B.Tech –IV Sem.

(Statistical Techniques - II)

MATHEMATICS (EAS-301)

A.    Binomial Distribution

1.	(i)           In 800 families with 5 children each, how many families would be expected to have (i) 3 boys (ii) 5 girls (iii) either 2 or 3boys (250,25,500) (ii)         In 800 families with 4 children each, how many families would be expected to have (i) 2 boys and 2 girls (ii) at least one boy (iii) no girl (v) at most two girls  (300, 750, 50,550) (iii)        In 800 families with 5 children each, how many families would be expected to have (i) 3 boys and 2 girls (ii) 2 boys and 3 girls (iii) no girl (v) at most two girls  (250,250,25,400)
2.	Fit a binomial distribution to the following frequency data: X 0 1 3 4 Y 28 62 10 4
3.	Find the mean and variance of the Binomial Distribution.
4.	In sampling a large number of parts manufactured by a machine, the mean number of defectives in a sample of 20 is 2. Out of 1000 such samples, how many would be expected to contain at least 3 defective parts. (323)
4.	(i)           Assuming that 20% of the population of  a city are literate, so that the chance of an individual being literate is 1/5 and assuming that 100 investigators each take 10 individuals to see whether they are literate,  how many investigators would be expect to report 3 or less were literate ? (87.9≈ 88) (ii)         Assuming half the population of a town consumes chocolates and that 100 investigators each take 10 individuals to see whether they are consumers, how many investigators would you expected to report that three people or less were consumers? (17.2≈ 17)
5.	In bombing action, there is 50% chance that any bomb will strike the target. Two direct hits are needed to destroy the target completely. How many bombs are required to be dropped to give 99% chance or better of completely destroying the target? (11)
6.	Six dice are thrown 729 times. How many times do you expect at least three dice to show a five or six? (233)
7.	If the probability of a defective bolt is 0.1, find the mean and standard deviation for the distribution bolts in a total of 400. (40,6)
8.	The probability that a bomb dropped from a plane will strike the target is. If six bombs are dropped, (i) exactly two will strike the target. (ii)  At least two will strike the target.   (0.246,0.345)
9.	Following results were obtained when 100 batches of seeds were allowed to germinate on damp filter paper in a laboratory:  , Determine the binomial distribution. Calculate the expected frequency for x=8 assuming p>q.
10.	A policeman fires 6 bullets on a dacoit. The probability that the dacoit will be killed by a bullet is 0.6. what is the probability dacoit is still alive.
11.	If the probability of hitting a target is 10 % and 10 shots are fired independently. what is the probability that the target will be hit at least once? (0.6513)
12.	The probability of man hitting a target is 1/3. how many times must he fire so that the probability of his hitting the target at least once is more than 90% (6)
13.	The sum and product of the mean and variance of a binomial distribution are 25/3 and 50/3 respectively. Find the distribution.
14.	Four persons in a group of 20 are graduates. If 4 persons are selected at random from 20, find the probability that  (i) all are graduates (ii) at least one is graduate.  (0.0016,0.5904)
15.	Rohit takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at the end of 11 step, he is one step away from the starting point. (0.36787)

