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Handout 1

 

SHOW YOUR WORK in a CLEAR and ORGANIZED format if you expect to receive full or partial credit. IDENTIFY YOUR

 

ANSWERS. Use 4 decimal places in your computations.

 

1. A wireless phone service provider is concerned about the population proportion of customers who will renew their contract when

 

their term expires. They have determined that in the past, 40% of all customers have renewed their contract; that is, the

 

population proportion of customers who renew their contract is p=.40. In an attempt to get a larger share of the customers

 

renewing their contracts, the company decides to invest in upgrades for the customer service operation to provide easier and

 

faster billing assistance and information on service options. After 6 months of upgraded service, the company will select a

 

random sample of 400 customers and determine the sample proportion of customers who indicate they will renew their contract.

 

If the sample proportion is .45 or greater, the company will consider this to be an indication that the upgraded customer

 

service is having a positive effect on the renewal rate and that they should continue to provide the upgraded service. Otherwise,

 

the upgraded service will be considered to have no effect on the proportion of customers who renew their contract and will be

 

discontinued.

 

(a) (10 points) If the population proportion of customers who will renew their contracts after 6 months with the upgraded

 

customer service is still p=.40 (that is, there was no improvement in the renewal rate), what is the probability that the

 

company will conclude that, based on the sample proportion they have computed, the upgraded customer service

 

operation should be continued? (This would be the incorrect conclusion.) (b) (10 points) The customer service manager feels that it will take longer than 6 months for the majority of customers to

 

use the upgraded customer service operation, and as such she suggests that the company wait 12 months before taking a

 

sample of 400 customers and determining the sample proportion of customers who will renew their contracts. If at the

 

end of 12 months, the population proportion of customers who will renew their contracts is p=.48, what is the

 

probability that, based on the sample proportion that is computed, the company will conclude that the upgraded

 

customer service should be discontinued? (This would be the incorrect conclusion.) 2. Based on past performance, an appliance manufacturer has determined that the amount of time, X, needed to assemble one of

 

their wash machines is a normally distributed random variable with a population mean assembly time of =50 hours and a

 

standard deviation of =6 hours. The company?s operations manager feels that a lot of the variation in the assembly times is due

 

to the large number of workers involved in the assembly process, with each one working at a different rate and therefore causing

 

delays at certain points in the process. She decides to reduce the number of workers in the assembly process, and have each

 

individual perform more tasks than they currently are responsible for. To assess the new process, she will select a random

 

sample of 16 washers produced with fewer workers and compute the sample mean assembly time to see if it gives an indication

 

of an improvement (that is, reduction) in the population mean assembly time.

 

(a) (10 points) If the sample mean assembly time is less than 48 hours, the operations manager will conclude that the

 

reduced number of workers has resulted in a decrease in the population mean assembly time and will continue to use a

 

reduced number of workers . If the reduction in the number of workers actually has no effect and the population mean

 

assembly time remains at =50 hours, what is the probability that the sample mean assembly time will be less than 48

 

hours and fewer workers will continue to be used? (b) (10 points) The union representative for the assembly workers feels that the reduction in the number of workers has no

 

effect on the population mean assembly time and wants the number of workers to be unchanged. She would like to see

 

a change in the size of the sample used to compute the sample mean assembly time using a reduced number of workers.

 

She wants to change the sample size in such a way that if there is no effect on the population mean assembly time, the

 

probability computed in part (a) will be smaller. Should the union representative insist that a larger sample size be

 

used, or a smaller sample size? You must provide numerical computations to support your answer. 3. An automobile dealer that specializes in hybrid cars is trying to decide whether she wants to locate a dealership in a community.

 

She first wants to get an idea of the value of the population proportion of consumers in the community who will purchase a

 

hybrid for their next new car.

 

(a) (9 points) She would like to compute a 95% confidence interval for the population proportion described above, and she

 

would like the interval to have a margin of error equal to .03. What sample size should she select in order to meet these

 

conditions? (b) (9 points) She has determined that she is willing to spend only enough money to collect a sample of 900 consumers.

 

Among the 900 consumers in the sample, 324 indicated that their next new car purchase will be a hybrid. Determine a

 

95% confidence interval for the population proportion of consumers in the community that will purchase a hybrid for

 

their next new car. (c) (7 points) She realizes that the margin of error for the interval in part (b) is larger than she would like, but she cannot

 

afford to spend any more money to increase the sample size. Using the same data provided in part (b), what confidence

 

level should she use if she wishes to have an interval with a margin of error of .03? 4. An economist is investigating the income level of residents in a city that is showing signs of growth and development. The

 

income of the residents, X, is assumed to be a normally distributed random variable. A random sample of 100 residents from the

 

city is selected and the income of each resident is obtained. The sample mean income and sample standard deviation of the

 

incomes are as follows: X=$62,000 and s=$5,000.

 

(a) (9 points) The economist presents her results by saying that a confidence interval for the population mean income is

 

given by: $62,000 1182.50. What confidence level did the economist use to arrive at this interval? (b) (4 points) Identify 2 things that the economist could do to reduce the margin of error in part (a)? Be specific.

 

1. 2. (c) (8 points) Determine the sample size the economist would need to use if she desires that the margin of error in a 99%

 

confidence interval for the population mean income is to be no greater than $1000? You may use the results given in the

 

first paragraph as a preliminary sample to obtain an estimate of the population standard deviation (this estimate would

 

be the sample standard deviation). (d) (4 points) Give 2 reasons why the sample size required in part (c) is larger than the sample size of 100 that was

 

originally used. Be specific.

 

1. 2. 5. (i) (5 points) Two quality control analysts are each asked to compute a confidence interval for the unknown population

 

proportion of defectives bolts, p, being produced by a machine. The 1st analyst decides that she will use a random sample of 50

 

bolts to compute the sample proportion of defectives, and she will then compute a 99% confidence interval for the population

 

proportion of defectives. The 2nd analyst decides that he will use a random sample of 200 bolts to compute the sample

 

proportion of defectives, and he will then compute a 95% confidence interval for the population proportion of defectives. Which analyst will have a higher probability that the confidence interval they compute will contain the true value of the unknown

 

population proportion of defectives? CIRCLE YOUR ANSWER. NO COMPUTATIONS REQUIRED.

 

(a) The 1st analyst

 

(b) The 2nd analyst

 

(c) Both analysts will have the same probability of having their interval contain the population proportion of defectives.

 

(d) The probabilities that the intervals will contain the population proportion of defectives cannot be determined. (ii) (5 points) Two analysts are going to compute confidence intervals for the unknown population mean lifetime, , of batteries being

 

used in a particular brand of cellphones. The standard deviation of the batteries lifetimes is known to be =4 hours. The 1st analyst

 

selects a random sample of batteries and computes a 95% confidence interval that has a margin of error of .85 hours. The 2 nd analyst

 

selects a random sample of batteries and computes a 99% confidence interval that has a margin of error of .70 hours. Which analyst will

 

have a wider interval? CIRCLE YOUR ANSWER. NO COMPUTATIONS REQUIRED.

 

(a) The 1st analyst

 

(b) The 2nd analyst

 

(c) Both confidence intervals will be the same width.

 

(d) The width of the confidence intervals cannot be determined.

 







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