PROBLEM 1 (16 pts) A survey was conducted to examine the relationship between pl

Question

PROBLEM 1 (16 pts) A survey was conducted to examine the relationship between place of residence and type of car owned. The results of the random sample of 500 car owners is given in the table below Location Type of Car Owned Own Fo Do not own Foreign Car Suburb 60 90 Rural 25 125 90 110 One of the above individuals is randomly selected for further questioning What is the probability of selecting an individual who owns a foreign car and does not live (5 pts) in a rural area? (Spts) What is the probability that the selected individual lives in a large city? a) b) c6pts)For the group of 500 car owners in the table above, are Location and Type of Car Owned independent? Show work. PROBLEM 2 (12 pts) A cereal manufacturer is interested in testing the effectiveness of two types of promotions on the mean weekly revenues of Brand K cereal. The manufacturer is interested in testing hether there is sufficient evidence to conclude that promotion 1 is more effective than promotion 2 irn increasing average weekly revenues at a 1% level of significance. Promotion 1 was used in 60 markets, resulting in average weekly revenues of $28,320 with a standard deviation of $3,240. Promotion 2 was conducted in 70 markets, resulting in average weekly revenues of $26,970 with a standard deviation of S2,830. Assume that the promotions can be considered as independent random samples from normal distributions. Conduct the appropriate hypothesis test. Include the null and alternative hypotheses and our decision. In addition, test whether there is sufficient evidence, at a 5% level of significance, to conclude that the mean sales in markets using promotion 1 exceeds the mean in markets exposed to promotion 2 by more than $1000. PROBLEM 3 (32 pts) A suburban bus company is considering the institution of a commuter bus route from a particular suburb into the central business district of the city. The company has determined that t route will be profitable if more than 25% of the commuters ride the route a)(12 pts) A random sample of 300 commuters from the relatively large suburb is selected. Suppose that 70 of the selected commuters indicate that they would use this bus route. Calculate a 99% confidence interval for the proportion of all commuters in this suburb who would utilize this potential bus route. b(13 pts) Suppose that the company conducts a sample of 300 customers and that 90 of the sampled individuals indicate that they would use the route. Is there sufficient evidence to indicate that the new route would be profitable, at a 5% level of significance? State the null and alternative hypotheses. c)(7 pts) Calculate the p value for your test of part c. Interpret. PROBLEM 4 (25 pts) Toby's Trucking Company (TTC) determined that on an annual basis the distance traveled per truck is distributed normally with a mean of 50,000 miles and a standard deviation of 12,000 miles. a(5 pts) Define the random variable and state its probability distribution. b10 pts) What is the probability that the distance traveled by a randomly selected TTC truck exceeds 57,000 miles? c)(10 pts) Forty percent of TTCs trucks travel at least how many miles in a year? PROBLEM 5 (16 pts) The customer service department of a local gas utility would like to estimate the average length of time between the entry of a service request and the connection of service. Suppose the length of time between request of service and actual connection is distributed normally. A random sample of fifteen requests was selected resulting in an average number of days between request and connection of 4.3 days with a standard deviation of 1.76 da a)(11 pts) Calculate a 98% confidence interval for the mean number of days between request for service and actual connection, based on this sample b)(3 pts) Interpret your interval of part a 2 pts) How would your answer to part a change if the distribution of time between request for service and connection was not distributed normally? Briefly explain. PROBLEM 6 (15 pts) Based on past records, the main line printer in a university computer center operates properly on 95% of all days. A random sample of fifteen days indicates that the printer was not operating properly on three of the observed days

Explanation / Answer

Solution(1)

(a) Let A be a event selecting an individual who owns a foreign can and does not live in rural area.

Then,

P(A)= 90/500+60/500=0.18+0.12=0.30

(b) Let B be a event of selecting individual lives in a large city

Then,

P(B)=90/500+110/500=200/500=0.40

(c) Using Chi-square test

m<-matrix(c(90,60,25,110,90,125),2,3,byrow=T)
> m
     [,1] [,2] [,3]
[1,]   90   60   25
[2,] 110   90 125
> chisq.test(m)

        Pearson's Chi-squared test

data: m
X-squared = 32.601, df = 2, p-value = 8.334e-08

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