Statistics and Numerical Methods

At the 0.05 significance level, do the data support the statistician’s belief that for employees in the manufacturing industry, the mean annual salary in Tamil Nadu differs from the mean annual salary in Maharastra by less than Rs 4105?

 

• A college conducts both day and night classes intended to be identical. A sample of 100 day students’ field estimation results as below:

x1 = 72.4

and

s 1 = 14.8

A sample of 200 night students’ field examination results as below:

x2 = 73.9

and

s 2 = 17.9

Are the two means statistically equal at 10% level of significance?

• For the last decade, a medical representative’s record shows a proportion of favorable calls of 2/5. He tries a new technique and achieves 23 favorable calls out of 40 calls. Does this signify that the new technique is effective at 5% level of significance?

 

• In a survey of 600 adults who earn over Rs100,000 a month, 36 of them said that they feel that it is a necessity to fly first class when they travel. Do the data support the belief that more than 5 percent of those adults with a monthly income exceeding Rs100,000 find that it is a necessity to fly first class when traveling? Use a 10% significance level for the test.

• A manufacturer claimed that least 95% of the equipment which he supplied to a factory conformed to specifications. An examination of a sample of 200 pieces of equipment revealed that 18 were faulty. Test his claim at a significance level of

0.05 of 0.01.

• Before an increase in excise duty, on tea, 400 people out 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. State whether there is a significant decrease in the consumption of tea.

 

• A manufacturer of storm windows samples 250 new homes and found that 142 of them had storm windows. Another sample of size 320 of older homes was taken; 150 of them had storm windows. The manufacturer believes that the proportion of new home that have storm windows is larger than the proportion of older homes that have storm windows. Do the sample statistics support the manufacturer’s claim at the 0.05 significance level?

• On a certain day 74 trains were arrives on time at Delhi and 83 were late. At New Delhi 65 on time, 107 late. Is there any difference in the proportions arriving on time at the 2 stations?

• Records at western power show that 13% of customers pay their monthly bills with a postal money order. If you take a sample of 300 accounts, what is the probability that a sample proportion of those paying with a postal money order will be within 0.03 of the true proportion?

• Records at western power show that 57% of customers consume more than 275kw hours per month. If you take a sample of 300 accounts, what is the probability that a sample proportion of those consuming 275kw hours per month will be within 0.03 of the true proportion?

 

• Last week, the 12,983 employees at Quantity Two Corporation worked a mean of

38.2 hours, with a S.D of 5.8 hours, as head of personal; you take a random sample of 50 time awards. What is the probability that the sample mean will be within 1 hour of the true mean?

• The home mortgages at First Home savings and loan have a mean balance due of

$ 37,594 and a of 100 mortgage accounts. What is the probability that the sample mean will be?

• Within $ 1000 of the true mean?
• Within $ 800 of the true mean?

23. A sample of 30 light tubes fielded a standard deviation of 90 hours running time where as the long experience with the particular brand showed standard deviation of 105 hours using a = 0.05 , test it there is any difference is Standard deviation

26. Two impendent samples of signs 9 and 8 gave the sum of squares of deviation from their respective means as 160 and 91 can the samples be regarded as drawn from the normal population with equal variance
F0.05 (8,7) = 3.73
F0.05 (7,8) = 3.50

27. The following data gives the number of aircraft accidents that occurred during the various days of the week. Find whether the accidents uniformly distributes over the week

Days	SUN	MON	TUE	WED	THU	FRI	SAT
No	Of    14	16	8	20	11	9	14

accidents

 

29. A lawn-equipment shop is considering adding a brand of lawn movers to its merchandise. The manager of the shop believes that the highest quality lawn movers are Trooper, lawn eater and Nipper, and he needs to decide whether it makes a difference which of these three shop adds to its existing merchandise.

Twenty owners of each of these three types of lawn movers are randomly sampled and asked how satisfied they are with their lawn movers

Lawn mover	Very satisfied	Satisfied	Not satisfied	total
Trooper	11	6	3	20
Lawn eater	13	4	3	20
Nipper	13	6	1	20

Are the owners of the lawn movers homogeneous in their response of the survey? Use a 5% significance level.