B.    Poisson distribution

16.	The distribution of road accident per day in a city is Poisson with mean 4. Find the no. of days out of 100 days when there will be :(i) No accident (ii) at least 2 accident (iii) at most 3 accident (iv) between 2 and 5 accident.
17.	The probability that a managed 50yr. will die within a year is 0.01125. What is the probability that 12 such men, at least 11 will celebrate their 51st birthday? .(
18.	If the variance of the Poisson distribution is 2, find the probabilities for r= 1, 2, 3, 4 from the recurrence relation of the Poisson distribution. Also find P (x≥4).   (0.2706,0.2706, 0.1804 ,0.902 and 0.1431)
19.	An insurance company finds that 0.005% of the population dies from a certain kind of accident each year. What is the probability that the company must pay off no more than 3 of 10,000 insured risks against such incident in a given year? ( 0.0016)
20.	find the mean and variance of Poisson distribution
21.	Fit a Poisson distribution to the following data and calculate theoretical frequencies Deaths 0 1 2 3 4 Frequencies 122 60 15 2 1
22.	The frequency of accidents per shift in a factory is shown in the following table.                         UPTU 2004 Accident/Shift 0 1 2 3 4 Frequencies 192 100 24 3 1
23.	Suppose that a book of 600 pages contains 40 printing mistakes. Assume that these errors are randomly distributed throughout the book and r, the number of errors per page has a Poisson distribution. What is the probability that 10 pages selected at random will be free from errors? (9512)
24.	If the probability that a man aged 50 years will die within a year is 0.01125. What is the probability that of 12 such men, at least 11 will reach their 51 birthday?
25.	Suppose the number of telephone calls on an operator received from 9:00 to 9:05 follow a Poisson distribution with a mean 3.Find the probability that (i) The operator will receive no calls in that time interval tomorrow. (ii) In the next three days, the operator will receive a total of 1 call in that time interval. (0.4978,0.00111)
26.	The no. of arrivals of customers during any day follows Poisson distribution with mean of 5. What is the probability that the total no. of customers on two days selected at random is less than 2? (0.0004994)
27.	6 coins are tossed 6400 times. Using the Poisson distribution, determine the approximate probability of getting six heads times.
28.	Show that in the Poisson distribution  with unit mean, mean deviation about mean is  times the S.D.
29.	For a Poisson distribution with mean  Show that  where
30.	An insurance company finds that 0.005% of the population dies from a certain kind of accident each year. What is the probability that the company must pay off no more than three of 10,000 insured risk against such incident in a given year? (0.0016)
31.	Suppose that a book of 585 pages contains 43 printing mistakes. Assume that these errors are randomly distributed throughout the book and r, the number of errors per page has a Poisson distribution. What is the probability that 10 pages selected at random will be free from errors? (0.4795)
32.	A certain screw making machine produces on average 2 defective screws out of 100, and packs them in boxes of 500. Find the probability that a box contains 15 defective screws. (0.035)
33.	If there are 3 misprint in a book of 100 pages, find the probability that a given page will contain (i) no misprint (ii) more than 2 misprint.

C.    Normal  Distribution

34.	The income of a group of 10000 persons was found to be normally distributed with mean Rs. 750 p.m. and S.D. of Rs. 50. Show that, of this group, about 95% had income exceeding Rs. 668 and only 5% had income exceeding Rs. 832. Also find the lowest income among the richest 100. (income exceeding 668= 95% , income exceeding Rs.832= 5% , among richest 100= 866.5)
35.	In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and standard deviation.)
36.	In a simple sample of 600 men from a certain large city, 400 are found to be smokers; in one of 900 from another city 450 are smokers, do the data indicates that cities are significantly different with Respect to prevalence of smoking among men?
37.	A sample of 100 dry battery cells tested to find the length of life produced the following results. Mean 12Hrs and Standard deviation 3 Hrs. Assuming the data the data to be normally distributed what percentage of battery cell expected to have life. (i) More than 15 Hrs. (ii) Less than 6 Hrs. (iii) B/w 10 and 14 Hrs.        (15.87%, 2.28%, 49.70%)
38.	The life of army shoe is ‘normally’ distributed with mean 8 months and standard deviation 2 months. If 5000 pairs are issued how many pairs would be expected to need replacement after 12 months? Given   .  (4886)
39.	In a normal distribution, 7% of the items are under 35 and 89% are under 63. Find the mean and standard deviation.()
40.	In a test on 2000 electric bulbs, it was found that the life of a particular make, was normally distributed with an average life of 2040 hrs and S.D. of 60 Hrs. estimate the number of bulbs likely to burn for (i) More than 2150 Hrs. (ii) Less than 1950 Hrs. (iii) More than 2150 Hrs but Less than 1950 Hrs.                                                      (67,184,1909)
41.	In an examination taken by 500 candidates, the average and the standard deviation of marks obtained are 40% and 10%. Find approximately (i) how many will pass, if 50 % is fixed as a minimum? (ii) What should be minimum if 350 candidate to pass? (iii)  How many have scored marks above 60 %.  (79, 35%, 11)
42.	Student of class were given a mechanical aptitude test. Their marks were found to be normally distributed with mean 60 and S.D. 5. What percentage of student s scored? (i) More than 60 marks. (ii) Less than 56 marks. (iii) B/w 45and 65 marks.   (50%, 21.2%, 84%)
43.	Assuming that the diameter of 1000 brass plugs taken consecutively from a machine, form a normal distribution with mean  0.7515 cm and S.D. 0.002 cm, how many of the plugs are likely to be rejected if the approved diameter is 0.752±0.004 cm. (52)
44.	Prove that for normal distribution the mean deviation from the mean equals to 4/5 of the standard deviation.

