Business Statistics

Q.1 Ans

 
Public transportation    Automobile    d[X-32]    d²    d1[X-32]    d1²       
28    29    -4    16    -3    9       
29    31    -3    9    -1    1       
32    33    0    0    +1    1       
37    32    5    25    0    0       
33    34    1    1    +2    4       
25    30    -7    49    -2    4       
29    31    -3    9    -1    1       
32    32    0    0    0    0       
41    35    9    81    +3    9       
34    33    2    4    +1    1       
= 320    = 320        = 194        =30     

(a)   Compute the sample mean time to get to work for each option.

(b)   Compute the sample standard deviation for each option.

(c)   Which method of transportation is more consistent?

Q.2 Ans.
 
Salesman    Average number of calls per day    (X)    X – 9.5
     x    X²    4
fx       
A    8    -1.5    2.25    5.0625       
B    10    +0.5    0.25    0.0625       
C    12    +2.5    6.25    39.0625       
D    15    5.5    30.25    915.0625       
E    7    -2.5    6.25    39.0625       
F    5    -4.5    20.25    410.0625       
Total    57        65.5    1408.375     
 
(a)   Compute a measure of skewness. Is the distribution symmetrical?

                                 
(b)   Compute a measure of kurtosis. What does this measure mean?

Q.3 
Ans

 
Years of service    Income (Rs. In 1000’s)    [X-8]
dx    dx²    [Y-8]
dy    dy²    dxdy       
11    10    +3    9    +2    4    +6       
7    8    -1    1    0    0    0       
9    6    +1    1    -2    4    -2       
5    5    -3    9    -3    9    +9       
8    9    0    0    +1    1    0       
6    7    -2    4    -1    1    +2       
10    11    +2    4    +3    9    +6       
56    56        28        28    21     

                                     
(a)   What is the degree of association (correlation) between years of service and income?

(b)   Find the regression equation of income on years of service.

(c)   What initial start would you recommend for a person applying for the job having served in a similar capacity in another company for 13 years?

5. Answer any three of the following.
a)   Define Statistics. What are the important functions and limitations of Statistics? 

c) What are the various types of Correlation?

d) What are the various components of a time series?

Case study 
ACC – A pioneer in the Indian cement industry

 
Year    Sales
(in million rupees)(X)    Advertisement
(in million rupees) (Y)    [X-33000]
dx    dx²    [Y-200]
dy    dy²    dxdy       
1995    20,427    58    -12,573    158080329    -142    20164    1785366       
1996    23,294    72    -9706    94206436    -128    16384    1242368       
1997    24,510    122    -8490    72080100    -78    6884    662220       
1998    23,731    61    -9269    85914361    -139    19321    1288391       
1999    25,858    144    -7142    51008164    -56    3136    399952       
2000    26,792    132    -6208    38539264    -68    4624    422144       
2001    29,361    172    -3639    13242321    -28    728    101892       
2002    32,260    184    -740    547600    -16    256    11840       
2003    33,718    259    +718    515524    +59    3481    42362       
2004    39,003    334    +6003    36036009    +134    17956    804402       
2005    45,498    321    +12498    156200004    +121    14641    1512258       
2006    37,235    336    +4235    17935225    +136    18596    575960       
2007    64,680    442    +31680    1003622400    +242    58564    7666560       
Total    4,26,367    2,637    -2,633    1727927737    +37    183835    16515715     

1. Develop an appropriate regression model to predict sales from advertisement. 

2. Calculate the coefficient of correlation and state its interpretation.

3. Predict the sales when advertisement is Rs. 500 million.

1. The algebraic sum of the deviations from mean is:  

Options    
Maximum    
Minimum    
Zero    
None of the above

2. The arithmetic mean of the first n natural numbers 1, 2, ……,n is:    

Options    
n/2    
(n+1)/2    
n(n+1)/2    
None of the above

3. Which of the following relationship is true for a asymmetrical distribution:    

Options    
mean – mode = 3(mean – median)    
mode = 3medain – 2mean    
3medain = 2mean + mode    
All of the above

4. If the mean and coefficient of variation of a set of data is 10 and 5 respectively, then the standard deviation is:    

Options    
10    
50    
5    
None of the above

5. If the first and third quartiles are 22.16 and 56.36 respectively, then the quartile deviation is:    

Options    
17.1    
34.2    
51.3    
None of the above

6. The relationship between mean deviation and quartile deviation is:    

Options    
MD = 5/6 QD    
MD = 6/5 QD    
MD = 4/5 QD    
MD = 5/4 QD

7. If the mean deviation is 8, then the value of the standard deviation will be:    

Options    
15    
12    
10    
None of the above

8. If quartile deviation is 8, then the value of standard deviation will be:    

Options    
12    
16    
24    
None of the above    

9. If events are mutually exclusive, then:    

Options    
their probabilities are less than one    
their probabilities sum to one    
both events cannot occur at the same time    
both of them contain every possible outcome of an experiment.

