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Pailan College of Management and Technology B.B.A Lessons Plan 2nd Semester 2012 – 2013 STATISTICS II Faculty Nabanita Maity Course Title B.B.A Course Code BBA-203 Objective of the subject The central objective of the undergraduate students in Statistics is to equip them with requisite quantitative skills that they can employ and build on in flexible ways. Students are expected to learn concepts and tools for working with data and have experience in analyzing real data that goes beyond the content of a service course in statistical methods. Teaching Methodology Black board , Class exercises, Assignments. Assessment of the Students will be on the basis of the following weightage :- 1. End Term Exam 70 2. Attendance 5 3. Assignment 5 4. Mid Term. I Exam 20 5. Mid Term. II Exam 20 Total best of two 100 Marks Text Books N.G.Das (Volume – 2) , Business Statistics (Kalyani publication). Reference books A.M Goon, M.K Gupta & B, Dasgupta : Basic Statistics : World Press. G. C. Beri – Business Statistics, Second Edition: Tata McGraw- Hill Course Duration 40 lectures. Time per 1 hour each Lecture 1 Module 1 Detailed of Course to be covered Topic Theory of Probability: Probability as a concept; Basic probability rules; Tree diagrams; Conditional probability; Mutually exclusive events and independent events; Bayes’ Theorem or Inverse probability rule No. of Lectures. 10 Assignment Questions: 1. State and prove : a) The theorem of Total Probability (for both m.e. and not m.e. events) b) Compound Theorem c) Baye’s Theorem 2. What do you mean by independent events. 3. If A and B are independent events ; prove that a) A and Bc are also independent b) Ac and B are also independent c) Ac and Bc are also independent 4. 40% of the students in a class are girls.If 60% and 70% of the boys and girls respectively of the class pass a certain test, then what is the probability that a randomly selected student from the class will have passed the test. 5. Three identical urns I, II and III contain 4 white and 3 red balls , 3 white and 7 red balls and 2 white and 3 red balls respectively.An urn is chosen at random and a ball is drawn out from it.If the ball is found to be white , what is the probability that urn II was selected. 6. If A and B are independent events and P(A) = 2/3 , P(B) = 3/5, then find P(A+B) , P(AcB) , P(Ac/B) 7.The probability that a contractor will get a plumbing contract is 2/3 and that he will not get an electric contract is 5/9.If the probability of getting atleast one contract is 4/5 then what will be the probability that he will get both the contracts. 8. A picnic is arranged to be held on a particular day.The weather forecast says that there is 80% of rain on that day.If it rains the probability of a good picnic is 0.3 and if it does not the probability is 0.9.What is the probability that the picnic is good. 9. Three cards are drawn one after another from a full pack of playing cards.What is the probability that i) the first two are spades and the third is a heart. ii) two are spades and one is heart. 10. If A and B are two events prove that P(A/B) = 1 – P(Ac/B) , P(B)>0 11. two bags contain respectively 3 white and 2 red balls ,2 white and 4 red balls.One ball is drawn at random from the first bag into the second, then a ball is drawn from the second bag.What is the probability that the ball is drawn from the second bag. 2 Module 2 Detailed of Course to be covered No. of Lectures Discrete and Continuous random variables; Expectation value; Mean and Variance of a Random Variable; Theorems on expectation; Marginal and joint probability distributions. 6 Topic Probability distribution of a Random Variable Assignment Questions : 1. What do you mean by p.m.f. and p.d.f. 2. The random variable X has the following p.m.f X 0 1 2 P(X =i) 0 K 2k Determine the constant k and find P(X>6) 3 2k 4 3k 5 k2 6 2k2 7 7k2+k 3. With the usual notation prove that i) E(X+Y) = E(X) +E(Y) ii) E(XY) = E(X).E(Y) when X and Y are independent events. 4. If X be a random variable and ‘a’ is a constant , then prove that E[aΨ(X)] = aE[Ψ(X)] 5. Let X be a random variable with the following probability distribution: X P(X=x) -3 1/6 6 ½ 9 1/3 Find [(2X+1)2] 6. A random variable X has the density function f(x) given by f(x) = a/(x2 +1) , -∞ <x < ∞ Find i) the constant a. ii) the probability that x2 lies between 1/3 and 1. 