D.    Statistic Quality Control

45.	The following table gives the sample mean and the range for IQ samples each of size 6, in the production of the certain component. Construct the central chart for mean and range and comment on the nature of control. Sample no. 1 2 3 4 5 6 7 8 9 10 Mean 37.5 49.8 51.5 59.2 54.7 34.7 51.4 61.4 70.7 75.3 Range R 9.5 12.8 10.0 9.1 7.8 5.8 14.5 2.8 3.7 8.0
46.	Draw and R-chart from the following data Sample no. 1 2 3 4 5 6 7 8 9 11 10.4 10.8 11.2 11.8 11.6 9.6 9.6 10 R 4 5 9 4 4 7 7 8 8
47.	Construct p-chart from the following data No. of samples (each of 100 items) 1 2 3 4 5 6 7 8 9 10 No. of defectives 12 10 6 8 9 9 7 10 11 8
48.	The following data of defective of 10 samples of size 100 each, construct np-chart Sample no. 1 2 3 4 5 6 7 8 9 10 No. of defectives 4 8 11 3 11 7 7 16 12 6
49.	The data given below gives the no. of blemishes on the lamination on limitation glass of 22 samples. Construct a c-chart and comment on the production process. No. of blemishes per product are        6, 6, 6, 7, 7, 6, 6, 7, 8, 7, 6, 5, 7, 9, 9, 8, 8, 8, 9, 7, 8, 8.
50.	The following are the mean lengths and range of length of a finished product from 10 samples each of size 5. the specification limit for length are 200±5 cm. Construct   and R Chart and Examine whether the process is under control and state your recommendation.(n=5, A2=0.577, D3=0 and D4=2.115) Sample No. 1 2 3 4 5 6 7 8 9 10 Mean 201 198 202 200 203 204 199 196 199 201 Range 5 0 7 3 4 7 2 8 5 6 (uncontrolled process, control process)
51.	In a blade manufacturing factory, 1000 blades are examined daily. Following information shows number of defective blades obtained there. Draw the np chart and given your findings. Date 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 No. of defective 9 10 12 8 7 15 10 12 10 8 7 13 14 15 16 (control process)
52.	Distinguish b/w the np chart and p chart. The following data of defective of 10 samples of size 100 each, construct np-chart and given your comment Sample no. 1 2 3 4 5 6 7 8 9 10 No. of defectives 6 9 12 5 12 8 8 16 13 7
53.	What are Statistic Quality Control techniques? Discuss the objective and advantage of Statistic Quality Control?
54.	The following fig. given number of defectives in 20 samples, containing the 2000 items. 425,430,216,341,225,322,280,306,337,305,356,402,216,264,126,409,193,280,326,389. Calculate the central line and the control limit for p-chart. (CL=0.1537, LCL=0.12952,UCL=0.1779)
55.	What is the cause of variation and control chart .

