Set No. Code No: 220105 II B.Tech., II-Semester Supplementary Examinations, November-2003 PROBABILITY AND STATISTICS (Common to Civil Engineering, Production Engineering, Computer Science and Engineering, Chemical Engineering, Computer Science and Information Technology, Computer Science and Systems Engineering, and Electronics and Computer Engineering) Time: 3hours Max.Marks:80 Answer any FIVE questions All questions carry equal marks ---1. In a sample of 446 cars, stopped at a road block, only 67 of the drivers had their seat belts fastened. Estimate the probability that a driver stopped on that road, will have his or her seat belt fastened. 2. 1 A merchant can buy an item for Rs.2.10 and sell it for Rs.4.50. The probabilities for a demand of 0, 1, 2, 3, 4 or “5 or more” items are, respectively, 0.05, 0.15, 0.30, 0.25, 0.15 and 0.10. Calculate the expected profit resulting from stocking 0, 1, 2, 3, 4 or 5 items and determine how many the merchant should stock so as to maximize expected profit. A continuous random variable has the pdf f(x) = k x e-λx if x ≥0 , λ > 0 =0 other wise. Determine the constant k, and find mean and variance. A random sample of size 12 is taken from a normal population N (µ, 32) Find the probability that the variance of the sample lies between 3.4 and 14.8. b) A random sample of size 10 taken from a normal population N (µ,σ2) has standard deviation 4. Find 'a' and 'b' such that P (a < σ2 < b) = 0.95. A pet store sells pet food in 2-pound bags that are weighed on the platform of an old 25-pound scale. Suppose it is known that the standard deviation of weights is σ = 0.12 pound. If a sample of n = 16 bags of pet food were weighed carefully in a laboratory and the average weight was x = 2.09 pounds, find an approximate 95% confidence interval for μ, the mean weight of pet food in the 2-pound bags sold by the pet store. Let X equal the weight in grams of a 52-gram snack pack of candies. Assume that the distribution of X is N (μ, 4). A random sample of n =10 observations of X yielded the following data: 55.95 56.54 57.58 55.13 57.48 56.06 59.93 . 58.30 52.57 58.46 (i) Give a point estimate for μ. (ii) Find the endpoints for a 95% confidence interval for μ. (iii) Based on these very limited data, what is the probability that an individual snack pack selected at random is filled with less than 52 grams of candy? Contd…..2. 3. 4.a) 5.a) b) Code No: 220105 6.a) :: 2 :: Set No:1 From each of the two consignments of apples, a sample of size 200 is drawn, and the number of rotten apples counted. Test whether the proportion of rotten apples in the two consignments significantly different or not given that Consignment Sample from Consignment A Sample from Consignment B Sample Size 200 Number of rotten apples 44 200 30 b) Tests are made on the proportion of defective castings produced by five different molds. If there were 14 defectives among 100 castings made with mold-I, 33 defectives among 200 castings made with mold-II, 21 defectives among 180 castings made with mold-III, 17 defectives among 120 castings made with moldIV, and 25 defectives among 150 castings made with mold-V test (at the 0.05 level of significance) whether the true proportion of defectives is the same for each mold. Predict y at x=5 by fitting a least squares straight line to the following data: x 2 4 6 8 10 12 y 1.8 1.5 1.4 1.1 1.1 0.9 b) Construct a 95% confidence internal for α c) Test null hypotheris β = -0.12 against β > -0.12 at 0.01 level of significance. 7.a) 8. Find the correlation coefficient for the following bivariate frequency distribution. X 10-20 20-30 30-40 40-50 50-60 60-70 18 4 5 6 4 19 2 4 8 4 2 2 20 2 6 10 6 4 3 21 4 11 8 4 1 Y @@@@@ Set No. Code No: 220105 II B.Tech., II-Semester Supplementary Examinations, November-2003 PROBABILITY AND STATISTICS (Common to Civil Engineering, Production Engineering, Computer Science and Engineering, Chemical Engineering, Computer Science and Information Technology, Computer Science and Systems Engineering, and Electronics and Computer Engineering) Time: 3hours Max.Marks:80 Answer any FIVE questions All questions carry equal marks ---1. A card is drawn from a well shuffled pack of cards. What is the probability that it is either a spade or an ace? 2. 2 The two finalists in a golf tournament play 18 holes with the winner getting Rs.50,000 and the runner-up getting Rs.30,000. What are the two player’s mathematical expectation if (i) they are evenly matched. (ii) One of the two players should be favoured by odds of 3 to 1. A continuous random variable X is defined by f(x) = (3 + x)2 / 16 if -3 ≤ x < -1 = (6 - 2x2) / 16 if -1 ≤ x < 1 = (3 - x)2 / 16 if 1 ≤ x ≤ 3. Verify that f(x) is a density function and find also the mean of X. The probability that a man aged 50 years will die with in a year is .01125. What is the probability that of 12 such men at least eleven will reach their 51st birth day ? A random sample is taken from N (30, 12). How large a sample should be taken if the sample mean is to lie between 25 and 35 with probability 0.95. A random sample of size 100 is taken from an infinite population with mean 76 and variance 256. Fin the probability that the mean of the sample is in the interval (75. 