A process manufacturing resistors whose specifications are 1 kΩ ± 0.2 kΩ. The resistance of these parts is known to follow a normal distribution with a mean of 1.03 kΩ and a standard deviation of 0.08 kΩ.

a. What is the probability that a randomly selected resistance does not meet the specifications? (5pts)

b. If 10 resistors are selected at random, what is the probability that there will be at least 1 out of specification? (3pts)

c. If 16 resistors are selected at random, what is the probability that the average of the 16 resistors is less than 1.05 kΩ? (7pts)

d. If 99.73% is desired to be within specifications, by how much will the standard deviation need to be reduced? (5 pts)

Question 7

Suppose Alana has personal wealth of $10,000 and there is a probability of 0.2 of losing her car worth $6,400 in an accident.   Her utility (of wealth) function is given by  u(w) =  w^0.5, where  w  is wealth.     

(a) What is Alana's expected wealth, expected utility, and utility of expected wealth? If she can insure "fully", and if this insurance is fair, how much would it cost her?

(b) What is the maximum amount Alana would be prepared to pay for full insurance? What is the certainty equivalent and the risk premium associated with the uncertain situation she is in if she does not have any insurance? What difference would it make if her utility of wealth function were instead  u(w) = 5w?

P.0.2 Show that (a) the diagonal entries of a Hermitian matrix are real; (b) the diagonal entries of a skew-Hermitian matrix purely imaginary; c) the diagonal entries of a skew-symmetric matrix are zero.

P.0.5 Let A ∈ Mn be invertible. Use mathematical induction to prove that (A-1)k = (Ak)-1 for all integers k.

P.0.25 Let A ∈ Mn be idempotent. Show that A is invertible if and only if A = I

P.0.26 Let A,B ∈ Mn be idempotent. Show that tr((A-B)3) = tr(A-B).

A process manufacturing resistors whose specifications are 1 kΩ ± 0.2 kΩ. The resistance of these parts is known to follow a normal distribution with a mean of 1.03 kΩ and a standard deviation of 0.08 kΩ. to.

What is the probability that a randomly selected resistance does not meet the specifications?

b. If 10 resistors are selected at random, what is the probability that there will be at least 1 out of specification?

c. If 16 resistors are selected at random, what is the probability that the average of the 16 resistors is less than 1.05 kΩ?

d. If 99.73% is desired to be within specifications, by how much will the standard deviation need to be reduced?

A certain disease has an incidence rate of 0.2%. If the false-negative rate is 6% and the false positive rate is 5%, compute the probability that a person who tests positive actually has the disease.

Somebody speculates that a person will test COVID 19 positive with probability 0.2. Then for finding the probability for exactly 100 patients testing positive in a sample of 400 patients, how do you do continuity correction? In the above question, what is the required probability using normal approximation to binomial? What is the probability that at least 70 patients test COVID 19 positive?

In the inspection of tin plate produced by a continuous electrolytic process, 0.2 imperfection is spotted per minute on average. Find the probabilities of spotting.

(a) one imperfection in 3 minutes.

(b) at least two imperfections in 5 minutes.

What is the pH of the solution made by mixing 0.2 mol NaH2PO4 and 0.5 mol NaOH with water to make 1.00 L of solution? The pKa values for H3PO4 are 2.12 7.20 and 12.

What diameter of cast iron pipe would be required to ensure that a discharge of 0.2 m3/s would not cause a head loss in excess of 0.01 m/100 m of pipe length, assume water temperature is 20 oC