   E Anova table , test and Time Series

56.	Five dices were thrown 192 times and the number of times 4, 5 or 6 were as follows- Calculate.  number of dice throwing 4, 5  6 5 4 3 2 1 0 f 6 46 70 48 20 2
57.	In 120throwsof a single die, the following distribution of faces was obtained – Faces 1 2 3 4 5 6 Total 30 25 18 10 22 15 120 Do these results constitute of reputation of the equal probability null hypothesis.
58.	Two horses A and B were tested according to the time (in sec.) to run a particular track with the following results – Horse A 28 30 32 33 33 29 34 Horse B 29 30 30 24 27 29   Test whether you can discriminate between two horses. You can use the fact that 5% value of t for 11 degrees of freedom is 2.20.
59.	What is the analysis of variance and where is it used?
60.	What are the assumptions under analysis of variance?
61.	Obtain the 5 yearly moving average of the following data Year 1982 1983 1984 1985 1986 1987 1988 1989 1990 Sales(in crore of Rs.) 36 43 43 34 44 54 52 24 15
62.	To test the effectiveness of inoculation against chotera, the following table was obtained. Attacked Not Attacked Total Inoculated 30 160 190 Not Inoculated 140 460 60 Total 170 620 790 Use  - test to Defend or refute the statement. The inoculation prevents attack from cholera.
63.	The demand for a particular spare part in a factory was found to vary from day to day. In a sample study, the following information was obtained: Days Mon Tue Wed Thurs Fri Sat No. of parts Demanded 1124 1125 1110 1120 1125 1116 Use  – test to test the hypothesis that number of parts demanded does not depend on the day of the week at 5% level of signification.
64.	A die is thrown 270 times and the results of these throws are given below: No. appeared on the die 1 2 3 4 5 6 Frequency 40 32 29 59 57 59 Test whether the die is biased or not. (biased)
65.	The theory predicts the proportion of beans in the four groups, G1, G2, G3, G4 should be in the ratio 9:3:3:1. In an experiment with 1600 beans the numbers in the four groups were 882,313,287 and 118. Does the experimental result support the theory? (Accepted)
66.	From the following table regarding the colour of eyes of father and son, test if the colour of son,s eye is associated with that of the father. Father            Son Light Not  Light Light 471 51 Not  Light 148 230 (261.498)

KANPUR INSTITUTE OF TECHNOLOGY, KANPUR

  Assingment

 B.Tech –IV Sem.

(Statistical Techniques - II)

MATHEMATICS (EAS-301)

1.	In 800 families with 4 children each, how many families would be expected to have (i) 2 boys and 2 girls (ii) at least one boy (iii) no girl (v) at most two girls.  In 800 families with 5 children each, how many families would be expected to have (i) 3 boys and 2 girls (ii) 2 boys and 3 girls (iii) no girl (v) at most two girls .
2.	Fit a binomial distribution to the following frequency data: X 0 1 3 4 Y 28 62 10 4
3.	Find the mean and variance of the Binomial Distribution.
4.	In sampling a large number of parts manufactured by a machine, the mean number of defectives in a sample of 20 is 2. Out of 1000 such samples, how many would be expected to contain at least 3 defective parts. (323)
5.	Assuming that 20% of the population of  a city are literate, so that the chance of an individual being literate is 1/5 and assuming that 100 investigators each take 10 individuals to see whether they are literate,  how many investigators would be expect to report 3 or less were literate ? (87.9≈ 88) Assuming half the population of a town consumes chocolates and that 100 investigators each take 10 individuals to see whether they are consumers, how many investigators would you expected to report that three people or less were consumers? (17.2≈ 17)
6.	In bombing action, there is 50% chance that any bomb will strike the target. Two direct hits are needed to destroy the target completely. How many bombs are required to be dropped to give 99% chance or better of completely destroying the target? (11)
7.	Six dice are thrown 729 times. How many times do you expect at least three dice to show a five or six? (233)
8.	An insurance company finds that 0.005% of the population dies from a certain kind of accident each year. What is the probability that the company must pay off no more than 3 of 10,000 insured risks against such incident in a given year?
9.	If the probability of a defective bolt is 0.1, find the mean and standard deviation for the distribution bolts in a total of 400.
10.	Following results were obtained when 100 batches of seeds were allowed to germinate on damp filter paper in a laboratory:  , Determine the binomial distribution. Calculate the expected frequency for x=8 assuming p>q.
11.	A policeman fires 6 bullets on a dacoit. The probability that the dacoit will be killed by a bullet is 0.6. what is the probability dacoit is still alive.
12.	The distribution of road accident per day in a city is Poisson with mean 4. Find the no. of days out of 100 days when there will be :(i) No accident (ii) at least 2 accident (iii) at most 3 accident (iv) between 2 and 5 accident.
13.	The probability of man hitting a target is 1/3. how many times must he fire so that the probability of his hitting the target at least once is more than 90% (6)
14.	If the variance of the Poisson distribution is 2, find the probabilities for r= 1, 2, 3, 4 from the recurrence relation of the Poisson distribution. Also find P (x≥4).
15.	Rohit  takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at the end of 11 steps, he is one step away from the starting point. (0.36787)

