Find the probability that in five years' time : (a) both machines will be performing a usual function, (b) neither will be operating, (c) only machine B will be operating, and (d) at least one of the machines will be operating.
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- A car company conducts crash tests of its new airbag system to evaluate its safety. One hundred cars are run into a solid concrete barrier at a speed of 35 miles per hour, and the chest compression of a crash test dummy is measured each time. They hope that the average compression if the test were repeated over and over (which we could think of as the population) would be about 2 inches, corresponding to an injury that would be painful but not fatal. In the hundred tests, the average chest compression in the crash test dummies was 2.1 inches, with an SD of 0.2 inches.(a) State the null and alternative hypotheses in terms of the problem.(b) What kind of box model will we use for this problem? A 0-1 box, or a box with other values on the tickets?(c) State the null and alternative hypotheses in terms of the box model (d) Compute the z test statistic and the p-value, using the sample average as your statistic as you compute z (e) What do you conclude?3. Suppose Chelsea has the following utility function over sure wealth: v (y) = In y (a) Consider the lottery that yields the outcome $2 with probability and the outcome $8 with probability . What is Chelsea's certainty equivalent for this lottery? 2 3* (b) For the lottery in (b), how much would Chelsea pay to have her risk taken away? (c) What is Chelsea's probability premium at $25 for a lottery that could swing her wealth $20 up or down? What about $15 up or down?In a certain year, it was estimated that 1,000,000 of the 307,000,000 residents of the United States are HIV-positive. (HIV is the virus that is believed to cause AIDS.) The Repeatedly Reactive Immunoassay (RRIA) diagnostic test correctly diagnoses the presence of AIDS/HIV 99.7% of the time and correctly diagnoses its absence 98.5% of the time. (Round your answers to four decimal places.) (a) Find the probability that a person whose test results are positive actually has HIV.(b) Find the probability that a person whose test results are negative does not have HIV.(c) Find the probability that a person whose test results are positive does not have HIV.(d) Find the probability that a person whose test results are negative actually has HIV.(e) Which of the probabilities would you be interested in if you or someone close to you tested positive? the probability that a person whose test results are positive actually has HIVthe probability that a person whose test results are negative does not…
- Consider a service facility with four clerks. Clerk one works at rate 14/hour; Clerk two at rate 18/hour; Clerk three at rate 20/hour; and Clerk four at rate 16/hour. All service times are independent and exponential. If when you arrive there is already one person waiting, what is your expected waiting time?Suppose X-U (-54, 60) and F(t) is the cumulative distribution function. What is the probability that X is in the interval [-51, -21] or in the interval [-36, 57]? O (F(-21)-F(-51))+ (F(57) - F(-36)) O (F(-21)-F(-51)) x (F(57) - F(-36)) O F(-21)-F(-36) O F(57) - F(-51)6. A small company can employ 3 workers. Each worker independently stays on the job for an exponentially distributed time with a mean of one year and then quits. When a worker quits, it takes the company an exponentially distributed time with a mean of 1/10 of a year to hire a replacement for that worker (independently of how long it takes to replace other workers). If the company takes in $1000 per day in revenue when it has 3 workers, $800 per day when it has 2 workers, and $500 per day when it has 0 or 1 workers, what is the company's long-run average daily revenue?
- A golfer practices different parts of their game for hours. Specifically, the golfer practices:(1) putting, (2) their short game, and (3) their iron game. The amount of time spent on each partof their game is exponentially distributed. If the golfer is practicing their putting, they will alwaysswitch to their short game after an average of 30 minutes. When the golfer practices their short game,they do so for an average of 45 minutes, and then switches to putting with probability 0.6 or to theiriron game with probability 0.4. When they practice their iron game, they do so for an average of 60minutes. After practicing their iron game, the golfer will switch to putting or to their short game withequal probability.(a) Draw the CTMC rate diagram, clearly label your states and transition rates (qij ’s)(b) Find the transition rates out of the states (vi’s) and transition probabilities (Pij ’s).(c) Find the limiting probabilities (make sure to write out the balance equations).(d) In the long…Consider the following scenario: • Let P(C) = 0.3• Let P(D) = 0.8• Let P(C|D) = 0.3 A. P(C AND D) = [ Select ] ["0.30", "0.24", "0.26", "0.11"] B. Are C and D Mutually Exclusive? [ Select ] ["No, they are not Mutually Exclusive.", "Yes, they are Mutually Exclusive."] C. Are C and D independent events?[ Select ]["No, they are Dependent.", "Yes, they are Independent."] D. P(C OR D) = [ Select ] ["0.92", "0.86", "0.60", "1.1"] E. P(D|C) = [ Select ]["0.30", "0.95", "0.24", "0.80"]AJ’s sleep pattern had been fluctuating due to stress and diet, so he started to keep a diary of how many hours he slept each night. On the 4th night after he started the diary, AJ slept 9.4 hours, which was the most sleep he got any night. Gradually, he got less sleep each successive night until, on the 18th night after he started the diary, AJ slept 6.2 hours, which was the least sleep he got any night. After keeping his diary for 6 months, he realized that the number of hours he got each night followed a sinusoidal curve. Let t be the number of nights since he started his diary, and s be how many hours he slept that night. It may be helpful to draw a crude sketch of the sleep time function and label the known t − and s − values on it. POSITIVE or NEGATIVE SIN or COS Maximum s − value = Minimum s − value = ______________ ______________ Middle s − value = ______________ = D Amplitude = ______________ = A ⇒ B = ______________ Given the crude sketch of the height function, the…
- You randomly recruit 23 students and tell each of them to come to the lab in two different mornings: each morning they will be offered some breakfast and will take a test. Each student will receive the light breakfast one time and the nutritious breakfast the other time (some will receive the light breakfast first, the others will receive the nutritious breakfast first, and the second time they switch). The average difference in a student’s score after eating the nutritious breakfast and their score after eating a light breakfast is 4.5, and the standard deviation of this difference is 3.6. (a) Construct a 95% confidence interval for the difference in the average score of students after having a nutritious breakfast and the average score of students after having a light breakfast.Two real numbers, x and y, are randomly selected, both from the interval [0, 10]. What is the probability that, at the same time, their sum is less than 7 and the absolute value of their difference is greater than 2?a. Find the probability that the device A functions but the device B does not. b. Find the probability that the device A or the device B or both function. c. Find the probability that the circuit operates. d. It is known that the circuit is operating, find the probability that device A is functional.