Statistics and Probability Answers

Find the area under the normal curve in each of the following cases.

1. between z = 0 and z = 1.63
2. between z= 0 and z = 1.78
3. between z = -2.46 and z = 1.55
4. to the right of z = 2.35
5. to right of z = -1.31
6. to the left of z = 0.35
7. to the left of z - 1.85
8. to the right of z = -0.95

A basket contains 10 ripe and 4 unripe bananas. If these bananas are taken from the basket one after the other, determine the possible values of the randon variable R representing the number of ripe bananas.

*List the sample space

*Count the number of ripe in each outcome and assign this number to this outcome

*Contstruct the frequency distribution of the values of the random variable R

*Contstruct the probability distribution of the random variable R by getting the probability of occurrence of each value of the random variable

*Construct the probability histogram

 State whether the following statements are true or false. Give a short proof or a counter 

example in support of your answers: (10) 

 (a) Poisson distribution is a limiting case of binomial distribution for n p, 1

→ → ∞ and 

np ∞.

→

 (b) For two independent events A and B, if P(A) = 2.0 and P(B) = ,4.0 then 

(A∩ B) = .6.0

 (c) If H0

: P ≤ 6.0 and X ~ B(n, p)n -known and p unknown and 1 0 H :µ = µ where 

X ~ N

2 2

(µ,σ )σ unknown, then H0

 and H1

 are simple null hypothesis. 

 (d) Frequency density of a class for any distribution is the ration of total frequency to 

class width. 

 (e) If X and Y are independent r.v.s with M (t) X

 and M (t) Y

 as their m.gf’s 

respectively, then M (t) M (t)M t).( X +Y = X Y

2) A,B and C are three events. Express the following 

A machine produces ball bearings with diameter of 2.5 inches. It is known that the standard deviation of the ball bearings is 0.005 inches. A sample of 100 ball bearings is selected and average diameter is found to 0.498 inches. Determine the 99 per cent confidence interval.

A, B and C are three events. Express the following events in set notations .

(i) simultaneous occurrence of A,B and C.

(ii) occurrence of at least one of them.

(iii) Both A and B occur and C does not occur .

(iv) The event B but not A occur .

(v) Not more than one of the events A,B and C occur .

Costumeers arrive at a checkout counter in a department store according to a Poisson distribution

at an average of seven per hour. During a given hour, what are the probabilities that

(a) no more than three customers arrive?

(b) at least two customers arrive?

(c) exactly four customers arrive?

Ten motors are packaged for sale in a certain warehouse. The motors sell for $100 each, but a

double-your-money-back guarantee is in effect for any defectives the purchaser may receive. Find

the expected net gain for the seller if the probability of any one motor being defective is .08.

(Assume that the quality of any one motor is independent of that of the others.)

9. An oil prospector will drill a succession of holes in a given area to find a productive well. The

probability that he is successful on a given trial is .2.

(a) What is the probability that the third hole drilled is the first to yield a productive well

A zoom be 999missile protection system consists of n radar sets operating independently, each with a

probability of .9 of detecting a missile entering a zone that is covered by all of the units.

(a) If n = 5 and a missile enters the zone, what is the probability that exactly four sets detect the

missile? At least one set?

(b) How large must n be if we require that the probability of detecting a missile that enters the

zone be .999?

8. Ten motors are packaged for sale in a certain warehouse. The motors sell for $100 each, but a zoom be 999.

stions correctly?

6. A complex electronic system is built with a certain number of backup components in its

subsystems. One subsystem has four identical components, each with a probability of .2 of failing

in less than 1000 hours. The subsystem will operate if any two of the four components are

operating. Assume that the components operate independently. Find the probability that;

(a) Exactly two of the four components last longer than 1000 hours

(B) The system operates longer than 1000 hours

A multiple-choice examination has 15 questions, each with five possible answers, only one of which

is correct. Suppose that one of the students who takes the examination answers each of the

questions with an independent random guess. What is the probability that he answers at least ten

questions correctly?