16.	Suppose that a book of 600 pages contains 40 printing mistakes. Assume that these errors are randomly distributed throughout the book and r, the number of errors per page has a Poisson distribution. What is the probability that 10 pages selected at random will be free from errors?
17.	The probability that a managed 50yr. will die within a year is 0.01125. What is the probability that 12 such men, at least 11 will celebrate their 51st birthday? .(
18.	If the probability that a man aged 50 years will die within a year is 0.01125. What is the probability that of 12 such men, at least 11 will reach their 51 birthday?
19.	find the mean and variance of Poisson distribution
20.	The no. of arrivals of customers during any day follows Poisson distribution with mean of 5. What is the probability that the total no. of customers on two days selected at random is less than 2?
21.	The frequency of accidents per shift in a factory is shown in the following table.                         UPTU 2004 Accident/Shift 0 1 2 3 4 Frequencies 192 100 24 3 1
22.	6 coins are tossed 6400 times. Using the Poisson distribution, determine the approximate probability of getting six heads times.
23.	Show that in the Poisson distribution  with unit mean, mean deviation about mean is  times the S.D.
24.	For a Poisson distribution with mean  Show that  where
25.	A certain screw making machine produces on average 2 defective screws out of 100, and packs them in boxes of 500. Find the probability that a box contains 15 defective screws. (0.035)
26.	If there are 3 misprint in a book of 100 pages, find the probability that a given page will contain (i) no misprint (ii) more than 2 misprint.

27.	The income of a group of 10000 persons was found to be normally distributed with mean Rs. 750 p.m. and S.D. of Rs. 50. Show that, of this group, about 95% had income exceeding Rs. 668 and only 5% had income exceeding Rs. 832. Also find the lowest income among the richest 100. (income exceeding 668= 95% , income exceeding Rs.832= 5% , among richest 100= 866.5)
28.	In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and standard deviation.)
29.	In a simple sample of 600 men from a certain large city, 400 are found to be smokers; in one of 900 from another city 450 are smokers, do the data indicates that cities are significantly different with Respect to prevalence of smoking among men?
30.	In a normal distribution, 7% of the items are under 35 and 89% are under 63. Find the mean and standard deviation.()
31.	In a test on 2000 electric bulbs, it was found that the life of a particular make, was normally distributed with an average life of 2040 hrs and S.D. of 60 Hrs. estimate the number of bulbs likely to burn for (i) More than 2150 Hrs. (ii) Less than 1950 Hrs. (iii) More than 2150 Hrs but Less than 1950 Hrs.                                                      (67,184,1909)
33.	In an examination taken by 500 candidates, the average and the standard deviation of marks obtained are 40% and 10%. Find approximately (i) how many will pass, if 50 % is fixed as a minimum? (ii) What should be minimum if 350 candidate to pass? (iii)  How many have scored marks above 60 %.  (79, 35%, 11)
34.
35.	Assuming that the diameter of 1000 brass plugs taken consecutively from a machine, form a normal distribution with mean  0.7515 cm and S.D. 0.002 cm, how many of the plugs are likely to be rejected if the approved diameter is 0.752±0.004 cm. (52)
36.	Prove that for normal distribution the mean deviation from the mean equals to 4/5 of the standard deviation.

