sampling questions

8/11/2019 Sampling Questions

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UNIT - III

SAMPLING THEORY

SAMPLING DISTRIBUTION OF SINGLE MEAN:

1.

Individual filing of income tax returns prior to 30 June had an average refund of Rs. 1200.Consider the population of last minute filers who file their returns during the last week of June.

For a random sample of 400 individuals who filed a return between 25 and 30 June, the sample

mean refund was Rs 1054 and the sample standard deviation was Rs 1600. Using 5 percent level

of significance, test the belief that the individuals who wait until the last week of June to file their

returns to get a higher refund than early the filers. (Answer: Z = -1.825.)

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2.

A packaging device is set to fill detergent powder packets with a mean weight of 5 kg, with a

standard deviation of 0.21 kg. The weight of packets can be assumed to be normally distributed.

The weight of packets is known to drift upwards over a period of time due to machine fault, which

is not tolerable. A random sample of 100 packets is taken and weighed. This sample has a mean

weight of 5.03 kg. Can we conclude that the mean weight produced by the machine has increased?

Use a 5 percent level of significance. (Answer: Z = 1.428.)

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3.

A sample of 400 boys is found to have a mean height of 67.47. Can it reasonably regarded as a

sample from a large population with mean height 67.39 and standard deviation 1.30? (Test at

5% significance level). (Answer: Z = 1.231.)

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4.

The management of a company claims that the average weekly income of their employees is Rs.

1900/-. The trade union disputes this claim stressing that it is rather less. An independent survey

of 150 randomly selected employees showed an average of Rs. 1850/- with a standard deviation

of Rs. 300. Would you accept the view of the management or the trade union? (Answer: Z = -2.04.)

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5.

A manufacturer of fluorescent tubes claims that his tubes have a life time on an average 2000

burning hours, a sample of 100 tubes was taken at random and tested for burning life. It was

found to have a mean life of 1950 hours with a standard deviation of 150 hours, can the claim of

the manufacturer be accepted at 5% level of significance. (Answer: Z = -3.33.)

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6.

The mean life time of a sample of 400 fluorescent light bulbs produced by a company is found to

be 1600 hours with a standard deviation of 150 hours. Test the hypothesis that the mean life timeof the bulbs produced in general is higher than the mean life of 1570 hours at = 0.01 level of

significance. (Answer: Z = 4.)

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7.

CSO estimates that the average Annual income of all the wage earners in India is Rs. 75,000 with a

(Standard Deviation) of Rs. 6,000. A private research organizations was asked to test the official

estimate at 5% level of significance. A random sample of 3,600 wage earners was taken across the

country and found the sample mean as Rs. 73,250. (Answer: Z = -17.5.)


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SAMPLING DISTRIBUTION OF TWO MEAN:

1. Intelligence test on two groups of boys and girls gave the following results.

Mean S.D N

Girls 75 15 150

Boys 70 20 250

Is there a significant difference in the mean scores obtained by boys and girls?

(Answer: Z = 2.8398.)***********************************************************************************************************

2. You are given the position in a factory before and after the settlement of an industrial dispute.

Comment on the gains or losses from the point of view of workers and that of management.

Before After

No. of workers 4800 4700

Mean wages(Rs.) 900 950

Median wages(Rs.) 960 900

Standard deviation(Rs.) 240 200

(Answer: Z = -11.04.)

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3.

In order to determine whether two differently priced bands of flash light batteries are equally

effective, a consumer testing bureau tested 45 batteries of each brand for length of life. The results

are given in the table below.

Brand-I Brand-II

Mean 165 Hours 177 Hours

S.D 15 Hours 19 Hours

Determine whether there is a difference between the effectiveness of two brands of batteries

(=0.01). (Answer: Z = -3.32.)

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4.

In a sample of 1000, the mean is 17.5 and the standard deviation is 2.5. In another sample of 800,

the mean is 18 and the standard deviation is 2.7. Assuming that the samples are independent,

discuss whether the two samples could have come from a population which have the same

standard deviation. (Answer: Z = 2.283.)

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5. The mean production of wheat from a sample of 100 fields is 200 lbs per acre with a standard

deviation of 10 lbs. Another sample of 150 fields gives the mean at 220 lbs per acre with a

standard deviation of 12 lbs. Assuming the standard deviation of the universe as 11 lbs, find at 1

percent level of significance, whether the two results are consistent. (Answer: Z = -1.992.)

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6.

A factory uses process A and process B for manufacturing the same item. The average weight in a

sample of 250 items produced from process A is found to be 120 grams with a standard deviation

of 12 grams, while for the process B, average weight in a sample of 400 items is found to be 124

grams with a standard deviation of 14 grams. Test whether the average weights of both the

processes are same? (Answer: Z = -3.874.)

