Sampling Methods
1 Where items are included in a sample, based on the judgement of the person who is selecting the sample, it is called non-random.
(True/False)
2 The statistic is a characteristic of a population (True/False)
3 A sampling plan that selects members from a population at uniform intervals in time order or space is called stratified sampling.
(True/False)
4 As a general rule, it is not necessary to include a finite population multiplier in a computation for standard error of the mean when
the size of the samples is greater than 50. (True/False)
5 The probability distribution of all the possible means of samples is known as the sample distribution of the mean. (True/False)
6 The principles of simple random sampling are the theoretical foundation of statistical inference. (True/False)
Z The standard error of the mean is the standard deviation of the distribution of sample means (True/False)
8 A sampling plan that divides the population into well defined groups from which random samples are drawn is called as cluster
sampling. (True/False)
9 With increasing sample size, the sampling distribution of the mean approaches normality, regardless of the distribution of the
population (True/False)
10 The standard error of the means decreases in direct proportion to sample size. (True/False)
11 To perform a complete enumeration, one would need to examine every item in a population (True/False)
12 In everyday life, we see many examples of infinite populations of physical objects. (True/False)
13 To obtain a theoretical sampling distribution, we consider all the samples of a given size. (True/False)
14 large samples are always a good idea because they decrease the standard error. (True/False)
15 if the mean for a certain population were 15, it is likely that most of the samples we could take from that population would
have means of 15 (True/False)
16 the precision of a sample is determined by the no. of items in the sample and not the proportion of the total population that is
sampled. (True/False)
17 the standard error of a sample statistic is the standard deviation of its sampling distribution. (True/False)
18 judgement sampling has the disadvantage that it may lose some representativeness of a sample (True/False)
19 the sampling fraction compares the size of a sample to the size of the population. (True/False)
20 Any sampling distribution can be totally described by its mean and standard deviation. (True/False)
21 the precision with which the sample mean can be used to estimate the population mean decreases as the standard error
increases. (True/False)
22 Which of the following is a method of selecting samples from a population
a: judgement sampling b: random sampling c: probability sampling d: a and b but not c
23 Choose the pair of symbols that best completes this sentence : is a parameter, whereas is a statistics:
a: N, p b: a, S c: N, n d: all the above
24 In random sampling, we can describe mathematically how objective our estimates are. Why is this ?
a: we always know the chance that any population element will be included in the sample
b: every sample always has an equal chance of being selected
c: all the samples are exactly the same size and can be counted
d: none of the above e: a and b but not c
25 Suppose you are performing stratified sampling on a particular population and have divided it into strata of different sizes. How
· can you make your sample selection?
a: • select at random an equal no. of elements from each stratum
b: draw equal no. of elements from each stratum and weigh the results
c: draw no. of elements from each stratum proportional to their weights in the population
d: a and b only e: b and c only
26 In which of the following situations would a g=cr/Vn be the correct formula to use for computing:
a: sampling is from an infinite population
b: sampling is from a finite population with replacement
c: sampling is from a finite population without replacement
d: a and b only e: b and c only.
27 The dispersion among sample means is less than the dispersion among the sampled items themselves because:
a: each sample is smaller than the population from which it is drawn.
b: very large values are averaged down and very small values are averaged up
c: the sampled items are all drawn from the same population
d: none of the above e: b and c and not a.
28 Suppose a population with N = 144 has p = 24. What is the mean of the sampling distribution of the mean for samples of size 25:
a: 24 b: 2 c: 4.8 d: cannot be determined from the information given
29 The central limit theorem assures us that the sampling distribution of the mean :
a: is always normal
b: is always normal for large sample sizes
c: approaches normality as sample size increases
d: appears normal only when Nis greater than 1000
30 Suppose that for a certain population a it is calculated as 20 when samples of size 25 are taken and as 10 when samples of size 100 are
taken. A quadrupling of sample size, then, only halved a R. We can conclude that increasing sample size is:
a: always cost-effective b: sometimes cost•ef€ective
c: never cost-effective d: none of the above
31 For the above questions, what must the value of a for this infinite population
a: 1000 b: 500 c: 377.5 d: 100
32 The finite population multiplier does not have to be used when the sampling fraction is :
a: greater than 0.05 b: greater than 0.50 c: less than 0.50
d: greater than 0.90 e: none of the above
33 The standard error of the mean for a sample size of two or more is:
a: always greater than the standard deviation of the population
b: generally greater than the standard deviation of the population
c: usually less than the standard deviation of the population
d: none of the above
34 A broader patrol check point that stops every passenger van is using:
a: simple random sampling b: systematic sampling c: stratified sampling d: complete enumeration
35 In a normally distributed population, the sampling distribution of the mean:
a: is normally distributed
a: has a mean equal to the population mean
b: has a standard deviation equal to the population standard deviation divided by the square root of the sample size
c: both a and b
36 The central limit theorem:
a: requires some knowledge of the frequency distribution
b: permits us to use sample statistics to make inferences about population parameters
c: relates the shape of a sampling distribution of the mean to the mean of the sample
d: requires a sample to contain fewer than 30 observations.
