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Homework answers / question archive / 1) In this example, we show a property of a simple queue, which we will discuss later, as an application of a Poisson process

1) In this example, we show a property of a simple queue, which we will discuss later, as an application of a Poisson process

Statistics

1) In this example, we show a property of a simple queue, which we will discuss later, as an application of a Poisson process. We consider arrival of tasks to a system to be a Poisson process with parameter λ . We denote the inter-arrival times by T1, T2,…… that is, Tn is the time elapsed between the (n −1)st and nth arrivals, or the time between consecutive arrivals of tasks. We want to find the cumulative density function (cdf) of the inter-arrival times.

2) A manufacture of a certain candy brand says that in each of the bag of candies, there is an equal number of candies of red, green, yellow, and white colors. A randomly selected bag consists of 15, 25, 13, and 16 red, green, yellow, and white color candies, respectively. Is there a truth of what candy manufacture says as far as the distribution of colors of candies is concerned? Use the α = 0.05 to test the above.

3) Consider a car seller who sells four brands (A, B, C, and D) of cars. He says that he does not give any priority to any brand of cars and considers all brands equally important. In a given day, the number of cars available with brands A, B, C, and D is 50, 40, 45, and 60, respectively. Using significance α = 0.05, test to find out if there is sufficient evidence to conclude that the car seller is right with his claim.

4) Does the mode of teaching influence the student’s academic success? An instructor wants to check whether his mode of teaching has an impact on student’s success in introductory statistics course. He randomly selects 35 students from his three sections of the introductory statistics classes that are taught face-to-face, hybrid, and online. He records the final grades for each student in each section as the measurement of student’s academic success. Does this data show that the mode of teaching impact the student’s performance, using the level of confidence of 0.01?

5) An engineer is going to compare the average densities of two types of wood. In order to test this, he randomly selected six samples of the first wood, and the average density and the standard deviation are 22.5 and 0.16, respectively. He randomly selected five samples of the second type of wood and found the average density and the standard deviation as 21.9 and 0.24, respectively. Assuming the populations are normally distributed, answer the following questions:(i)Construct a 95% CI for the difference in population means and interpret the interval.(ii)Use hypothesis test to confirm the above findings in part (i)

6) Consider Example 5.41 about the average amount of carbon dioxide. Test whether the average amount of carbon dioxide emitted by the first brand is higher than that of the second brand using the level of significance 0.05.

7) As a project in a statistics course at a university, a coin is randomly chosen. Two students are to test the fairness of the coin. One student believes the coin is “fair” and the other believes it is “biased toward heads”. To test this difference, use α = 0.05 and conduct a hypothesis test about the fairness of the coin. Suppose two students are to test the fairness of a coin in hand. One student believes the coin is “fair” and the other believes it is “biased toward heads”. Use α = 0.05, choose a random sample by tossing the coin 30 times, and test a hypothesis to indicate whether or not the first student’s claim is supported by the results.

8) A university is experiencing problems with its first mathematics course that all entering students must enroll in. The mathematics department decides to implement a plan consisting of mandatory attending a tutorial laboratory by students in a special tutorial program on a course section with 25 students who have randomly registered in. The department chair hypothesizes that with all tutorial activities, the class average on the final examination score will increase to be >70 out of 100. We want to test the chair’s claim with a significant level of 0.05.

9) Hypothesis testing is for the acquisition of information about the population parameters based on the sample statistic; that is, no generalization is involved. For instance, in assigning dose level to a cancer treatment, a doctor needs to know if the average dose level assigned was effective or noneffective in order to make a decision about the next dose level. So, if the doctor wants to change the dose level, he needs to make some assumption, test, and then decide.

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