10. Posterior probabilities for certain events are equal to their prior probabilities provided:    
Options    
all the prior probabilities are less than zero.    
Events are mutually exclusive    
Events are statistically independent    
None of the above

11. What is the probability that a value chosen at random from a population is larger than the median of the population?    

Options    
0.25    
0.50    
0.75    
1

12. Bayes’ theorem is useful in    

Options    
Revising probability estimates    
Computing conditional probabilities    
Computing sequential probabilities    
None of the above
13. A probability of getting the digit 2 in a throw of unbiased dice is    

Options    
0    
½    
1/6    
¾

14. A bag contains 3 red, 6 white and 7 blue balls. If two balls are drawn at random, then the probability of getting both white balls is    

Options    
5/40    
6/40    
7/40    
14/40

15. What is the probability of getting more than 4 in rolling a dice?    

Options    
1/6    
1/3    
½    
1

If the outcome is an odd number when a die is rolled, then the probability that it is a prime number is    

16. Options    
1/3    
2/3    
1/6    
5/6

17.      

Options    
Independent    
Dependent    
Equally likely    
None of the above    

18. 

Options    
0.10    
0.90    
1.00    
0.75

19. In a binomial distribution if n is fixed and p > 0.5, then    

Options    
The distribution will be skewed to left    
The distribution will be skewed to right    
The distribution will be symmetric    
Cannot say anything

20. The binomial distribution is symmetric when    
Solve by solvezone 

Options    
p < 0.5    
p > 0.5    
p = 0.5    
p has any value

21. The standard deviation of the binomial distribution is:

Options    
np    
√np
npq    
√npq

22. All normal distribution are    

Options    
Bell shaped    
Symmetrical    
Defined by its parameter    
All of the above

23.  
Options    
2    
5    
10    
15    

24. For a standard normal probability distribution, the mean µ and standard deviation are:    

Options    
µ = 0 , s = 1    
µ = 16 , s = 4    
µ = 25 , s = 5    
µ = 100 , s = 10

25. For a normal distribution if mean is 30, then its mode value is    

Options    
15    
30    
50    
None of the above

26. Which of the following is a necessary condition for using a t distribution?    

Options    
Small sample size    
Unknown population standard deviation    
Both (a) and (b)    
Infinite population

27. Sampling distribution is usually the distribution of    

Options    
Parameter    
Statistic    
Mean    
Variance

28. The process of selecting a subset of a population for a survey is known as    

Options    
Survey research    
Representation    
Triangulation    
Sampling

29. What is sampling for groups with considerable variation but similar to each other called?    
Options    
Cluster    
Stratified    
Systematic    
Random

30. If the relationship between x and y is positive, as variable y decreases, variable x    

Options    
Increases    
Decreases    
Remains same    
Changes linearly    
31. The line of best fit to measure the variation of observed value of dependent variable in the sample data is        

Options    
Regression line    
Correlation coefficient    
Standard error    
None of these

32.    
 
Options    
Less than one    
More than one    
Equal to one    
None of these    

33. Linear programming is a    

Options    
Constrained optimization technique    
Technique for economic allocation of limited resources    
Mathematical technique    
All of the above

34. A constraint in a linear programming model restricts    

Options    
Value of objective function    
Value of a decision variable    
Use of the available resources    
All of the above

35. The best use of linear programming technique is to find an optimal use of    

Options    
Money    
Manpower    
Machine    
All of the above    

36. While solving a LP model graphically, the area bounded by the constraints is called    

Options    
Feasible region    
Infeasible region    
Unbounded solution    
Unbounded solution

37. A feasible solution to an LP problem    

Options    
Must satisfy all the problem’s constraints simultaneously    
Need not satisfy all the constraints, only some of them    
Must be a corner point of the feasible region    
Must optimize the value of the objective function

38. The standard deviation of first n natural numbers is:    

 

39. Two events A and B are statistically independent when 
Options    

 

40. For a standard normal probability distribution, the mean µ and standard deviation are    

Options    

µ = 0  ,  σ = 1
µ = 16  ,  σ = 4
µ = 25  ,  σ = 5
µ = 100  ,  σ = 10