Fill-in the blanks:
37 A portion of the elements in a population chosen for direct examination or measurement is a
38 the proportion of the population contained in a sample is the
39 is the process by which inferences about a population are made from information about a sample
40 The is the distribution obtained by finding the sampling distribution of all samples of a given size of a population.
41 sampling should be used when each group considered has small variation within itself but there Is wide variation between
different groups.
42 A method of random sampling in which elements are selected from the population at uniform intervals is called sampling.
43 ________is the degree of accuracy with which the sample mean can estimate the population mean.
44 Within a population, groups that are similar to each other (although the groups themselves have wide internal variation) are called
45 A sampling distribution of the proportion is a probability distribution of the
State whether the following is true or false
46: A rupee to be received in future has less value than a rupee available today (T/F)
47: Money has been invested in capital of a firm instead of investment in bonds. The return on bonds is called opportunity cost
of capital (T/F)
48: When there are inflationary conditions the discount rates are low (T/F)
49: Present value of a perpetuity is expressed as = 1/r (T/F)
46: The difference between cash outflows and present value of inflows is called net present value
(T/F) Answers
01 T 02 F 03 F 04 F 05 T
06 T 07 T 08 F 09 T 10 F
11 T 12 F 13 T 14 F 15 F
16 T 17 T 18 T 19 T 20 F
21 T 22 d 23 b 24 e 25 e
26 d 27 b 28 a 29 c 30 b
31 d 32 u 33 d Si d 35 d
36 b 37 sample 38 sample fraction 39 statistical
40 theoretical sampling distribution inference
41 stratified 42. systematic 43 precision 44 cluster 45 sample
46 T 47 T 48 F 49 T 50 T proportion
MCQs ON CORRELATION & REGRESSION
1 Coefficient of correlation is always , according to Karl Pearson:
a: plus or minus 1 b: plus 1
c: minus 1 d: 0
2 r = { xy /N ax ay refers to:
a: price index b: standard deviation
c: coefficient of correlation d: mean deviation
3 Where there is no correlation, the coefficient of correlation will be:
a: +1 b: -1
c: both d: none of the above
4 If two variables changes in the same ratio and in the upward direction, the correlation is said to be:
a: perfectly negative
a: perfectly positive
a: zero
b: negative
5 +1 refers to the coefficient of correlation being:
a: no correlation
b: negative correlation
a: positive correlation
c: perfect positive correlation
6 Correlation means which of the following:
a: measuring relationship between two variables
b: measures changes in prices
c: measuring average correlation between the variables
d: prediction
7 If r M -1, this means that there is:
e: positive correlation
b: negative correlation
b: perfectly positive correlation
c: perfectly negative correlation
8 If correlation is between + 0.9 and + 1, it means there is:
a: high degree of positive correlation
b: high degree of negative correlation
c: perfect degree of positive correlation
d: perfect degree of negative correlation
9 If the value of two series move in the opposite direction, there is:
a: perfect correlation b: positive correlation
c: negative correlation d: lack of correlation
10 Scatter diagram is used to :
a: determine the extent of correlation
b: know the direction of correlation
c: estimate the extent of rank correlation
d: none of the above
11 What is the correct value of coefficient of correlation:
a: there is nn limit of it
a: it is between plus or minus 1
b: it is greater than 1
c: it is less than 1
12 There is positive correlation between X and Y when:
a: X and Y both increase
b: X and Y both decrease
c: X increases and Y falls
d: Y increases and X falls
13 The probable error of correlation is:
a: 0.6745 of standard error
b: 0.6745 of correlation co-efficient
c: 0.6745 of standard deviation
d: none of the above
14 If two lines intersect each other at 90 degree angle, the co-efficient of correlation will be:
a: 0 b: -1 c: +1 d: none of the above
15 The relationship between price and supply is generally:
a: negative b: neutral c: positive d: linear
16 Between income and consumption there is generally:
a: no correlation b: positive correlation c: negative correlation d: partial correlation
