onent per day. Each repairman successfully fixes the component with probability 70% regardless of whether he has worked on it previous days. Model the system as a Markov chain Write down equations for determining lon

onent per day. Each repairman successfully fixes the component with probability 70% regardless of whether he has worked on it previous days. Model the system as a Markov chain Write down equations for determining lon

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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An aerosolized vaccine for measles was developed in Mexico and has been used on 44 million children since 1980. Aerosolized vaccines have the advantages of being able to be administered by people without clinical training and do not cause injection‐associated infections. The percentage of children developing an immune response to measles after receiving the subcutaneous injection of the vaccine is 95%,95%, and for those receiving the aerosolized vaccine, it is 85%85% . There are 2020 children to be vaccinated for measles using the aerosolized vaccine in a small rural village in India. We are going to count the number who developed an immune response to measles after the vaccination. What would be the probability that at least one child does not develop an immune response to measles if all children were vaccinated using the subcutaneous injection of the vaccine? (Enter your answer as a proportion rounded to four decimal places.)
The data set Pain contains hypothetical data for a clinical trial of a drug therapy to control pain. The clinical trial investigates whether adverse responses increase with larger drug doses. Subjects receive either a placebo or one of four drug doses (1, 2, 3, or 4 units). An adverse response is recorded as Adverse = 'Yes'; otherwise, it is recorded as Adverse = 'No'. The number of subjects for each drug dose and response combination is contained in the variable Count. (a) Construct a contingency table that corresponds to the data set created above. What type of variable is Dose? (B) Compute the sample proportions of adverse responses at each dose level. Do you observe any trend in the proportion of adverse responses with respect to dose level? (C) Conduct a Cochran-Armitage trend test at the 5% significance level to address the interests of the trial.
For a parallel structure of identical components, the system can succeed if at least one of the components succeeds. Assume that components fail independently of each other and that each component has a 0.25 probability of failure. Complete parts (a) through (c) below. (a) Would it be unusual to observe one component fail? Two components? It be unusual to observe one component fail, since the probability that one component fails, is than 0.05. It be unusual to observe two components fail, since the probability that two components fail, is than 0.05. (Type integers or decimals. Do not round.)
At Suburban Community College, 40% of all business majors switched to another major the next semester, while the remaining 60% continued as business majors. Of all non-business majors, 20% switched to a business major the following semester, while the rest did not. Set up these data as a Markov transition matrix. (Let 1 = business majors, and 2 = non-business majors.) calculate the probability that a business major will no longer be a business major in two semesters' time.
The owner of a chain of mini-markets wants to compare the sales performance of two of her stores, Store 1 and Store 2. Sales can vary considerably depending on the day of the week and the season of the year, so she decides to eliminate such effects by making sure to record each store's sales on the same 8 days, chosen at random. She records the sales (in dollars) for each store on these days, as shown in the table below. Day 1 2 3 4 5 6 7 8 Store 1 889 699 534 398 432 213 252 929 Store 2 479 525 252 364 160 32 234 632 Difference(Store 1 - Store 2) 410 174 282 34 272 181 18 297 Send data to calculator Based on these data, can the owner conclude, at the 0.10 level of significance, that the mean daily sales of the two stores differ? Answer this question by performing a hypothesis test regarding μd (which is μ…
The owner of a chain of mini-markets wants to compare the sales performance of two of her stores, Store 1 and Store 2. Sales can vary considerably depending on the day of the week and the season of the year, so she decides to eliminate such effects by making sure to record each store's sales on the same 12 days, chosen at random. She records the sales (in dollars) for each store on these days, as shown in the table below. 5 6 7 Day Store 1 Store 2 1 Difference (Store 1 - Store 2) Send data to calculator ✓ 2 3 786 878 827 645 708 500 699 140 Yes No 4 646 802 712 579 527 367 76 115 66 181 133 8 (a) State the null hypothesis H and the alternative hypothesis H₁. H:D H₁:0 (b) Determine the type of test statistic to use. Type of test statistic: (Choose one) ▼ 970 9 10 11 12 618 679 762 635 271 572 324 215 698 530 Based on these data, can the owner conclude, at the 0.05 level of significance, that the mean daily sales of the two stores differ? Answer this question by performing a hypothesis…
(Devore: Section 3.2 #15) Many manufacturers have quality control programs that include inspec- tion of incoming materials for defects. Suppose a computer manufacturer receives circuit boards in batches of five. Two boards are selected from each batch of inspection. We can represent possible outcomes of the selection process by pairs. For example, the pair (1,2) represents the selection of boards 1 and 2 for inspection. (a) List the ten different possible outcomes. (b) Suppose that boards 1 and 2 are the only defective boards in a batch. Two boards are to be chosen at random. Define X to be the number of defective boards observed among those inspected. Find the probability distribution (pmf) of X. (c) Let F denote the cdf of X. Determine F(0), F(1) and F(2); then obtain F(x) for all other x.
Anne and Barry take turns rolling a pair of dice, with Anne going first. Anne’s goal is to obtain a sum of 3, while Barry’s goal is to obtain a sum of 4. The game ends when either player reaches his goal,and the one reaching the goal is the winner. Define a Markov Chain to model the problem.
Write the payoff matrix for the given game, use Rachel as the row player. Two players, Rachel and Charlie, each have two cards. Rachel has one black card with the number 3 and one red card with the number 5. Charlie has a black card with a 6 written on it and a red card with a 1. They each select one of their cards and simultaneously show the cards. If the cards are the same color, Rachel gets, in dollars, the sum of the two numbers shown. If the cards are different colors, Charlie gets, in dollars, the difference of the two numbers shown.
Sarah believes that completely cutting caffeine out of a person's diet will allow him or her more restful sleep at night. In fact, she believes that, on average, adults will have more than two additional nights of restful sleep in a four-week period after removing caffeine from their diets. She randomly selects 8 adults to help her test this theory. Each person is asked to consume two caffeinated beverages per day for 28 days, and then cut back to no caffeinated beverages for the following 28 days. During each period, the participants record the numbers of nights of restful sleep that they had. The following table gives the results of the study. Test Sarah's claim at the 0.05 level of significance assuming that the population distribution of the paired differences is approximately normal. Let the period before removing caffeine be Population 1 and let the period after removing caffeine be Population 2. Numbers of Nights of Restful Sleep in a Four-Week Period With Caffeine 24 22 24 20…
Either a mixed column or mixed row strategy is given. Use the given payoff matrix and find the optimal pure strategy (or strategies) the other player should use. Express the answer as a row or column matrix. Also determine the resulting expected payoff.
A medical study is conducted to determine which migraine treatment, A or B, provides faster relief. The study uses 10 volunteers who claim to suffer from migraines. Half of the volunteers are randomly assigned to use treatment A when they experience their first migraine. The other half are assigned to use treatment B. Then, after no treatment for one month, the treatments are reversed. The volunteers each record the amount of time it takes, in minutes, to experience relief from their migraine under each treatment. The data are displayed in the table. A 3-column table with 10 rows. Column 1 is labeled volunteer with entries 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Column 2 is labeled Treatment A with entries 10, 13, 13, 9, 13, 12, 14, 10, 8, 7. Column 3 is labeled Treatment B with entries 19, 18, 19, 15, 20, 16, 16, 16, 13, 17. A 99% confidence interval for the mean difference (A – B) in the time it takes to experience relief is –8.37 minutes to 0.732 minutes. Based on the confidence interval,…