Alice is trying to communicate with Bob, by sending a message (encoded in binary)

across a channel.

(a) Suppose for this part that she sends only one bit (a 0 or 1), with equal probabilities.

If she sends a 0, there is a 5% chance of an error occurring, resulting in Bob receiving a

1; if she sends a 1, there is a 10% chance of an error occurring, resulting in Bob receiving

a 0. Given that Bob receives a 1, what is the probability that Alice actually sent a 1?

(b) To reduce the chance of miscommunication, Alice and Bob decide to use a repetition

code. Again Alice wants to convey a 0 or a 1, but this time she repeats it two more times,

so that she sends 000 to convey 0 and 111 to convey 1. Bob will decode the message by

going with what the majority of the bits were. Assume that the error probabilities are

as in (a), with error events for di?erent bits independent of each other. Given that Bob

receives 110, what is the probability that Alice intended to convey a 1?

13. Company A has just developed a diagnostic test for a certain disease. The disease

a?icts 1% of the population. As defined in Example 2.3.9, the sensitivity of the test is

the probability of someone testing positive, given that they have the disease, and the

specificity of the test is the probability that of someone testing negative, given that they

don't have the disease. Assume that, as in Example 2.3.9, the sensitivity and specificity

are both 0.95.

Company B, which is a rival of Company A, o?ers a competing test for the disease.

Company B claims that their test is faster and less expensive to perform than Company

A's test, is less painful (Company A's test requires an incision), and yet has a higher

overall success rate, where overall success rate is defined as the probability that a random

person gets diagnosed correctly.

(a) It turns out that Company B's test can be described and performed very simply: no

matter who the patient is, diagnose that they do not have the disease. Check whether

Company B's claim about overall success rates is true.

(b) Explain why Company A's test may still be useful.

(c) Company A wants to develop a new test such that the overall success rate is higher

than that of Company B's test. If the sensitivity and specificity are equal, how high

does the sensitivity have to be to achieve their goal? If (amazingly) they can get the

sensitivity equal to 1, how high does the specificity have to be to achieve their goal? If

(amazingly) they can get the specificity equal to 1, how high does the sensitivity have

to be to achieve their goal?

[3+4+4+2=13 points] Problem 3. Consider a Markov chain with three states {51, 53, 53}, and the following transition matrix: If? 1,!2 1'} A: a 1:3 233 2,33 a 1,?3 1} Draw the state diagram, annotated with transition probabilities for Markov chain. 2} l[liven that the initial distribution on = {1,,}, what is the probability,' that the chain is in state .32 after 2 steps. 3} Find the stead}.r state distribution for this Markov chain. 4} Suppose that for a single visit to state 5, you receive f[s] dollars, where sl) = 1,1152} 2 2, and f[53} = 3. When the Markov chain reaches steady state, what's the value of E[ f [1(2)]? An article presents a study of the effect of the subbase thickness on the amount of surface deflection caused by aircraft landing on an airport runway. In six applications of a 159 kN load on a runway with a subbase thickness of 407 mm, the average surface deflection was 3.22 mm with a standard deviation of 0.48 mm. a. Can you conclude that the mean surface deflection is greater than 3 mm? Round the test statistic to four decimal places. b. Can you conclude that the mean surface deflection is greater than 2.5 mm? Round the test statistic to four decimal places.Calculate the p-value for each situation below. Choose right-, left-, or two-tailed. Show your calculation. 105. z-test statistic = - 1.98 Right-, Left-, or Two-tailed H, : p <.45 n="250" z-test statistic="-" right- left- or two-tailed h1: h t-test h. : u s>