• In an industry, 200 workers, employed for a specific job, were classified according their performance and training received /not received to test independence of a specific training and performance. The dates is

 

Performance

	Good	Not Good	Total
Trained	100	50	150
Untrained	20	30	50
Total	120	80	200

5% level of significance

•   To see whether silicon chip sales are independent of where the U.S economy is in the business cycle, data have been collected on the weekly sales of Zippy Chippy, a silicon Valley firm , and on whether the U.S economy was rising to a cycle peak, falling to a cycle through, of at a cycle through the results are Economically weakly chip sales:

	High	Medium	Low	Total
At Peak	8	3	7	18
Rising	4	8	5	17
Falling	8	4	3	15
Total	20	15	15	50

• A completely randomized design experiments with 10 plota and 3 treatments gave the following results. Analyze the results for treatment effects.

Plot No :	1	2	3	4	5	6	7	8	9	10
Treatment :	A	B	C	A	C	C	A	B	A	B
Yield :	5	4	3	7	5	1	3	4	1	7

• The following figures relate to production in kgs. of three variables A, B, C of wheat sown on 12 plots.

A	14	16	18
B	14	13	15	22
C	18	16	19	19	20

Is there any significant difference in the production of the varieties

DEPARTMENT OF MATHEMATICS UNIT- II DESIGN OF EXPERIMENTS

Part A

• What is ANOVA ?
• What are the uses of analysis of variance?
• Explain the word “treatment” in analysis of variance.
• Define the term completely randomized design?
• Define experimental error , Explain design of experiments.
• Name the basic principles of experimental design.
• Is a 2x 2 Latin square design possible? Why?
• When do you apply analysis of variance techniques ?
• Explain the meaning and use of analysis of variance.

Part : B

 1. The following are the numbers of mistakes made in 5 successive days of 4 technicians working for a photographic laboratory.
Test at the level of significance
 = 0.01 whether the difference among the 4 sample means can be attributed to chance.

 

 

 Five varieties of wheat, a, b, c, d and e were tried. The gross size of the plots was 18 feet x 22 feet, the net being 14 feet x 18 feet. Thus the whole experiment        occupied an area 90 feet x 110 feet. the plan, the varieties  shown  in  each  plot  and


yields obtained in kg. are given in the following table.
Carry out an analysis of variance. What inference can you draw from the given data.




 




B 90	E 80	C 134	A 112	D 92
E 85	D 84	B 70	C 141	A 82
C 110	A 90	D 87	B 84	E 69
A 81	C 125	E 85	D 76	B 72
D 82	B 60	A 94	E 85	C 88

 

• The following table show the lives in hours of four brands of electric lamps.

Brand A : 1610       1610	1650	1680	1700	1720	1800
Brand B : 1580       1640	1640	1700	1750
Brand C : 1460       1550	1600	1620	1640	1660	1740	1820
Brand D : 1510       1520	1530	1570	1600	1680

Perform an analysis of variance test the homogeneity of the mean lives of the four brands of lamps.

• Analyze the data given below and interpret the results.

A13	B09	C21	D07	E06
D09	E08	A15	B07	C16
B11	C17	D08	E10	A17
E08	A15	B07	C10	D07
C11	D09	E08	A11	B15

 



• An experiment was designed to study the performance of four different detergents for cleaning fuel injectors. The following cleanness readings were obtained with specially designed equipments for 12 tanks of gas distributed over three different models of engines

Looking on the detergents as treatments and the engines as blocks, obtain the appropriate analysis of variance table and




test at the 0.01 level of significance whether there are differences in detergent or in the engines.

	E I	E II	E III
Detergent A	45	43	51
Detergent B	47	46	52
Detergent C	48	50	55
Detergent D	42	37	49



• A farmer wishes to test the effects o four different fertilizers A, B, C, D on the yield of wheat. In order to eliminate sources of error due to variability in soil fatality he uses the fertilizers in a Latin Square arrangement as indicated in the following table, where the numbers indicate yields in bushels per unit area.

