are 81 to 13 against having their phones tapped without a court order. The people in the survey, by 68 to 27, were opposed to letting the government use a wiretap on citizens suspected of crimes, except with a court order. The survey was conducted for 1,495 house- holds and also found the following results: — The people surveyed are 80 to 12 against the use of any kind of electronic spying device without a court order. — Citizens are 77 to 14 against allowing the government to open their mail without court orders. — They oppose, by 80 to 12, letting the telephone company disclose records of long- distance phone calls, except by court order. For each of the questions, a few of those in the survey had no responses. Env. 4.44 A survey is conducted to estimate the percentage of pine trees in a forest that are infected by the pine shoot moth. A grid is placed over a map of the forest, dividing the area into 25-foot by 25-foot square sections. One hundred of the squares are randomly selected and the number of infected trees is recorded for each square. Is this a binomial experiment? Env. 4.45 In an inspection of automobiles in Los Angeles, 60 of all automobiles had emissions that did not meet EPA regulations. For a random sample of 10 automobiles, compute the following probabilities:

a. All 10 automobiles failed the inspection. b. Exactly 6 of the 10 failed the inspection.

c. Six or more failed the inspection. d. All 10 passed the inspection.

Use the following Minitab output to answer the questions. Note that with Minitab, the binomial probability p is denoted by p and the binomial variable y is represented by x. Bus. 4.46 Over a long period of time in a large multinational corporation, 10 of all sales trainees are rated as outstanding, 75 are rated as excellent good, 10 are rated as satisfactory, and 5 are considered unsatisfactory. Find the following probabilities for a sample of 10 trainees selected at random:

a. Two are rated as outstanding. b. Two or more are rated as outstanding.

c. Eight of the ten are rated either outstanding or excellent good. d. None of the trainees is rated as unsatisfactory. Med. 4.47 A relatively new technique, balloon angioplasty, is widely used to open clogged heart valves and vessels. The balloon is inserted via a catheter and is inflated, opening the vessel; thus, no surgery is required. Left untreated, 50 of the people with heart-valve disease die within about 2 years. If experience with this technique suggests that approximately 70 live for more than 2 years, would the next five patients of the patients treated with balloon angioplasty at a hospital constitute a binomial experiment with n ⫽ 5, p ⫽ .70? Why or why not? Bus. 4.48 A random sample of 50 price changes is selected from the many listed for a large super- market during a reporting period. If the probability that a price change is posted correctly is .93, a. Write an expression for the probability that three or fewer changes are posted incorrectly. b. What assumptions were made for part a? 4.49 Suppose the random variable y has a Poisson distribution. Use Table 15 in the Appendix to compute the following probabilities:

a. Py ⫽ 1 given m ⫽ 3.0 b. Py ⬎ 1 given m ⫽ 2.5

c. Py ⬍ 5 given m ⫽ 2.0 4.50 Cars arrive at a toll booth at a rate of six per 10 seconds during rush hours. Let N be the number of cars arriving during any 10-second period during rush hours. Use Table 15 in the Appendix to compute the probability of the following events:

a. No cars arrive. b. More than one car arrives.

c. At least two cars arrive. 4.51 A firm is considering using the Internet to supplement its traditional sales methods. From the data of similar firms, it is estimated that one of every 1,000 Internet hits result in a sale. Sup- pose the firm has 2,500 hits in a single day.

a. Write an expression for the probability that there are less than six sales, do not com-

plete the calculations. b. What assumptions are needed to write the expression in part a? Binomial Distribution with n 10 and p 0.6 x PX x PX x 0.00 0.0001 0.0001 1.00 0.0016 0.0017 2.00 0.0106 0.0123 3.00 0.0425 0.0548 4.00 0.1115 0.1662 5.00 0.2007 0.3669 6.00 0.2508 0.6177 7.00 0.2150 0.8327 8.00 0.1209 0.9536 9.00 0.0403 0.9940 10.00 0.0060 1.0000