# Due Date: Friday, Apr. 11, 2003, turn in after lecture

Correct solutions to any seven problems carry full credit. See the HW submission rules. On the front page include the following header.

• (HW_1_1)  [Sec. 1.2,  #26]  The clearness index was determined for the skies over Baghdad for each of the 365 days during a particular year. It was a contribution to the study of the solar radiation of the Baghdad environment. The following table summarised the resulting data.

 Class interval for the clearness index Number of days Relative Frequency Cumulative Relative Frequency 0.16 - 0.20 3 0.008219 0.00822 0.21 - 0.25 5 0.013699 0.02192 0.26 - 0.30 6 0.016438 0.03836 0.31 - 0.35 8 0.021918 0.06027 0.36 - 0.40 12 0.032877 0.09315 0.41 - 0.45 16 0.043836 0.13699 0.46 - 0.50 24 0.065753 0.20274 0.51 - 0.55 39 0.106849 0.30959 0.56 - 0.60 51 0.139726 0.44932 0.61 - 0.65 106 0.290411 0.73973 0.66 - 0.70 84 0.230137 0.96986 0.71 - 0.75 11 0.030137 1.00000

• Determine the Relative Frequency and the Cumulative Relative Frequency (fill in the two last columns of the table)
• Sketch the Relative Frequency histogram and comment on it.
• Cloudy days are those with the clearness index 0.35. What proportion of the days were cloudy?
• Clear days are those for which the clearness index is at least 0.66. What proportion of the days were clear?

• (HW_1_2)  [Sec. 1.3,  #36] A sample of 26 offshore oil workers took part in a simulated escape exercise, resulting in the accompanying data on time (sec) to complete the escape ("Oxygen Consumption and Ventilation During Escape from an Offshore Platform." Ergonomics, 1997: 281-292). The escape times are shown below. The operators of the platform have requested your assistance in improving the platform. Specifically, the operators want to know what escape time corresponds with a 1% chance of being exceeded.
 389 356 359 363 375 424 325 394 402 373 373 370 364 366 364 325 339 393 392 369 374 359 356 403 334 397

• Construct a stem-and-leaf display of the data. How does it suggest that the sample mean and median will compare?
• Calculate the sample mean and median
• By how much can the largest time be increaded without effecting the value of the sample median?
• What are the values of the sample mean and median when the observations are reexpressed in minutes?

• (HW_1_3) [Sec. 1.4,  #50] The accompanying data on bearing load-life (million revolutions) for bearings of certain type when subjected to a 9.56 kN load are (Lubric. Eng., 1984, 153-159)
 14.5 25.6 52.4 66.3 69.3 69.8 76.2
• Calculate the values of the sample mean and median. What do their values, relative to one another, tell you about the sample?
• Calculate the sample variance ans standard deviation.

• (HW_1_4) [Sec. 2.1,  #8] An engineering construction firm is currently working on power plants at three different sites. Let Ai denote the event that the plant at site i is completed by the contract date. Use the operations of union, intersection, and complementation to describe each of the following events in terms of A1A2, and A3. Draw a Venn diagram, and shade the region corresponding to each one.
• At least one plant is completed by the contract date.
• All plants are completed by the contract date.
• Only the plant at site 1 is completed by the contract date.
• Exactly one plant is completed by the contract date.
• Either the plant at site 1 or both of the other two plants are completed by the contract date.

• (HW_1_5) [Sec. 2.2,  #13] A computer consulting firm presently has bids out on three projects. Let Ai = { awarded project i } , for i = 1, 2, 3, and suppose that P(A1) = 0.22, P(A2) = 0.25, P(A3) = 0.28, P(A1 ∩ A2) = 0.11, P(A1 ∩ A3) = 0.05, P(A2 ∩ A3) = 0.07, P(A1 ∩ A2 ∩ A3) = 0.01. Express in words each of the following events and compute the probability of each event.
(a) A1 ∪ A
(b) A'1 ∩ A'2
(c) A1 ∪ A2 ∪ A3
(d) A'1 ∩ A'2 ∩ A'3
(e) A'1 ∩ A'2 ∩ A3
(f) (A'1 ∩ A'2) ∪ A3

• (HW_1_6) [Sec. 2.2,  #21] An insurance company offers four different deductible levels – none, low, medium, and high – for its homeowner’s policyholders and three different levels – low, medium, and high – for its automobile policyholders. The accompanying table gives proportions for the various categories of policyholders who have both
types of insurance. For example, the proportion of individuals with both low homeowner’s deductible and low auto deductible is 0.06 (6% of all such individuals). Suppose an individual having both types of policies is randomply selected.
 Homeowner’s

 Auto N L M H L 0.04 0.06 0.05 0.03 M 0.07 0.10 0.20 0.10 H 0.02 0.03 0.15 0.15

(a) What is the probability that the individual has a medium auto deductible and a high homeowner’s deductible?
(b) What is the probability that the individual has a low auto deductible? A low homeowner’s deductible?
(c) What is the probability that the individual is in the same category for both auto and homeowner’s deductibles?
(d) Based on your answer in part (c), what is the probability that the two categories are different?
(e) What is the probability that the individual has at least one low de-ductible level?
(f) Using the answer in part (e), what is the probability that neither deductible level is low?

• (HW_1_7) [Sec. 2.2,  #25] The three major popular options on a certain type of new car are automatic transmission (A), a sunroof (B), and a stereo with a compact disc player (C). If 70% of all purchasers request A, 80% request B, 75% re-quest C, 85% request A or B, 90% request A or C, 95% request B or C, and 98% request A or B or C, compute the probabilities of the following events.
(a) The next purchaser will select at least one of the three options.
(b) The next purchaser will select none of the three options.
(c) The next purchaser will select only a stereo with a compact disc player.
(d) The next purchaser will select exactly one of the three options.

• (HW_1_8) [Sec. 2.2,  #26] A certain system can experience three different types of defects. Let A1, i = 1, 2, 3, denote the event that the system has a defect of type i. Suppose that P(A1) = 0.12, P(A2) = 0.07, P(A3) = 0.05, P(A1 ∪ A2) = 0.13, P(A1 ∪ A3) = 0.14, P(A2 ∪ A3) = 0.10, P(A1 ∩ A2 ∩ A3) = 0.01.
(a) What is the probability that the system does not have a type 1 defect?
(b) What is the probability that the system has both type 1 and type 2 defects?
(c) What is the probability that the system has both type 1 and type 2 defects, but not type 3 defects?
(d) What is the probability that the system has at most two of these defects?

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Ivo D. Dinov, Ph.D., Departments of Statistics and Neurology, UCLA School of Medicine