A18          C21          D25         B11 D22 B12                 A15          C19 B15  A20           C23                    D24 C22          D21          B10           A17
Perform an analysis of variance to determine if there is a significant difference between the fertilization at     = 0.05 level of significance.

• A completely randomizes design experiment with 10 plots and 3 treatments gave the following result , analysis the result for treatment effects.

 

Plot no:	1	2	3	4	5	6	7	8	9	10
Treatment :	A	B	C	A	C	C	A	B	A	B
Yield	5	4	3	7	5	1	3	4	1	7

 

Question Bank
STATISTICS & NUMERICAL METHIODS

Unit III : Solution of equations and eigen value problems

Part A

• By Newton’s method find an iterative formula to find 1/N.
• Find the positive root of x3 + 5x – 3 = 0 usind Newton’s method with 0.6 as first approximation.



æ1

• Find inverse of A = ç

è 2




3 ö
÷ by Gauss – Jordan method. 7 ø



• State the convergence and order of convergence for method of false position.
• Why Gauss Seidel iteration is a method of successive corrections.
• Compare Gauss Jacobi and Gauss Siedel methods for solving linear system of the form AX = B.
• State the conditions for convergence of Gauss Siedel method for solving a system of equations.
• Find an iterative formula to find          where N is a positive number.
• Compare Gaussian elimination method and Gauss-Jordan method.
• What type of eigen value can be obtained using power method.
• State the order of convergence and convergence condition for Newton Raphson,s method.

ê      ú

 Find the dominant eigen value of A = é1 2 ù by power method.
3 4
ë      û
• How is the numerically smallest eigen value of A obtained.
• State two difference between direct and iterative methods for solving system of equations.

Part B

• Consider the non – linear system x2 – 2x – y +0.5 =0 and x2+4y2 – 4 = 0. Use Newton – Raphson method with the starting value (x0, y0) = (2.00, 0.25) and compute (x1, y1), (x2, y2) and (x3, y3) .

 

• Fins the real root of xex – 3 = 0
• Find the real root for x3 – 2x – 5 = 0 correct to three decimal places.

 




 

 

PART-A

• (a) Define sampling,
• Standard error,




 

(e) one tailed test and two tailed test



• Null hypothsis and alternative hypothsis,
• Error in sampling


• level of significance
• small sample test and Large sample test.


• What is the essential difference between confidence limits and tolerance limits?

Confidence Limit: To estimate a parameter a population
Tolerance Limit : To indicate between what limits one can find a proposition of population .
4.(a) Define student’s t-test

• Write down the formula of test statistic “t” to test the significance of difference between the means,
• What is the assumption of t-test?
• State the uses of Student’s t distribution

5. If two sample are taken from two population of unequal variances can we apply t-test to test the difference of




mean. 6. (a) Define c 2




test of goodness of fit      (b) Give the main use (application) of




c 2 test




(c) Write the condition for the application of c 2




test,



• What are the expected frequencies of 2 ´ 2 contingency table given below.

 