37.	The following table gives the sample mean and the range for IQ samples each of size 6, in the production of the certain component. Construct the central chart for mean and range and comment on the nature of control. Sample no. 1 2 3 4 5 6 7 8 9 10 Mean 37.5 49.8 51.5 59.2 54.7 34.7 51.4 61.4 70.7 75.3 Range R 9.5 12.8 10.0 9.1 7.8 5.8 14.5 2.8 3.7 8.0
38.	The following data of defective of 10 samples of size 100 each, construct np-chart Sample no. 1 2 3 4 5 6 7 8 9 10 No. of defectives 4 8 11 3 11 7 7 16 12 6
39.	The data given below gives the no. of blemishes on the lamination on limitation glass of 22 samples. Construct a c-chart and comment on the production process. No. of blemishes per product are        6, 6, 6, 7, 7, 6, 6, 7, 8, 7, 6, 5, 7, 9, 9, 8, 8, 8, 9, 7, 8, 8.
40.	The following are the mean lengths and range of length of a finished product from 10 samples each of size 5. the specification limit for length are 200±5 cm. Construct   and R Chart and Examine whether the process is under control and state your recommendation.(n=5, A2=0.577, D3=0 and D4=2.115) Sample No. 1 2 3 4 5 6 7 8 9 10 Mean 201 198 202 200 203 204 199 196 199 201 Range 5 0 7 3 4 7 2 8 5 6 (uncontrolled process, control process)
41.	In a blade manufacturing factory, 1000 blades are examined daily. Following information shows number of defective blades obtained there. Draw the np chart and given your findings. Date 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 No. of defective 9 10 12 8 7 15 10 12 10 8 7 13 14 15 16 (control process)
42.	Distinguish b/w the np chart and p chart. The following data of defective of 10 samples of size 100 each, construct np-chart and given your comment Sample no. 1 2 3 4 5 6 7 8 9 10 No. of defectives 6 9 12 5 12 8 8 16 13 7
43.
44.	The following fig. given number of defectives in 20 samples, containing the 2000 items. 425,430,216,341,225,322,280,306,337,305,356,402,216,264,126,409,193,280,326,389. Calculate the central line and the control limit for p-chart. (CL=0.1537, LCL=0.12952,UCL=0.1779)

45.	Five dices were thrown 192 times and the number of times 4, 5 or 6 were as follows- Calculate.  number of dice throwing 4, 5  6 5 4 3 2 1 0 f 6 46 70 48 20 2
46.	In 120throwsof a single die, the following distribution of faces was obtained – Faces 1 2 3 4 5 6 Total 30 25 18 10 22 15 120 Do these results constitute of reputation of the equal probability null hypothesis.
47.	What is the analysis of variance and where is it used?
48.	What are the assumptions under analysis of variance?
49.	To test the effectiveness of inoculation against chotera, the following table was obtained. Attacked Not Attacked Total Inoculated 30 160 190 Not Inoculated 140 460 60 Total 170 620 790 Use  - test to Defend or refute the statement. The inoculation prevents attack from cholera.
50.	The demand for a particular spare part in a factory was found to vary from day to day. In a sample study, the following information was obtained: Days Mon Tue Wed Thurs Fri Sat No. of parts Demanded 1124 1125 1110 1120 1125 1116 Use  – test to test the hypothesis that number of parts demanded does not depend on the day of the week at 5% level of signification.
51.	The theory predicts the proportion of beans in the four groups, G1, G2, G3, G4 should be in the ratio 9:3:3:1. In an experiment with 1600 beans the numbers in the four groups were 882,313,287 and 118. Does the experimental result support the theory? (Accepted)