tut problems week 8 (thursday%2cfriday sessions) (1)

7/25/2019 Tut Problems Week 8 (Thursday%2cFriday Sessions) (1)

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Text Book: Complete Business Statistics (Aczel, Sounderpandian, Saravanan, Joshi)-7e

Hypothesis testing:

(109) NS Software Systems and Services (popularly known as N4S) have recently recruited Komala, a

quality expert from Institute of International Management, Bilekahalli (IIMB). She decided to train the

employees of N4S in quality awareness as well as the techniques to be used through a new innovative

method, developed specially by her. Dr. Raghav who is the VP, Corporate Training agreed to experiment

with the new method of training, provided the effectiveness of the training can be measured. Komala

and Dr. Raghav agreed to use a special instrument (as some of the OB experts prefer to call it), which

has to be administered to the trainees before and after the training. Based on the scores of the

instrument, before and after the training, a Quality Capability Motivation Improvement Index (QCM)

can be computed for each of the trainees. They have agreed to use this index as a measure for the

effectiveness of the training and a positive value of the index indicates effectiveness of the training.

I. It was proposed that a sample of 20 employees would be selected for training under the new method.

For obvious reasons, they decided to do a one sided hypothesis test. The sample standard deviation of

the QCM obtained from these 20 observations was 12.5 .

(109 a) Dr. Raghav informed Komala that he is willing to go ahead with the new method if the sample

average of QCM is greater than equal to 4.8327. Based on this decision, calculate the Type I error that

Dr. Raghav is willing to tolerate.

(109 b) Based on the above, formulate the null and alternate hypothesis.

(109 c) At this stage, Dr. Rahav decided to change thevalue to 0.025 and suitably modified the

decision rule. The average QCM of the 20 trainees turned out to be 5.3. Should Dr. Raghav go ahead

with the new method of training?

(109 d) Assume that the true average QCM is 0.4673. What is the Type II error that Dr. Raghav will be

committing based on the decision in question (109 c) above?

II.Dr. Raghav decided to compare the new method with the old method of training that the company

had been using. He selected a random sample of 60 employees for training based on the old method

and obtained the QCM scores for these 60 employees. The average for this sample was 0.1 with a

sample standard deviation of 9.3. Dr. Raghav insisted that 0.025.

(109 e)Test whether the old method is effective or not?

(109 f) Test whether there is a significant difference between the two methods.

(109 g) Consider the results from Question I part (109 c) and Question II parts (109 e) and (109 f). Are

these results consistent? Justify your answer.


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(110)In a recent readership survey of business magazines, 200 business executives from Delhi, Mumbai,

Calcutta, Chennai, and Bangalore were asked about their most favorite business magazines. The only

contenders were Business for Tomorrow (BFT), Business Asia-Pacific (BAP) and Business Universe (BU).

The survey reporter who does not believe is straight-forward reporting, narrated the survey findings as:

1)

40% of the interviewed executives in Delhi prefer BFT; the same was the percentage ofexecutives interviewed in Calcutta who preferred BAP as well the percentage of executives

interviewed in Chennai who preferred BU. The same was the overall percentage of

executives who preferred BAP.

2)

The same number of executives was interviewed from Calcutta, Chennai and Bangalore and

in each of these three cities one-third of the executives interviewed preferred BFT.

3)

10 executives from each of the three cities Delhi, Mumbai and Bangalore preferred BU.

4)

In Mumbai, the ratio of the number of executives who chose BAP to those who chose BFT

was 3:2.

5)

In Bangalore, the three magazines were in a deadlock for the most preferred spot.

6)

The proportion of executives who were from Mumbai was the same as the proportion of

executives who preferred BU.

(110 a) Let md

be the proportion of all executives in Delhi (Mumbai) whose first preference for

Business magazines is BFT. Find a 95% confidence interval formd

.

(110 b) Test the hypothesismd

at 5% level of significance against the alternative thatmd

.

