IGNOU AST-01 - Statistical techniques

1. a) The distribution of a discrete random variable X is given as : Find : (i) The value of K (ii) P (X >2) (iii) E (X) (iv) Draw a rough sketch of the distribution. b) For overall quality improvement of cloth, a textile manufacturer decides to monitor the number of defects in each bolt of cloth. The data from 10 inspections is reported as follows : Draw an appropriate control chart to check whether the process is under statistical control. 2. a) Three dice are rolled. If no two dice show the same number on the top face, what is the probability that one dice shows ‘1’ on the top face? b) If 2% of the books bound at a certain binary have defective bindings, determine the probability that 5 of 400 books bound by this binary will have defective bindings. [Given that e-8 = 0.00034] 3. a) Out of 20,000 customers’ leader accounts, a sample of 600 accounts was taken to test the accuracy of posting and balancing wherein 45 mistakes were found. Determine the 95% confidence interval for number of defective cases. b) In 16 one-hour test runs, the gasoline consumption of an engine on an average is found to be 16⋅ 4 gallons with a standard deviation of 2⋅1 gallons. At 5% level of significance, test the claim that the average gasoline consumption of this engine is 12⋅0 gallons per hour c) A sample of 100 employees is to be drawn from a population of collages A and B. The population means and population mean squares of their monthly wages are given below: Determine the sample size to be taken using (i) proportional allocation, and (ii) Neyman allocation methods. 4. a) A company has sales (y) which shows profit on the cost of production (x) of furniture when sold to government agencies. The regression lines are y = x + 5 and 16x = 9y + 95. Find : (i) Mean of sale and production cost. (ii) Correlation coefficient between sales and production cost. (iii) If σy = 4, find σx. (iv) Predict the production cost when sale is 2,000. b) An oil company claims that less than 20% of all the car owners have not tried its oil. Test the claim at the 0⋅01 level of significance if a random check reveals that 22 of 200 car owners not tried its oil. 5. a) The odds that a research monograph will be accepted by 3 independent referees are 3 to 2, 4 to 3 and 2 to 3, respectively. Find the probability that of the three reports : (i) all will be accepted. (ii) more than 1 of the reports will be favorable. b) A company installed 10000 electric bulls in a metro. If these bulbs have an average life of 1000 hours with S. D of 200 hours. Assuming normality, what number of bulls might be expected to fail (i) in the first 800 hours (iii) between 800 and 1200 hours. c) Cite two situations where systematic sampling is appropriate. Explain, how it is different from stratified sampling. Justify. 6. a) A random sample is selected from each of three types of ropes and their breaking strength (in pounds) are measured with the following results : Test whether the breaking strength of the ropes differs significantly at 5% level of significance. b) Works out the 5-yearly moving average for the data of a number of commercial industrial failures in a country during 7985 to 2000: 7. a) A cigarette company interested in the effect of gender on the type of cigarettes smoked and has collected the following data from a random sample of 150 persons : At 95% level of significance, test whether the type of cigarette smoked and gender are independent. b) The lifetime (in ‘000 hours) of five LED bulbs of 10 watts are 46, 40 48, 50 and 42. Draw all possible random samples of size 3 without replacement and estimate the average lifetime of the LED bulbs. 8. a) Following frequency distribution presents the lifetimes of 200 incandescent lamps : Draw the histogram, ‘more than’ and ‘less than’ ogive. Also, find the median from the ogives. b) Draw all the possible samples of sizes 2 without replacement from the population {8, 12, 20} and verify that the sample mean is the unbiased estimator of population mean. Find the variance of the estimate the population mean. 9. a) Suppose that a testing procedure A results in 20 unacceptable transistors out of 100 produced whereas another testing procedure B results in 12 unacceptable transistors out of 100 produced. Can we conclude at 5% level of significance that the two methods are equivalent? b) Two different sampling techniques were adopted while investigating the same group of students to find the number of students falling in different intelligence level. The results are tabulated as follows : At 5% of level of significance, test whether the sampling techniques adopted significantly different ? 10. Which of the following statements are True and False? Justify your answers. a) If bxy = 0⋅9 and byx = −0⋅9, then r = 0⋅81. b) If a pair of dice is thrown, then the probability that the sum of the number on the top faces is greater than 12 is 0. c) If the probability of being left-handed is 0⋅1, then the probability that none of the 3 persons selected randomly is left-handed is 0⋅729. d) Analysis of variance is a technique to test the equality of variation in several populations. e) If the level of significance is the same, the area of the rejection in a 2-tailed test is less than in a 1-tailed test.