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7. A detergent soap manufacturer claims that his soap brand A outsellshis soap brand B on the

average by Rs. 500 per month. A study is undertaken to test this claim with a sample of 200 retail


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shop. They had an average sale of brand A and B worth Rs. 5,500/- and Rs. 4925/- per month with

standard deviation respectively as Rs. 400 and Rs. 350. Use appropriate statistic to test the claim

of the manufacturer and comment at = 0.05. (Answer: Z = 15.3.)

SAMPLING DISTRIBUTION OF SINGLE PROPORTION:

1. The manufacturer of a Spot Remover claims that his product removes at least 90% of all spots.

What can be concluded about the claim if the spot remover removed only 174 spots out of 200

spots chosen at random from spots on cloth from a dry cleaning unit? (Answer: Z = -1.41.)

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2. A manufacturer of spark plugs claims that 3% of the items supplied by him are defective. Random

samples of 500 plugs are found to have 20 defective items. Test the claim of the manufacturer at

95% confidence limits, i.e., p 1.96 SE (P). (Answer: Z = 1.31.)

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3. In a sample 400 parts manufactured by a factory, the number of defective parts was found to be

30. The company, however, claimed that almost 5% of their products is defective. Is the claimtenable? (Answer: Z = 2.27.)

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4. A candidate in an election from a large constituency thinks that he will win the election if at least

45% of the electorate vote for him. He, therefore, conducts a sample survey to enable him to

decide whether he should stand for the election or not. The survey covers 10,000 voters and it is

found that 4,420 voters would vote for him. Advise him as to whether he should stand for the

election, stating clearly the level of significance and other assumptions on which you base your

conclusion. (Answer: Z = -1.61.)

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5.

A manufacturer claims that at least 95% of the equipments which he supplied to a factory

conformed to the specification. An examination of the sample of 200 pieces of equipment revealed

that 18 were faulty. Test the claim of the manufacturer. (Answer: Z = -2.67.)

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6. An auditor claims that 10% of a companys invoices are incorrect. To test this claim a random

sample of 200 invoices is checked and 24 are found to be incorrect. At 1% significance level, test

whether the auditors claim is supported by the sample evidence. (Answer: Z = -1.77.)

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7.

A company manufacturing a certain type of breakfast cereal claims that 60% of all housewives

prefer that type to any other. A random sample of 300 housewives contains 165 who do preferthat type. At 5% level of significance, test the claim of the company. (Answer: Z = 0.943.)

SAMPLING DISTRIBUTION OF TWO PROPORTION:

1.

Historically, it is known that machine produces 16 imperfect articles in a batch of 500. After a

particular overhaul, the machine produced 3 imperfect articles in a batch of 100. Has the

machine improved? (Answer: Z = 0.10436.)


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2.

Two suppliers A and B supply an item to a company. Supplier A claimed that he supplies 3%

deflective less than its competitor B. To test his claim, the company selected 400 items from A and

found that 35 components are defective whereas 300 items from B give rise to 25 defective. What

can be said about the claim of the supplier A? Use =0.05 (Answer: Z = 0.1963.)

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3. A medical researcher testing the effectiveness of a new drug found that 70% of a random sample

of 280 patients improved under this drug. In a control group, 140 patients improved under this

drug. In a control group, 140 patients were given a PLACEBO, 50% of these patients improved.

Test at =0.05, the effectiveness of the new drug. (Answer: Z = 4.)

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4.

Before an increase in excise duty on tea, 400 people out of a sample of 500 persons were found to

be tea drinkers. After an increase in duty, 400 people were tea drinkers in a sample of 600 people.

Using standard error of proportion, state whether there is a significant decrease in the

consumption of tea. Take (i) =0.05 (ii) =0.01 (Answer: Z = 4.81.)

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5. In two large populations there are 30% and 25% respectively of fair haired people. Is this

difference likely to be hidden in samples of 1200 and 900 respectively from the two populations?

(Answer: Z = 2.55.)

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6.

The subject under investigation is the measure of dependence of Tamil on words of Sanskrit

origin. One newspaper article reporting the proceedings of the constituent assembly contained

2025 words of which 729 words were declared by literacy critic to be of Sanskrit origin. A second

article by the same author describing atomic research contained 1600 words of which 640 words

were declared by the same critic to be of Sanskrit origin. Assuming that simple sampling

conditions hold, estimate the limits for the proportion of Sanskrit words in the writers vocabulary

and examine whether there is any significant difference in the dependence of this writer on wordsof Sanskrit origin in writing the two articles. Take =0.05 (Answer: Z = -2.469.)

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7. In a simple random sample of 600 men taken from a big city, 400 are found to be smokers. In

another simple random sample of 900 men taken from another city 450 are smokers. Do the data

indicate that there is a significant difference in the habit of smoking in the two cities?

(Answer: Z = 6.423.)

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