You have been contracted by the City of Sydney Council to inspect an issue that has been reported by residents living in a western suburb. The residents claim that one of the roads in their suburb has lately had issues with traffic speeding. The City of Sydney Council wants you to investigate if traffic speeding is an issue and, if this is the case, they will invest in building asphalt speed bumps to enforce reduced traffic speeds on that particular road. In this problem you may assume that the speed of a vehicle follows a normal distribution (c.) (3 marks) Before collecting data, you want to ensure that you will take a sample size large enough to conduct a reliable hypothesis test. Given that there is a kindergarten nearby, you would like your test to have a 0.99 probability to conclude that traffic speeding is an issue if the mean speed is 30km per hour. Moreover, if traffic speeding is not an issue, you want a probability of 0.05 to falsely conclude that speeding is an issue. Based on previous observations of speeds on similar roads in Sydney, you may assume a known standard deviation of 5 km per hour for the speeds. How many observations should be included in your sample to meet these requirements? Carefully state the Type I and II error probabilities implied from the question (a.) (3 marks) The reason the residents complained is because they expect that the mean speed is at most 25km per hour, especially considering that there is a kindergarten nearby. The council thinks that this is a reasonable figure. Write down the null and alternative hypothesis that you would test to carry out the task the City of Sydney Council has assigned you. (b.) (3 marks) Given your hypothesis in (a.), can you explain in words the two different errors that may result from your investigation? Which of these errors do you think is worse to commit in this case? Explain. (d.) (4 marks) You obtain the following n = 16 measurements of car speeds (km/h): 35.13, 25.37, 27.15, 18.39, 31.56, 26.63, 19.73, 23.29, 35.73, 29.73, 37.55, 29.67, 25.63, 19.04, 31.38, 25.76. For these data, !-fi = 441.74 and El-ti - 7)" = 512.5844. Compute the sample mean and sample variance from the information given. Test your hypothesis in (a.) using the significance level implied from your answer in (c.). What is your advice to the City of Sydney Council? Carefully state your assumptions, however, note that you may not assume that the variance is known, but you may still assume that the speeds of vehicles follow a normal distribution (e.) (4 marks) During a meeting to present your results to the council committee, you are asked to provide an interval for the true mean of the traffic speed. Provide such an interval and interpret it in words for the committee members. Keep in mind that they do not know statistical terminology, so you have to express yourself using non-technical terms. Can you use this interval to test the hypothesis in (a.)? Explain why/ why not (f.) (3 marks) One of the committee members points out that the real danger for the residents is if a single vehicle speeds, and not that the average speed is large. Provide the committee members with an interval that can be used to study the speed of a future single vehicle. You have been contracted by the City of Sydney Council to inspect an issue that has been reported by residents living in a western suburb. The residents claim that one of the roads in their suburb has lately had issues with traffic speeding. The City of Sydney Council wants you to investigate if traffic speeding is an issue and, if this is the case, they will invest in building asphalt speed bumps to enforce reduced traffic speeds on that particular road. In this problem you may assume that the speed of a vehicle follows a normal distribution (c.) (3 marks) Before collecting data, you want to ensure that you will take a sample size large enough to conduct a reliable hypothesis test. Given that there is a kindergarten nearby, you would like your test to have a 0.99 probability to conclude that traffic speeding is an issue if the mean speed is 30km per hour. Moreover, if traffic speeding is not an issue, you want a probability of 0.05 to falsely conclude that speeding is an issue. Based on previous observations of speeds on similar roads in Sydney, you may assume a known standard deviation of 5 km per hour for the speeds. How many observations should be included in your sample to meet these requirements? Carefully state the Type I and II error probabilities implied from the question (a.) (3 marks) The reason the residents complained is because they expect that the mean speed is at most 25km per hour, especially considering that there is a kindergarten nearby. The council thinks that this is a reasonable figure. Write down the null and alternative hypothesis that you would test to carry out the task the City of Sydney Council has assigned you. (b.) (3 marks) Given your hypothesis in (a.), can you explain in words the two different errors that may result from your investigation? Which of these errors do you think is worse to commit in this case? Explain. (d.) (4 marks) You obtain the following n = 16 measurements of car speeds (km/h): 35.13, 25.37, 27.15, 18.39, 31.56, 26.63, 19.73, 23.29, 35.73, 29.73, 37.55, 29.67, 25.63, 19.04, 31.38, 25.76. For these data, !-fi = 441.74 and El-ti - 7)" = 512.5844. Compute the sample mean and sample variance from the information given. Test your hypothesis in (a.) using the significance level implied from your answer in (c.). What is your advice to the City of Sydney Council? Carefully state your assumptions, however, note that you may not assume that the variance is known, but you may still assume that the speeds of vehicles follow a normal distribution (e.) (4 marks) During a meeting to present your results to the council committee, you are asked to provide an interval for the true mean of the traffic speed. Provide such an interval and interpret it in words for the committee members. Keep in mind that they do not know statistical terminology, so you have to express yourself using non-technical terms. Can you use this interval to test the hypothesis in (a.)? Explain why/ why not (f.) (3 marks) One of the committee members points out that the real danger for the residents is if a single vehicle speeds, and not that the average speed is large. Provide the committee members with an interval that can be used to study the speed of a future single vehicle