(111)Upon taking the oath as the Head of the State of Shangla, Shuddhubabu sits down to identify the

main problems of the State. The work culture of the people of Shangla has been on the downswing for

some time and now it has become a very serious issue. Shuddhubabus own party may have contributed

substantially to this; it made employees more aware of their rights for which now they are prepared to

fight, but adequate care has not been taken to teach them the associated responsibility. In any case,Shuddhubabu decides to address this concern and take actions against government employees who

routinely report to their work late. Of course, the first problem was to identify if this indeed is a serious

problem, and if so whether severity is more or less uniform among different types of employees. To

start with, he decides to limit his survey to secretaries and officers in government offices; but decided to

get comparable survey done among similar employees in non-government offices who are supposed to

bear roughly similar responsibilities. The survey is done on a fixed date (so that the effect of other

factors will be removed to the extent possible) from a randomly chosen selection on employees. To start

with, it is noted if the person reported to his/her work in time; if (s) he did not, the delay in reporting to

work is recorded. The summary of the survey is included below:

Govt/Non-Govt

EmployeeType

No. ofemployees

checked

No. ofemployees

reported in

time

Delay for those who came lateMean Standard

deviation

Government

Offices

Secretaries 155 34 46 14

Officers 150 69 43 15

Non-govt

Offices

Secretaries 200 139 27 12

Officers 40 15 20 16


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Note: For questions (a), (b), and (c) you should consider only those who did not report in time.

(111 a)Based on the survey, calculate two-sided 95% confidence intervals for the population variance of

delay in reporting to work of the Officers in the non-government offices.

(111 b)Based on the survey, calculate two-sided 95% confidence intervals for the population variance of

delay in reporting to work of Secretaries in the government offices.

(111 c) From the above survey, can Shuddhubabu conclude that there is significantdifference between

the average delay in reporting to work between the secretaries and officers in the non-government

offices?

Shuddhubabu has decided that he would take disciplinary action against any employee if the sample

attendance records provide significant evidence that the employee comes late on more than 20%of the

days. Shuddhubabu feels that he cannot afford to take to chance of more than 1%to punish someone

who is late on at most 20%of the days. At the same time, Shuddhubabu is adamant to send a strong

message that indeed he means business. Thus he wants the scheme to take disciplinary action at least99%of the times correctly against someone who is late on more days than (s) he is on time.

(111 d)Formulate Shuddhubabus problem as a Statistical testing of hypothesis problem, by clearly

stating the null and the alternative hypothesis. What are the roles/terminology for the boldfaced

numbers given above, i.e. 20%, 1% and 99%, in your formulation?

(112) Ruma, Asst. Marketing Manager of NS Chemicals, who is responsible for marketing of Softwhite

detergent has been receiving complaints about the half-kilo pack. It was felt that the filling machines

are not filling the detergent powder very accurately. Ruma asked Komal (from the Quality Control) to

test a few packets. Komal selected 12 packets randomly and took the exact weight of the powder in the

packet. She calculated a 95% two-sided confidence interval for the variance ( 2 ). Since the lower limit

of this interval was only 110 Gms2 she told Ruma that this is quite acceptable. When Ruma asked for the

sample average, she did not remember, but she did remember that the upper and lower limits of the

confidence interval for . She told Ruma that these limits are 499.452 and 516.548.

(112 a) What is the sample variance of the detergent powder?

(112 b) Ruma feels that all the filling machines will have to be reset if the variance ( 2 ) is more than

300 Gms2. Carry out the appropriate hypothesis test to decide whether the filling machines need to be

reset ( = 0.10).

(112 c) Komal felt that it is not enough to control the variance, but she needs to look at the average

quantity of the detergent filled. She decided to test a hypothesis that = 500 Gms. She takes another

sample of 12 and gets a variance of 218 Gms2. What is the conclusion if = 0.05 i.e. for what range of

values of the sample mean should she accept the Null hypothesis?

(112 d) If the real value of is actually 510 gms, what is the probability of committing Type II error?


