In week 4 of the 2022 NFL season, the Miami Dolphins played against the Cincinnati Bengals Just before halftime, the Dolphins scored a touchdown to bring the score to Cincinnati 14, Miami 121 After scoring, they had the option to try for one point or two points. Suppose that teams that go into halftime losing 13 to 14 win the game with a probability of 0.46, and teams that go into halftime tied win games with a probability of 0.5. The probability that the Dolphins successfully convert their one-point try (and thus end the half with 13 points) is 0.938; the probability they convert their two-point try (and thus end the half tied) is 0.517. Of course, there is also the chance that they fail whichever they attempt, get no points, and go into halftime with only 12 points. Teams that go into halftime losing 12 to 14 win with a probability of 0.417, The Dolphins' head coach calls you and asks: should they go for one point or two? Go for one point, since it gives the best probability of winning. Go for two points, since it gives the best probability of winning. Since the probability of winning is within .01, it doesn't matter which the coach attempts. Since these are random processes, we cannot say which should be attempted A researcher investigated the size of farms. They took a random sample of 2010 farms in the United States and measured the size (in acres). Below is a summary of the results. Farm Size Frequency (in acres) 040 394 41-80 461 81-120 391 121-160 334 161-200 159 201>240 1 13 2241 148 (a) For a randomly selected farm, what is the probability it is less than 8]. acres? (3 decimal places) (b) For a randomly selected farm, given it is more than 160 acres, what is the probability the size is between 201 and 240 acres? (3 decimal places) What information does standard error provide? The standard error provides information about the variation in the sample. The standard error provides information about the variation in the population. The standard error provides information about the variation in the sampling distribution of the statistic. The standard error provides information about the variation in the data. DETAILS A barbershop in Cleveland has ve chairs for waiting customers. The number of customers waiting is random with the following probability distribution. Prohab y Distribution x P00 0 0.42 1 0.164 2 0.136 3 0.1 15 4 0.091 5 0.074 (a) What is the probability that at least 1 person is waiting are waiting in line? (3 decimal places) (b) In a sample of 5, what is the expected number of people that will be waiting in line? (3 decimal places) DETAILS Warning times for tornadoes are approximately bellshaped with mean 13.7 minutes and standard deviation 2.5 minutes. (a) What is the probability a randomly selected warning time is more than 10 minutes? (Use 3 decimals.) (b) What is the warning time of the longest 6% of all warning times? (Use 3 decimals.) minutes (c) If a random sample of 11 warning times is taken, what is the probability that the average time will be at least 12.5 minutes? Vecna is a dark wizard. One of his evil spells is to cast a curse on people, but those people must be susceptible to his curse. Suppose that in a certain town in the MidWest, roughly 37% of the residents are susceptible to Vecna's curse. If we were to take a random sample of 8 people from that town, what is the probability that at least 40% of our sample would be susceptible to Vecna's curse? We cannot determine this probability 0.569 0.309 0.430 The amount of growth, in a 15day period, for sunflower plants is normally distributed with mean 3.19 centimeters and standard deviation 052 centimeters (a) what is the probability a randomly selected sunflower will grow at most 2.5 centimeters? (Use 3 decimals.) (b) If a random sample of 5 sunowers, what is the probability that the average growth will be more than 3.5 centimeters? The holiday season is usually a time of giving, but public confidence in charitable organization has decreased over the years. The following table represents a survey of 124 randomly chosen charitable donors. These donors indicated whether their confidence in charitable groups was high or low condence High Low No opinion Total Male 18 20 12 50 Female 23 37 14 74 Total 41 S7 26 124 What is the probability that a randomly chosen female donor had high confidence? (Enter either as a fraction, or proportion to 4 decimals.) A In a certain population of mussels (Mytllus edulis), 80% of the individuals are infected with an intestinal parasite. A marine biologist plans to randomly select 140 mussels. (a) What is the variable? The variable is the sample proportion of mussels infected with the parasite. The variable is whether or not a mussel is infected with the parasite. The variable is the population proportion of mussels infected with the parasite. The variable is the number of mussels in the sample that are infected with the parasite. (b) Find the mean and standard deviation of the sampling distribution of f) for a random sample of 140 mussels. Mean = Standard Deviation - (c) Describe the effect of increasing ri on the sampling distribution of f). (Select ALL that apply.) The mean decreases. The standard deviation increases. The mean increases. The standard deviation stays the same. The mean stays the same. The standard deviation decreases. (d) Why is it reasonable to conclude the sampling distribution of f) is approximately normal? The sample size is greater than 30. The population is approximately normal. The expected number of successes and failures are both at least 15. The expected number of successes and failures are both at least 30. 10. Below are three histograms. Each histogram (A,B and C) corresponds to one of the following: o The distribution a! the population. I The sampling distribution 0' the sample mean when 4, . The sampling distribulion ol the sample mean when n = 6' A B C Which histogram corresponds to the distribution of the sample mean when n = 16? ' A ' B C l l l l l l i i i l l l l l i l i l l l l l i l i l l l l l l l i 1 1 There's not enough information. l Alter scoring a touchdown, a team can opt either for a one-point attempt or a two-point attempt (which give 1 or 2 points, respectively, if successful). From 2000-2014, there were only 997 two-point attempts (which translates to one attempt every four games, approximately). Lately, teams have been going for two points more frequently. Suppose the probability of converting a one-point attempt is 0.9742. The probability of converting a two-point attempt is around 0.606. a) What is the average number of points teams receive after going for a one-point attempt? (3 decimal places) b) What is the average number of points teams receive after going for a two-point attempt? (3 decimal places) c) Comparing the results from a) and b), should teams be going for one point or two points after scoring a touchdown? Since the purpose of the game is to score more points than the other team, and 2 is greater than 1, teams should go for the twopoint attempt all the time. Since these are averages and not guaranteed results from every attempt, the decision should be contextual and not pre-determined each time, Since you are expected to score more points on a twmpoint attempt, teams should take the tw0vpoint attempt all the time. Since you are expected to score more points on a one-point attempt, teams should take the one point all the time. The number of repair calls that an appliance repair shop receives during an hour can be modeled by the following distribution. Proba ty D'st 'bu 'on Repair Calls Proha 0 0.11 1 0.29 2 0.38 3 0.22 (a) Verify that this is a legitimate probability distribution. (Select all that apply.) All probabilities are between 0 and 1' None of the probabilities equal 0. All probabilities are between -1 and +1. Probabilities sum to 1. (b) What is the expected number of repair calls in an hour