7.The diameter of an electric cable say X is assumed to be a continuous random variable with p.d.f f(x) = 6x(1-x), 0< x <1 i) Check that f(x) is a p.d.f ii) determine a number b such that P(X<b) = P(X>b). 8. A continuous random variable X follows uniform distribution with p.d.f f(x) = ½ Find the probabilities: i) P(4 <x< 5) ii) P(x≤ 4.2) iii) P(x ≥ 5.5) 9. If X is a random variable , then prove that V(aX + b) = a2 V(X) 10. A random variable X has the density function f(x) given by f(x) = ½ - ax , 0 ≤x≤ 4 where ‘a’ is a constant i) Determine the value of ‘a’ and the probability that X lies between 2 and 3. ii) Calculate the probabilities : P(X ≥ 2.5) , P(-1.5 <X ≤3) and E(X). 11. A random variable X has mean m and SD σ , show that i) E (X-m / σ) = 0 ii) E (X-m / σ)2 = 1 3 (4 ≤ x ≤ 6) Module 3 Topic Theoretical Probability Distributions Detailed of Course to be covered No. of Lectures Probability mass function and density function; Discrete distributions – The Binomial distribution and its properties; Idea of geometrical and hypergeometric distributions. The Poisson distribution and its properties; Fitting a Binomial or Poison distribution to an observed distribution. 5 Continuous distributions –Uniform, Exponential and Normal distributions; Normal approximation to Binomial and Poisson distributions; Fitting a normal curve to an observed distribution 4 Assignment Questions 1. A discrete random variable X follows Poisson distribution such that P( X=1) = P(X=2). Find the variance of X. 2. The mean and the S.D. of a Binomial distribution are 20 and 6 respectively.Prove that these values are inconsistent. 3. A pair of dice is thrown 200 times.If getting a sum of 9 is considered a success, find the variance of the number of successes. 4. Deduce the mean and variance of i) Binomial Distribution ii) Poisson Distribution. 5. If a discrete random variable X follows Binomial Distribution with mean 5/3 and P(X =2) =P(X=1). Find P(X=atmost1) 6. A radioactive source emits on the average 2.5 particles per second. Calculate the probability that 2 or more particles will be emitted in an interval of 4 seconds. 7. If X is normally distributed with mean 3 and S.D. 2 find c such that P(X>c) =P(X≤ c). Given that ∫-α0.43 φ(t) dt = 0.6666. 8. If X is normally distributed with mean 18 and S.D. 5 , find the value of P(-31 < X < 67) and P( X <67 /X >18) Given that Ф (1.96) = 0.9750021 9. Using the formula for binomial distribution, find the probability of rolling atmost 2 sixes in 5 rolls of a dice. 4 Module 4 Detailed of Course to be covered Topic Sampling and Sampling Distributions Sampling versus complete enumeration;Random and nonrandom sampling; Different types of random sampling; Sample Statistic and Population Parameter; Practical methods of drawing a random sample. No. of Lectures 6 Sampling distributions – Standard error; sampling distribution of the sample mean and the sample proportion. Sampling from normal and non-normal populations; The Central Limit Theorem. Four Basic Distributions: Standard normal distribution; Chisquare distribution; t-distribution; F-distribution Assignment Questions 1. What are the advantages of sampling over census? 2. What are the different methods of sampling. 3. A simple random sample of size 36 is drawn from a finite population consisting of 101 units. If the population sd is 12.6 find the SD(sample mean) when sample is drawn i) with replacement ii) without replacement. 4. The values of a characteristic X of a population containing six units are given below – 2,6,5,1,7,3. Take all possible samples of size two and verify that the mean of the population is exactly equal to the mean of sample mean. 5. A population consists of four members : 3,7,11,15. Take all possible samples of size two drawn with replacement. Find i) population mean ii) population SD iii) mean of the sampling distributions of mean. 6. State and prove the Central Limit theorem. 