17 Between ages of husbands and wives, there is generally correlation:
a: negative b: neutral c: zero d: positive
18 If two series move in the same direction, the correlation is said to be:
a: positive b: negative c: zero d: none of these
19 Study of relationship between three or more variables is known as:
a: multiple correlation b: simple correlation c: neither of the two
20 When we consider more than two variables, but consider only two variables to be influencing each other, the effect of other
influencing variable being constant, it is called:
a: multiple correlation b: partial correlation c: simple correlationd: none of the above
21 If the amount of change in one variable tends to bear constant ratio to the amount of change in other variable then the
correlation is said to be: a: linear b: non-linear c: none of the above
22 Correlation analysis deals with measurement of:
a: association between two ore more variables
b: degree of relationship between variables
c: co-variation between two or more variables
a: all the above
23 Correlation analysis tells:
a: degree of relationship between variables
b: cause and effect relationship between variables
c: none of the above
24 Correlation observed between variables that cannot conceivably be casually related is called:
a: spurious correlations
a: nonsense correlation
b: both a and b
c: none of the above
25 The study of tea and coffee is known as :
a: partial correlation b: multiple correlation c: simple correlation d: none of the above
26 When we study the relationship between the yield of rice per acre and both the amount of rainfall and fertilizers used, it
is a problem of:
a: multiple correlation b: simple correlation c: none of the above
27 The coefficient of correlation:
a: cannot be negative b: cannot be positive
c: can be positive
d: can be both positive or negative
28 Negative correlation implies that on an average as one variable is decreasing the other is:
a: decreasing b: increasing
c: neutral d: none of the above
29 If the amount of change in one variable does not bear a constant ratio to the amount of change in other variable, this
type of correlation is known as: a: linear b: non-linear c: scattered d: none of the above
30 Which is the non-mathematical method of studying the correlation between the variables:
a: scatter diagram method
b: graphic method
c: both the above
d: none of the above
31 If 'r' is more than 6 times the probable error, the correlation is:
a: negative b: positive c: significant d: none of the above
32 The coefficient of correlation describes:
a: magnitude of correlation
a: direction of correlation
b: both the above
c: none of the above
33 If sum of the product of deviations of X and Y series from their mean is Zero, the co-efficient of correlation shall be:
a: +1 b: 0 c: 1 d: none of the above
Answers
01 a 02 c 03 c 04 b 05 d
06 a 07 d 08 a 09 c 10 b
11 b 12 a 13 a 14 a 15 c
16 b 17 d 18 a 19 a 20 b
21 a 22 d 23 a 24 c 25 c
26 a 27 d 28 b 29 b 30 c
31 c 32 c 33 b
QUESTIONS ON REGRESSION ANALYSIS
1 Greater the value of r
a: better are the estimates obtained through regression analysis
b: worst are the estimates
c: really makes no difference
d: none of the above
2 When one regression co-efficient is negative, the other would be:
a: negative b: positive c: zero d: none of the above
3 If the variables are independent of each other:
a: r is zero
a: lines of regression are at right angles
b: both a and b
c: none of the above
4 Nearer the two regression lines from each other:
a: higher the degree of correlation
b: lesser the degree of correlation
c: neither a nor b
5 The more and more distance between two regression lines means:
a: higher degree of correlation
b: lesser the degree of correlation
c: none of the above
d: both the above
6 Value of correlation coefficient is:
a: 4bxy and byx
b: + or - bxy and by:c
c: none of the above
7 The regression lines cut each other at the point of:
a: average of X and Y b: Average of X only
c: Average of ? only d: none of the above
8 The regression line of Y on X:
a: minimize total of squares of the horizontal deviations b: total of the squares of the vertical deviations
c: both vertical and horizontal deviations d: none of the above
9 There will be only one regression line in case of two variables if:
a: r = 0 b: r = +1
c: r = -1 d: r is either +1 or -1.