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(112 e)Ruma felt that it is appropriate that they should carry out a one sided hypothesis test and that

they should make sure that the customers should not get less than 500 Gms. She wanted the decision

rule to be such that the probability of the customers getting less than 500 Gms should be retained at

0.05. For what range of values of the sample mean should she accept the Null hypothesis?

(112 f) The Consumer Rights Forum (CRF) did a sample test of 25 packets and got a variance of 400

Gms2. Is there a significant difference at the 0.05 level between this and the variance in part (112 c)?

(113) The Indian army, after the recent clashes decided to evaluate a new gun called Bi-fours (it is so

called because its range is claimed to be at least 16 km). They have hired the services of Kanaka, who

recently graduated from IIMB. Kanaka selected a simple random sample of size 25 and calculated the

sample average. Based on the sample average and the as given by the manufacturers, she calculated

a 90%, two sided confidence interval for . When she increased the confidence level to 95% the width

of the interval increased by 504 meters

(113 a)What is the value of?

(113 b) What is the standard error of the sample mean?

(113 c)What should be sample size, if she wants to achieve a confidence level of 95% and at the same

time retain the width of the 90% confidence interval that she got with a sample size of 25?

(113 d) What percentage of the sample means will fall within1000 meters of the population mean ( )

(use sample size 25)?

(113 e) Kanaka decided to test the claim of the manufacturers using a one sided hypothesis test. If she

has obtained a sample mean (from the sample of 25 observations) of 17,600 meters, what is the

maximum possible value for ? State clearly, your null and alternate hypotheses?

(114) Pure Foods is facing allegations from the Consumer Forum that had tested 125 samples of their

one kilo packets. They claimed that the amount of adulteration was 7 unit (unit= parts per thousand per

kilo). The Food Regulation Council, by regulation, allowed at most 6.5 units of adulteration on average.

Past data showed that the packets had a standard deviation of 2 units and both the Consumer Forum

and the Pure Foods authorities agreed to use this value, if required and you should also do that in Parts

(a)-(e).

(114 a)Is Pure Foods following the Regulations if the Consumer Forums results are correct? Test at =

0.01. Clearly state the null and the alternative hypothesis, critical region and your decision.

The Company hires a PR firm for damage control. After collecting the samples from the Consumer

Forum, Pure Foods claims that of the 125 samples, 25 packets had gone beyond the expiry date on the

date of testing by the Consumer Forum. Hence these samples should not be considered. The average

adulteration in these 25 packets they said was 7.5 units and the standard deviation was 2.676 units.


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(114 b) Once the packets beyond the expiry date are discarded from the consideration, can you

conclude that the Pure Foods is not following the regulations (at = 0.01)?

(114 c) The Consumer Forum counters and says that the Food Regulation Council specifies = 0.05 and

hence Pure Foods is guilty. Find the p-value (probability value) of the above test (in part b) to find the

validity of the conclusion of the consumer forum.

The Consumer Forum decides that a Type II error is more serious as it is harmful for consumers.

(114 d) If the true average adulteration level of all Pure Foods products was 7 units, what is the

probability of committing a Type II error for the test carried at in Part (b)?

(114 e) If the Consumer Forum wants the Type II error to be less than 5% when the true average

adulteration level of all Pure Foods products was 7 units, then what is the minimum sample size it

should take for future testing?

The Quality Control Manager knows that everyone has been using 2 as the standard deviation. However,

he is not sure and wants to check this. He is also curious about the adulteration level of packets whichhad gone beyond the expiry date. Refer to the data summary before part (b).

(114 f) Assuming that the true standard deviation in adulteration level of packets which had gone

beyond the expiry date is as specified (assumed) for all packets (mentioned above), what is the

probability of the sample standard deviation in adulteration level of 25 packets which had gone beyond

the expiry date exceeding the observed value (2.676 units)? (It would be acceptable to get an

approximate value of this probability.)

(114 g) Find a 90% confidence interval for the true standard deviation in adulteration level of packets

which had gone beyond the expiry date.

(114 h) Find a 95% confidence interval for the true average adulteration level of packets which had gone

beyond the expiry date.

tut problems week 8 (thursday%2cfriday sessions) (1)

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