7. The ages of five persons are recorded as 14, 17,19,20,25. For random samples of size 3 drawn without replacement from this population show that the mean of the sample means is equal to the population mean. 5 Module 5 Detailed of Course to be covered Topic Estimation: point and interval estimation No. of Lectures Criteria of a good estimator; Methods of Point Estimation – The Method of Maximum Likelihood and The Method of Moments; Interval Estimates – Interval estimates and confidence intervals; confidence level and confidence interval; Calculating interval estimates of the mean and proportion from large samples; Finite correction factor. Interval estimates using the t distribution Determining the sample size in Estimation 6 Assignment Questions 1. What are the criteria of a good estimator? 2. What do you mean by an unbiased estimator? 3. Random samples of shirts from a consignment of 1000 shirts have an average price of Rs 140 and sd of Rs 12.50.Find 95% confidence interval for the average price of these 100 shirts. 4. Find the maximum likelihood estimates for population having Binomial distribution. 5. Show that the sample mean is consistent and unbiased estimate of the population mean but sample variance is consistent and biased estimate of population variance. 6. If T1 and T2 be statistics with expectations E(T1) = 2θ1+3θ2 and E(T2) = θ1+θ2. Find the unbiased estimators of parameters of θ1 and θ2. 7. A random samples of heights of 100 students from a large population of students is drawn. The average height of the students in the sample is 5.6 feet while the SD is 0.75 feet. Find i) 95% and ii) 99% confidence limits for the average height of all the students in the population. 6 Module 6 Detailed of Course to be covered No. of Lectures Concepts basic to the hypothesis testing procedure; Steps in Hypothesis testing; Type I and Type II errors; Two-tailed and one-tailed tests of hypotheses. Hypothesis testing of means when the population standard deviation is known / not known; Power of a Hypothesis Test; Hypothesis testing of proportions; Use of the t- distribution. Hypothesis testing for differences between means and proportions; two-tailed and one- tailed tests. 10 Topic Hypotheses Testing Assignment Questions 1. Define: Null hypothesis , Alternative Hypothesis , Level of Significance , Critical region , Type I error , Type II error. 2. What are the steps in Hypothesis Testing. 3. A random sample with observations 65,71,64,71,70,69,63,67,68 is drawn from a normal population with SD √7.056 . test the hypothesis that the population mean is 69 at 1% level of significance. 4. Random samples of 400 men and 600 women were asked whether they would like to have a flyover near their residence.200 men and 325 women were in favor of the proposal. Test the hypothesis that the proportions of men and women in favor of the proposal are same against that they are not at 5% level..Use test statistic Z . 5. The heights of 10 males of a given locality are found to be 70,67,62,68,61,68,70,64,64 and 66 inches.Is it reasonable to believe that the average height is greater than 64 inches?Test at 5% level.(t at 5% level is 1.833). 6. Random samples of size 500 and 400 have means 11.5 and 10.9 respectively.can it be regarded as drawn from the same population of SD 5?Find 99% confidence limits for the difference of means. 7. A certain diet newly introduced to each of 12 pigs resulted in the following increase of body weight : 6,3,8,2,3,0,-1,1,6,0,5,4. Can you conclude that the diet is effective in increasing the weight of the pigs? ( given t0.05 ,11 = 2.200 8. The average number of defective articles per day in a certain factory is claimed to be less than the average for all the factories. The average for all the factories is 30.5.A random sample of 100 days showed the following distribution: Class limits 16-20 21-25 26-30 31-35 36-40 No.of days 12 22 20 30 16 Test if the average is less than the figures for all factories at 5% leve. (Use Z test) 7 Total 100 Module 7 Topic Chi-Square and Analysis of Variance Detailed of Course to be covered No. of Lectures Chi-Square as a test of independence and as a test of goodness of fit. Analysis of Variance: Calculating the variance among the samples and within the samples. The F distribution and the F hypothesis test. 6 Assignment Questions : 1. Book – N.G.Das ; page no :- 287 , Question no :- 66 ,67, 68 ,69 ,70 ,71 ,72 ,73 ,74 ,75. 2. Book – N.G.Das ; page no :- 311 ,312 , Question no :-5 ,6 ,7 ,8 ,9 ,10 ,11. 8