Download accounting for managers - Pailan College of Management and

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