78) To determine whether bacteria count was lower in the west basin of Lake Manasasa sarovar than in the east basin, n = 37 samples of water were taken from the west basin, and the number of bacteria colonies in 100 milliliters of water was counted. The sample characteristics were ~x = 11.95 and s= 11.80 measured in hundreds of colonies. Find the approximate 95% confidence interval for the mean number of colonies, say μW in 100 milliliters of water in the west basin. Thirteen tons of cheeses are stored in some old gypsum mines, including 22pound wheels (label weight). A random sample of n = 9 of these wheels yielded the following weights in pounds: 21.50 18.95 18.55 19.40 19.15 22.35 22.90 22.20 23.10 3.a) b) 4.a) b) 5.a) b) Assuming that the distribution of the weights of the wheels of cheese is N( μ,σ), find a 95% confidence interval for μ. Contd…..2. Code No: 220105 6.a) :: 2 :: Set No:2 b) A manufacturer of electronic equipment subject’s samples of two competing brands of transistors to an accelerated performance test. If 45 of 180 transistors of the first kind and 34 of 120 transistors of the second kind fail the test, what can he conclude at the level of significance α = 0.05 about the difference between the corresponding sample proportions? On the basis of their total scores, 200 candidates of a civil service examination are divided into two groups, the upper 30 % and the remaining 70 %. Consider the first question of the examination. Among the first group, 40 had the correct answer, whereas among the second group, 80 had the correct answer. On the basis of these results, can one conclude that the first question is no good at discriminating ability of the type being examined here? Estimate r by fitting the ideal gas law PVr = c to the following data Pressure P (lb/in2) Volume V (IN3) 16.6 50 3 9.7 30 7 8.5 20 115.5 195.3 15 10 546.1 5 7. 8. Find the correlation coefficient for the following bivariate distribution. 20-30 30-40 40-50 50-60 60-70 1-3 2 5 6 2 1 3-5 8 9 7 6 5-7 14 6 5 3 7-9 3 1 1 9-11 11-13 @@@@@ Set No. Code No: 220105 II B.Tech., II-Semester Supplementary Examinations, November-2003 PROBABILITY AND STATISTICS (Common to Civil Engineering, Production Engineering, Computer Science and Engineering, Chemical Engineering, Computer Science and Information Technology, Computer Science and Systems Engineering, and Electronics and Computer Engineering) Time: 3hours Max.Marks:80 Answer any FIVE questions All questions carry equal marks ---1. A can hit a target 3 times in 5 shots, B, 2 times in 5 shots, and C, 3 times in 4 shots. Find the probability of the target being hit, when all of them try. 2. A game between the players is fair if each player has the same mathematical expectation. If someone gives us Rs.5 each time we roll a 1 or a 2 with a balanced die, how much must we pay that person each time we roll a 3, 4, 5 or 6 to make the game fair. In eight throws of a die 5 or 6 is considered a success. Find the mean number of success and the standard deviation. For the continuous random variable X whose pdf is given by f(x) = cx (2-x) if 0 ≤ x < 2 = 0 otherwise. Find c, mean and variance of X. The diameters of shafts made by a certain manufacturing process are known to be normally distributed with mean 2.500 ems and standard deviation 0.009 ems. What is the distribution of the sample mean of 9 such shafts selected at random? Calculate the proportion of such sample means which can be expected to exceed 2.505 ems ? A random sample of size 16 is taken from a normal population with mean 30 and variance 64. Find the probability that the sample variance S 2 will be less than the population variance Assume that the yield per acre for a particular variety of soybeans is N(μ,σ 2). For a random sample of n =5 plots, the yields in bushels per acre were 37.4, 48.8, 46.9, 55.0, and 44.0. (i) Give a point estimate for μ. (ii) Find a 90% confidence interval for μ. In a study of maximal aerobic capacity 12 women were used as subjects, and one measurement that was made was blood plasma volume. The following data give their blood plasma volumes in liters: 3.15 2.99 2.77 3.12 2.45 3.85 2.99 3.87 4.06 2.94 3.53 3.20 3 3.a) b) 4.a) b) 5.a) b) Assume that these are observations of a normally distributed random variable X that has mean ft and standard deviation σ. (i) Give the value of a point estimate of μ.