1.Find the sample mean x, median, variance s2, and standard deviation s of the following data:

1510251240

2.Compute a 95% confidence interval for the true population mean if the sample mean x =15.3 cm with a sample size of10 ceramic vessels and a sample standard deviation s = 3.5 cm of ceramics vessels.

3.Let be normally distributed with a mean of 120 and standard deviation of 25.

(a)What is the Probability that X will be less than 160.

(b)Suppose a sample of 26 students were selected, what is the probability that X will be greater than 125.

4.The average body temperature of a healthy adult is = 98.6 (in degrees Fahrenheit). Jenny is concerned that her temperature is elevated, and so she took her temperature on 5 randomly chosen occasions. Her average temperature x was 99.3. Let be a random variable that represents Jenny's body temperature. You may assume that has a normal distribution with standard deviation = 0.73. Test, at the 1% level of significance, whether the sample result provides strong evidence that Jenny's body temperature of = 98.6 is elevated.

5.A recent poll of students in a certain college revealed the following information about their class scheduling:

STUDENTS

HISTORY

MATHEMATICS

BIOLOGY

TOTALS

FRESHMAN

20

45

5

70

JUNIOR

25

35

30

90

SENIOR

10

15

15

40

TOTALS

55

95

50

200

A student is selected at random, and the following variables are defined:

B: Biology, H: History, M: Mathematics,F: Freshman,J: Junior,S: Senior.

Find these probabilities:

(a) P(S/B)(b) P ( J and M)(c) P(S or M)(d) P(B /S )

6.A coin is tossed 10 times.

(a)Find the probability of getting exactly 8 heads.

(Use the Binomial Probability Distribution table)

(b)Find the probability of getting at least 8 heads.

(c)Find the expected value ' '

(d)Find the standard deviation

7.An animal shelter has a 65% adoption rate for puppies. Of all the puppies in the shelter, 71%live to be 7 years older. Of all the puppies who lived to be 7 years older, 85% were adopted,

(a)What is the probability that a random selected puppy in the shelter will get adopted and lived 7 years older?

(b)Professor Jackson is in charge of a program to prepare people for a high school equivalency exam. Records show that 4.23% of the students need work in Math, 3.14% need work in English, and 2.03% need work in both areas. Compute the probability that a student selected at random needs Math or English.

8.A pair of dice is rolled, find these probabilities:

(a) P(sum is greater or equal to 10)

(b) P(sum is less than 6 )

(c) P(sum is 8 or 12 )

(d) P(sum is at most 5)

9.Draw the Normal Distribution Curve over the indicated area given and find these probabilities:

(a) Area between Z = -2.23 and Z = 2.33,

(b) Area to the right of Z - 3.45

10. From a deck of cards, a card is drawn. Find these probabilities:

(a)P(drawing a jack and red).

(b)P(drawing a spade giving that the card is black).

(c)P(drawing a diamond or black).

(d)P(drawing a jack and red).

(e)P(drawing a 10/ black).

[Question 1: 24 points] Consider a two-component redundant system. Let the two components be denoted as C1 and C2. C1 is a CPR component (A = 0.002 {hr} and C2 is an ER component modeled using the Weibull failure law ()3 = 2.2, 6' = 2000 hrs). Answer the following: (i) Write down the failure and reliability functions corresponding to C1 [4 points] (ii) Write down the failure and reliabilityr functions corresponding to CE [4 points] (iii) Write down the failure and reliability functions of the redundant system [4 points] (iv) Compute the individual component reliabilities at time t = 1000 111' [3 points] (v) Compute the system reliability at time t = 1000 hr [4 points] Delco, Inc. is not happy with the reliability of its security system and has decided to improve it. In the original system Part 1 has a reliability of 75.29%, and part 2 has a reliability of 9896. The company will add a backup component to part 1 of its security system. The backup component will have a reliability of 0.774. [1 )What is the reliability of the original system? % (2) What is the reliability of the improved security system? 9% (3) If a second backup is added with the same reliability as the first backup, what is the reliability of the system? 8%[ Question 2: 50 points] A component has a hazard rate of 0.005 failures per day. If a redundant and identical component is added to the existing component, what is the reliability of the two- component redundant system over 200 days? [15 points] Compute the percentage increase in the reliability of two-component redundant system over a single component for use over 200 days. [10 points] Assume that a reliability of 0.95 is required over 200 days of operation. One way to achieve this is by adding more redundant components. How many more identical and redundant components need to be added to the two-component system to achieve the desired reliability? [15 points]4. The term reliability refers to the probability that a device does not fail. Suppose a mechanical system consists of three components that function independently. It is known that component 1 has a reliability of .98, component 2 has a reliability of 95, and component 3 has a reliability of 0.99. If the system can function if at least one component functions, what is the reliability of the system? 5. The term reliability refers to the probability that a device does not fail. Suppose a mechanical system consists of three components that function independently. It is known that component 1 has a reliability of .98, component 2 has a reliability of 95, and component 3 has a reliability of 0.99. Suppose now you develop an alternative system with two identical components, each of which has the same reliability p. This system functions only if all components function. What is the minimum p needed to make sure this machine has a reliability greater than or equal to 0.95? 6. There are 25 pens in a drawer in your desk. Among them, 20 write well and 5 are defective. You will randomly select 4 pens to give to your classmate. Your classmate randomly selects one of the four pens that she received from you. Calculate the probability the pen she chose writes well. (Ctrl)