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

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 c. Use a normal approximation to compute the probability that less than six sales are made. d. Use a Poisson approximation to compute the probability that less than six sales are made.

e. Use a computer program if available to compute the exact probability that less than

six sales are made. Compare this result with your calculations in c and d. 4.52 A certain birth defect occurs in 1 of every 10,000 births. In the next 5,000 births at a major hospital, what is the probability that at least one baby will have the defect? What assumptions are required to calculate this probability? 4.10 A Continuous Probability Distribution: The Normal Distribution 4.53 Use Table 1 of the Appendix to find the area under the normal curve between these values:

a. z ⫽ 0 and z ⫽ 1.6 b. z ⫽ 0 and z ⫽ 2.3

4.54 Repeat Exercise 4.53 for these values: a. z ⫽ .7 and z ⫽ 1.7

b. z ⫽ ⫺1.2 and z ⫽ 0 4.55 Repeat Exercise 4.53 for these values:

a. z ⫽ ⫺1.29 and z ⫽ 0 b. z ⫽ ⫺.77 and z ⫽ 1.2

4.56 Repeat Exercise 4.53 for these values: a. z ⫽ ⫺1.35 and z ⫽ ⫺.21

b. z ⫽ ⫺.37 and z ⫽ ⫺1.20 4.57 Find the probability that z is greater than 1.75.

4.58 Find the probability that z is less than 1.14. 4.59 Find a value for z, say z

, such that Pz ⬎ z ⫽ .5.

4.60 Find a value for z, say z , such that Pz ⬎ z

⫽ .025.

4.61 Find a value for z, say z , such that Pz ⬎ z

⫽ .0089.

4.62 Find a value for z, say z , such that Pz ⬎ z

⫽ .05.

4.63 Find a value for z, say z , such that P⫺z

⬍ z ⬍ z ⫽ .95. 4.64 Let y be a normal random variable with mean equal to 100 and standard deviation equal to 8. Find the following probabilities:

a. Py ⬎ 100 b. Py ⬎ 105

c. Py ⬍ 110 d. P88 ⬍ y ⬍ 120

e. P100 ⬍ y ⬍ 108

4.65 Let y be a normal random variable with m ⫽ 500 and s ⫽ 100. Find the following proba- bilities:

a. P500 ⬍ y ⬍ 665 b. Py ⬎ 665

c. P304 ⬍ y ⬍ 665 d. k such that P500 ⫺ k ⬍ y ⬍ 500 ⫹ k ⫽ .60

4.66 Suppose that y is a normal random variable with m ⫽ 100 and s ⫽ 15. a. Show that y ⬍ 115 is equivalent to z ⬍ 1.

b. Convert y ⬎ 85 to the z-score equivalent. c. Find Py ⬍ 115 and Py ⬎ 85.

d. Find Py ⬎ 106, Py ⬍ 94, and P94 ⬍ y ⬍ 106. e. Find Py ⬍ 70, Py ⬎ 130, and P70 ⬍ y ⬍ 130.

4.67 Find the value of z for these areas. a. an area .025 to the right of z

b. an area .05 to the left of z