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THEORETICAL DISTRIBUTIONS Question 1 What is the probability that a standard normal variate Z will be (a) greater than 1.09 ; (b) less than -1.65 ; (c) lying between -1.00 and 1.96 ; (d) lying between 1.25 and 2.75 ? Answer : (a) P (b) P (c) P (d) P
(Z > 1.09) = 0.1379 (Z <= 1.65) = 0.0495 (-1.00 < Z < 1.96) = 0.8163 (1.25 < Z < 2.75) = 0.1026
Question 2 Find the value of Z such that the probability of a larger value is 0.78881 Answer : Z > - 0.80 Question 3 A certain type of wooden beam has a mean breaking strength of 1500 kgs and a standard deviation of 100 kgs. Find the relative frequency of all such beams whose breaking strengths lie between 1450 and 1600 kgs. Answer : P (1450 < Z < 1600) = 0.5328 ; 53.28% Question 4 Assume the mean height of children to be 68.22 cm with a variance of 10.8 cm. How many children in a school of 1,000 would you expect to be over 72 cm tall? Answer : P (X > 72) = 0.1251 Question 5 The life time of a certain kind of battery has a mean of 300 hours and a standard deviation of 3.5 hours. Assuming that the distribution of life times, which are measured to the nearest hour is normal, find the percentage of batteries which have life of more than 370 hours. Answer : P (X > 370) = 0.0228 ; 2.28%
Question 6 1000 light bulbs with a mean life of 120 days are installed in a new factory. What length of life is normally distributed with a standard deviation of 20 days. (a) How many bulbs will expire in less than 90 days? (b) If it is decided to replace all bulbs together, what interval should be allowed between replacement if not more than 10% should expire before replacement? Answer : (a) 66.8 ~ 67 bulbs (b) 94.4~94 days Question 7 The scores made by a candidate in a certain test are normally distributed with mean 500 and standard deviation 100. What percentage of candidate receives the scores between 400 and 600? Answer : 68.27% Question 8 In a normal distribution 31% of the items are under 45 and 8% are over 64. Find the mean and standard deviation of the distribution. Answer : µ = 49.96 ~ 50, σ = 10 Question 9 An auditor has found that the credit rewards of a large mail order have been approximately normally distributed and show an average account billing error of Rs. 10 and a standard deviation of Re. 1 (billing error may be positive or negative according to whether purchases were overcharged or undercharged). Suppose credit account is randomly selected from the files of the mail order house. Find the probability of a billing error between Rs. 10 and Rs. 11.50. Answer : 0.4332 Question 10 A random variable X is normally distributed with mean µ = 12 and s.d. σ = 2. Find P (9.6 < X < 13.8). Given that the area: A = 0.03159 for Z = 0.9 and A = 0.3849 for Z = 1.2, where Z is a standard normal variate. Answer : P (9.6 < X < 13.8) = 0.7008
Question 11 The income of a group of 10,000 persons was found to be normally distributed with mean equal to Rs. 750 and standard deviation equal to Rs. 50. What was the lowest income among the richest 250 ? Answer : Rs. 848 Question 12 The marks obtained in a certain examination follow normal distribution with mean 45 and standard deviation 10. If 1000 students appeared at the examination, calculate the number of students scoring (i) less than 40 marks and (ii) more than 60 marks. Answer : (i) (ii)
309 students (approx.) 67 students (approx.)
Question 13 Let X be a continuous variable and follows a normal distribution with mean 12 and standard deviation 2. What is the probability that the value of X selected at random lies in the interval [11, 14] ? Answer : P (11 <= X <= 14) = 0.5328 Question 14 The average test marks in a particular class is 79. The standard deviation is 5. If the marks are normally distributed, how many students in a class of 200 did not receive marks between 75 and 82 ? Answer : 97 students Question 15 The marks of the students in a certain examination are normally distributed with mean marks as 40% and standard deviation marks as 20%. On this basis, 60% of students passed. The result was moderated and 70% students passed. Find the pass marks before and after the moderation. Answer :
Pass marks before moderation = 45 Pass marks after moderation = 29.6 or 30
Question 16 The monthly income distribution of workers in a certain factory was found to be normal with mean Rs. 500 and standard deviation equal to Rs. 50. There were 228 persons getting income above Rs. 600 per month. How many workers were there in all ? Extract of area under standard normal is given below : Z : Area :
1 0.3413
2
2.5
0.4772
0.4938
3 0.4987
Answer : 10,000 workers Question 17 The life time of a certain kind of batteries has a mean life of 400 hours and standard deviation as 45 hours. Assuming the distribution of life time to be normal, find (i) (ii) (iii)
The percentage of batteries with a life time of atleast 470 hours. The proportion of batteries with life time between 385 and 415 hours. The minimum life of the best 5% of batteries.
Answer : (i) (ii) (iii)
5.94% or 6% 25.86% or 26% 474 hours
Question 18 The marks of the students are normally distributed. 10% get more than 75 marks and 20% get less than 40 marks. Find the mean and standard deviation of distribution. The relevant extract of Area Table (under the normal curve) is given below: Z : Area :
0.84
1.28
0.2995
0.3997
Answer : µ = 54, σ2=16.51
2.0 0.4772
Question 19 The distribution of wages of a group of workers is known to be normal with mean Rs. 500 and SD Rs. 100. If the wages of 100 workers in the group are less than Rs. 430, what is the total number of workers in the group? Answer : 413 workers Question 20 The mean of a normal distribution is 500 and 16 per cent of the values are greater than 600. What is the standard deviation of the distribution? ( Given that the area between z = o to z = 1 is 0.34 ) Answer : σ = 100 Question 21 1000 bulbs with a mean life of 120 days are installed in a new factory. Their length of life is normally distributed with standard deviation 20 days. (i) (ii)
How many bulbs will expire in less than 90 days? If it is decided to replace all the bulbs together, what interval should be allowed between replacement if not more than 10% should expire before replacement ?
Answer : (i) (ii)
66.8 ~ 67 bulbs 94 days
Question 22 A multiple choice quiz has 200 questions, each with 4 possible answers, of which only 1 is the correct answer. What is the probability (using normal approximation to binomial distribution) that sheer guesswork yields from 25 to 30 correct answers for 80 problems (out of the 200 problems) about which the student has no knowledge. [Area under the normal curve is 0.4996 from Z = 0 to Z = 2.71 and 0.3770 from Z = 0 to Z = 1.16] Answer : P (25 <= X <= 30) = 0.0936 Question 23 At a certain examination, 10% of the students who appeared for the paper in statistics got less than 30 marks and 97% of the students got less than 62 marks. Assuming the distribution to be normal, find the mean and the standard deviation of the distribution.