(ii) Determine point estimates of σ2 and σ. (iii) Find a 90% confidence interval for μ. Contd…..2. Code No: 220105 6.a) :: 2 :: Set No:3 b) A random sample of 6 steel beams has a mean compressive strength of 58,392 psi (pounds per square inch) with a standard deviation of 648 psi. Use this information and level of significance α = 0.05 to test whether the true average compressive strength of the steel from which this sample came is 58,000 psi. Measurements of the fat content of two kinds of ice cream, Brand A and Brand B, yielded the following sample data: Brand A (%) 13.5 14 13.6 12.9 13 Brand B (%) 12.9 13 12.4 13.5 12.7 Test the null hypothesis μ1 = μ2 (where μ1 and μ2 are the respective true average fat contents of the two kinds of ice cream) against the alternative hypothesis μ1≠ μ2 at the level of significance α =0.05. 7.a) b) c) d) 8. Estimate y at x=25 given that N=33 Σxi = 1104, Σyi = 1124, Σxi yi = 41355, = 41086. Determine a 95% confidence interal for α, β. Test the hypotheris β = 1.0 against β < 1.0 Test the hypothesis that α = 0 against α ≠ 0 at 0.05 level of significance. Find the correlation coefficient for the following bivariate distribution. 20-24 25-39 30-34 35-39 40-44 20-24 20 4 25-29 10 28 5 30-34 3 6 11 2 35-39 2 4 5 @@@@@ Σxi2 Set No. Code No: 220105 II B.Tech., II-Semester Supplementary Examinations, November-2003 PROBABILITY AND STATISTICS (Common to Civil Engineering, Production Engineering, Computer Science and Engineering, Chemical Engineering, Computer Science and Information Technology, Computer Science and Systems Engineering, and Electronics and Computer Engineering) Time: 3hours Max.Marks:80 Answer any FIVE questions All questions carry equal marks ---1. A problem in statistics is given to the three students A,B,C, whose chances of solving it are ½, ¾ ,and ¼ respectively. What is the probability that the problem is solved? 2. When the American league and national league champions are evenly matched, the probabilities that a world series will end in 4, 5, 6 or 7 games are, respectively, 1 1 5 5 , , and . What is the expected length of a world series when the two 8 4 16 16 teams are evenly matched? A continuous random variable X has the distribution function F(x) = 0 if x≤1 4 = k(x-1) if 1 < x ≤ 3 = 1 if x > 3. Find (i) k and (ii) the probability density function of X. A random sample of size 12 is taken from a normal population N (µ, 9) Find the probability that the variance of the sample lies between 3.4 and 14.8. (n − 1) σ 2 2 b) Show that E(s ) = where s2 is the variance of samples of size n from N n (µ, σ2) population . A leakage test was conducted to determine the effectiveness of a seal designed to keep the inside of a plug airtight. An air needle was inserted in the plug and this was placed under water. The pressure was then increased until leakage was observed. Let X equal the pressure in pounds per square inch. Assume that the distribution of X is N (μ, σ2). Using the following n = 10 observations of X: 3.1 3.3 4.5 2.8 3.5 3.5 3.7 4.2 3.9 3.3 (i) Find a point estimate of μ. (ii) Find a point estimate of σ. (iii) Find a 95% one-sided confidence interval for μ that provides an upper bound for μ. Contd…..2. 4 3. 4 a) 5.a) Code No: 220105 b) :: 2 :: Set No:4 When researching ground water it is often important to know the characteristics of the soil at a certain site. For example, let X equal the diameter of an individual grain of soil and assume that the distribution of X is N (μ,σ2) . The diameters of n = 30 individual grains of sand are given as: 1.24 1.36 1.28 1.31. 1.35 1.20 1.39 1.35 1.41 1.31 1.28 1.26 1.37 1.49 1.32 1.40 1.33 1.28 1.25 1.39 1.38 1.34 1.40 1.27 1.33 1.36 1.43 1.33 1.29 1.34 Find a 90% confidence interval for μ. Inspecting ceramic tiles prior to their shipment, a quality control engineer detects 2, 3, 6, 0, 4, and 9 defectives in six cartons, each containing 144 tiles, What can we assert with a probability of 0.99 about the possible size of his error, if he uses the mean of this sample to estimate the true average number of defective tiles per carton? An Oceanographer wants to check whether the average depth of the ocean in a certain region is 57.4 fathoms, as had previously been recorded. What can he conclude at the level of significance α = 0.05 if soundings taken at 40 random locations in the given region yielded a mean of 59.1 fathoms with a standard deviation of 5.2 fathoms? Estimate y at x=2.25 by fitting an indifference curve of the form xy=Ax + B to the following data. x y 1 3 2 1.5 3 6 4 7.5 6.a) b) 7. 8. Find the correlation coefficient for the following bivariate distribution. 15-25 25-35 35-45 45-55 55-65 65-75 15-25 1 2 1 25-35 1 12 4 35-45 1 10 3 45-55 1 6 2 55-65 65-75 1 4 1 2 2 @@@@@