PROBLEM I (2s PTS) When a new mac DS265 Sample Esam IL (125 pts defective. A wor

Question

PROBLEM I (2s PTS) When a new mac DS265 Sample Esam IL (125 pts defective. A worker is instructed to randomly select twenty items ( hine is functioning properly, only 1% of the items produced are by this machine, and determine the quality of each item (good or defective ect twenty items (out of a large batch of output) produced aX9 pts) Assuming the machine is operating properly, what is i g the machine is operating properly, what is the probability that the sample contains at least one defect? K7 pts) What is the expected number and variance of the number of defects, when the machine is operating properly? c)(9 pts) Suppose the machine is malfunctioning, resulting in a defective rate of 20% what is the probability that the sample of 20 items contains at least 14 good items PRO BLEM 2 (55 PTS) A large hotel is conducting a study concerning the time required to complete a room service request for guests in the hotel. Assume that the required time per request is distributed normally with a mean of 25 minutes and a standard deviation of 3 minutes a)(5 pts) Define the random variable and state its distribution. bXIl pts) What is the probability that a randomly selected room service request took at least 18 minutes to complete? Include a picture of the calculated probability eX 12 pts) 85% of all requests take at least how many manutes? Include a picture of your solution. dX4 pts) Suppose that the hotel is concerned with the efficiency of its staff. They are interested in conducting a test on the mean time to complete a room service request in order to determine whether it has increased from its historical mean of 25 minutes. State the null and alternative hypothesis in terms of the tested parameter, assuming that the historical mean time to complete a request is consistent with the null hypothesis. eX4 pts) Verbally describe Type II error in the context of this problem and part d 0(12 pts) A random sample of 60 room service requests is conducted, from a very large number of recent requests. The sample resulted in an average completion time of 26.1 minutes. Assume there is no change in the standard deviation of completion times. What is the sampling distribution of the sample mean? Find the probability of obtaining a sample average at least this large, if there has been no change in the mean completion time for room service requests? X4 pts) Given your answer to part f, does it seem likely that the mean time to complete a room service request has increased? Explain. b(3 pts) How would your answer to part f change if the distribution of the time required to completea room service request is no longer known to be normally distributed. PROBLEM 3 (39 pts) The Allied Express overnight delivery company claims an on-time delivery rate of %. A consumer group is interested in testing this claim to determine if the company is overstating the proportion of on-time deliveries. X3 pts) Assume that the company's claim is defined as the null hypothesis. State the null and alternative hypotheses in terms of the population parameter, the proportion of late deliveries. b)(4 pts) Verbally state the type I and II errors in the context of this problem. eX4 pts) A sample of 25 deliveries is conducted. The consumer group decides to reject the firm's claim if at least three late deliveries are found in the sample. State the decision rule using the appropriate random variable. Be sure to define this random variable and state its distribution. dX9 pts) Using the information of parts a and c, calculate the probability of a type l error. deliveries made by Allied Express). Twelve of the sampled deliveries were late. What is the distribution of the proportion of late deliveries in this sample, assuming that Allied's claim is true? 5 pts) Suppose the consumer group conducts a sample of 150 deliveries (out of a very large number of sampling 10 pts) Calculate the probability of obtaining a sample proportion of late deliveries at least as large as the one reported in part e, assuming that Allied's claim is true. 4 pts) Based on your anwer to part f, what conclusion can be drawn regarding the validity of Allied Express's on time delivery claim of 95%? Does the result provide support for the null or alternative yp PROBLEM 4 (6 pts) Suppose the amount of time it takes to assemble a plastic module ranges from 27 to 39 seconds and that assembly time is uniformly distributed. What is the probability that a randomly selected plastic module will take between 30 and 35 seconds to assemble? Draw a picture and shade your probability

Explanation / Answer

1) a) This is an instance of binomial probability, with probability of success, p=0.01, the number of trials, n=20 and number of success in n trials, r=1.

P(X>=1)=1-P(X<1)=1-P(X=0)=1-0.8179=0.1821

[P(X=r)=nCr(p)^r(q)^n-r

P(X=0)=20C0(0.01)^0(0.99)^20=0.8179]

b) E(X)=np=20*0.01=0.2

Var(X)=npq=20*0.01*0.99=0.198

c) The defect rate is p=0.20, the rate of producing correct output is 0.80.

P(X>=14)=1-P(X<14)=1-[P(X=0)+....+P(X=13)]=1-0.0867=0.9133

P(X=0)=0.0000,P(X=1)=0.0000, continue till P(X=13)=0.0545, and add all the probabilities to find the cumulative one, using the binomial formula as stated above.

2. a) The time required to complete a room service and the distribution is normal.

b) Xi=18, Xbar=25, s=3

z=(Xi-Xbar)/s=(18-25)/3=-2.33

P(X>=180=P(Z>=-2.33)=0.5+0.4901=0.9901 [add 0.5 to area between z=-2.33 and mean]

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