• For the




2 ´ 2 contingency table Write down the




corresponding c 2




value



• 
• Important properties of “F” test
PART-B
• The mean life time of a sample of 100 light bulbs produced by a company is computed to be 1570 hours with a standard deviation of 120 hours. If                                                        is the mean life time of all the bulbs produced by the company, test the hypothesis m = 1600 hours, against the alternative hypothesis m ¹ 1600 hours.
• A normal population has a mean of 6.48 and s.d of 1.5. In a sample of 400 members mean is 6.75. Is the difference significant?
• A simple sample of heights of 6400 Englishmen has a mass of 67.85 inches and a standard deviation of 2.56 inches, while a simple sample of heights of 1600 Australians has a mean of 68.55 inches and a standard deviation of 2.52 inches. Do the data indicate that Australians are on the average taller than Englishmen? process a sample of 150 bulbs gave the s.d of 95 hrs. Is the manufacturer justified in changing the process?
• The mean production of wheat of a sample of 100 plots is 200 kgs per acre with s.d of 10 kgs. Another sample of 150 plots gives the mean production of wheat at 220 kgs with s,d of 12 kgs. Assuming the s.d of the 11 kgs for the universe find at 1% level of significance, whether the two results are consistent.
• In a sample of 1000 people in Maharashtra, 540 are rice eaters and the rest are wheat eators. Can we assume that both rice and wheat are equally popular in this state at 1% level of significance.
• Random samples of 400 men and 600 women were asked whether they would like to have flyover near their residence. 200 men and 325 women were in favour of the proposal. Test the hypothesis that proportions of men and women in favour of the proposal, are same against that they are not, at 5% level.
• A cigarette manufacturing firm claims that its brand A of the cigarettes outsells its brand B by 8%. If it is found that 42 out of a sample of 200 smokers prefer brand A and 18 out of another random sample of 100 smokers prefer brand B, test whether the 8% difference is a valid claim. (Use 5% level of significance).
• Sandal powder is packed into packets by a machine. A random sample of 12 packets is drawn and their weights




are found to be (in kg) 0.49, 0.48, 0.47, 0.48, 0.49, 0.50, 0.51, 0.49, 0.48, 0.50, 0.51 and 0.48. Test if the average weight of the packing can be taken as 0.5 kg.

• The average breaking strength of steel rods is specified to be 18.5 thousand pounds. To test this a sample of 14 rods was tested. The mean and s.d obtained were 17.85 and 1.955 respectively. Is the result of the experiment significant with 95% confidence?
• The average breaking strength of steel rods is specified to be 17.5(in units of 1000 kg) to test this, sample of 14 rods tested and gave the following results 15,18,16,21,19,21,17,17,15,17,20,19,17,18. Is the result of the experiment is significant? Also obtain the 95% confidence interval for the average breaking strength?
• The average number of articles produced by two machine per day are 200 and 250 with s.d 20 and 25 respectively on the basis of records of 25 days production. Can you regard both the machines equally efficient at 1% level of significance.
• A group of 10 rats fed on diet A and another group of 8 rats fed on diet B, recorded the following increase in weight (gms).

Diet A: 5	6	8	1	12	4	3	9	6	10
Diet B: 2	3	6	8	10	1	1	2	8

Does it show superiority of diet A over diet B.

• In a certain experiment to compare two types of pig foods A and B, the following results of increase in weights were observed in pigs:

Pig number		1	2	3	4	5	6	7	8	Total
Increasein weight in lb	Food A	49	53	51	52	47	50	52	53	407
	Food B	52	55	52	53	50	54	54	53	423

• Assuming that the two samples of pigs are independent, can we conclude that food B is better that food A?
• Also examine the case when the same set of eight pigs were used in both the foods.
• The table below gives the number of aircraft accidents that occurred during the various days of the week. Test whether the accidents are uniformly distributed over the week.
• A random sample of size 25 from a population gives the sample standard deviation 8.5. Test the hypothesis that the population s.d is 10.
• It is believed that the precision (as measured by the variance) of an instrument is no more than 0.16. Write down the null and alternative hypothesis for testing this belief. Carry out the test at 1% level given 11 measurements of the same subject on the instrument.
• Five coins are tossed 256 times. The number of heads observed is given below. Examine if the coins are



unbiased, by employing c 2




goodness of fit.




No. of heads	0	1	2	3	4	5
Frequency	5	35	75	84	45	12

• A Sample analysis of examination results of 500 students was made. It was found that 200 students have failed, 170 have secured a third class, 90 have secured a second class and the rest, a first class. So these

figures support the general belief that the above categories are in the ratio 4:3:2:1 respectively?

• On the basis of information noted below, find out whether the new treatment is compartively superior to the

	Favourable	Non-favourable	Total
Conventional	40	70	90
New	60	30	110
Total	100	100	N=200

• Two independent samples of sizes 9 and 7 from a normal population had the following values of the variables.

Sample I:	18	13	12	15	12	14	16	14	15
Sample II:	16	19	13	16	18	13	15

Do the estimates of the population variance differ significantly at 5% level?