10 The regression line of Y on X gives:
a: the most probable values of X for given value of Y b: the most probable value of Y for given value of X
c: either a or b d: none of the above
11 X on Y gives:
a: the most probable values of X for given value of Y b: the most probable values of Y for given value of X
c: any of the above d: none of the above
12 If byx = 0.6 andbxy= 1,3 then r+
a: 0.75 b: 0.73 c: 0.88 d: none of the above
13 Regression is:
a: measure of average relationship between 2 ore more variables
b: to find a relation between 2 or more variables that are related casually
c: any of the above
d: none of the above
14 Coefficient of correlation is under-root of two:
a: standard deviations b: mean deviations a regression coefficients d: none of the above
15 The farther the two regression lines cut each other:
a: the greater the degree of correlation , b. the lesser the degree of correlation c: does not matter d: none of the
above
16 Where r = 0, the regression line cut each other at:
a: an angle of 45 degree b: an angle of 60 degree c: an angle of 90 degree d: none of the above
17 The regression equation X or Y is shown as:
a: X = a + by b: Y = a + bx c: Ex --- a +.b Ey d: EXY=aN-EbEy2
18 The regression equation Y or X is shown as:
a: Y = a + bx b: Y = a + by c: Y = Na + b EX d: none of the above
19 If two regression lines are nearer to each other thEn there would be:
a: low degree of correlation
b: higher degree of correlation
c: no correlation
d: none of the above
20 If coefficient of correlation between two variables is zero, the regression lines will be:
a: coinciding each other b: parallel to each other c: perpendicular to each other d: none of the above
21 If bxy = 1.3 and byx = 1.5, r would be:
a: 1.256 b: 1.296 c: 1.315 d: 1.396
Answers
01 a 02 a 03 c 04 a 05 b
06 a 07 a 08 a 09 d 10 b
11 a 12 c 13 b 14 c 15 b
16 c 17 a 18 a 19 b 20 c 21 d
OBJECTIVE TYPE QUESTIONS ON TIME SERIES
1 Ups and downs in the economy due to trade cycles are called:
a: cyclical variations b: seasonal variations c: secular tends d: none of the above
2 Which one of the following is the indication of seasonal variation:
a: improvement in techniques b: demand of woolen clothes in winter
c: purchase of books during the beginning of the academic sessions d: none of the above
3 Y = ? in series:
a: TxS+C+1 b: T-S+C-1 c: T-SxCx1 d: TxSxCx1
4 The fluctuations which complete the whole sequence of change within the span of a year or has some pattern year after year is
known as:
a: secular trend b: seasonal variation c: cyclical fluctuation d: none of the above
5 Cyclical fluctuations are also known as ;
e: oscillatory variations b: business cycle fluctuations c: either a or bd: none of the above
6 The period of moving average is to be decided on the basis of :
a: length of the cycle b: policies of the company c: both the above d: none of the above
7 Method of [east squares is:
a: flexible in nature b: non-flexible in nature c: semi-flexible in nature d: none of the above
8 Method of least squares cannot be used to fit:
a: Gompertz curve b: logistic curve c: both the above d: none of the above
9 In shifting the trend origin, the value of 'b':
a: changes b: remains unchanged c: badly fluctuates d: either a or b
10 The moving average may constitute a satisfactory trend for a series that is of:
a: linear duration b: whose cycles are regular in duration c: both the above d: none of the
above
11 To convert an annual trend equation to a monthly basis, when original data are given as totals:
a: 'a' is divided by 12 and 'b' by 144 b: 'a' is divided by 12 and 'b' by 12
c: 'a' is divided by 144 and 'b' by 12 d: 'a' is divided by 144 and 'b' by 144
12 Changes that have taken place as a result of such forces that could not be predicted like floods etc. are known as:
a: seasonal variations b: erratic variation c: cyclical variation d: none of the above
13 Seasonal variations repeat during a period of:
a: 12 months b: 24 months c: 36 months d: 48 months
14 The most important factors covering seasonal variations are:
a: growth of population b: technological improvements c: change in fashions d: weather
15 Seasonal moments are:
a: repetitive b: predictable around the trend time c: trend value d: none of the above
16 The general movement persisting over long period is called:
a: secular trend b: seasonal trend c: trend value d: none of the above
17 The most widely used method of measuring seasonal variations is :
a: ratio to moving averages method b: ratio to trend method
c: link relative method d: method of simple averages
18 Irregular variations are caused by:
a: floods b: strikes c: wars d: all the above
19 A time series is a set of observations taken at specified time, usually at:
a: equal intervals b: unequal intervals c: only at short-term intervals d: none of the above
20 Which one of the following is a component of time series:
a: secular trend b: seasonal variations c: cyclical variations
21 In Yc = a + bx, if b is positive, it shows:
a: rising trend b: seasonal variations c: cyclical variations d: all the above
22 We find irregular variations due to:
a: lock outs b: transport bottlenecks c: floods d: all the above
23 How can be remove seasonal variations:
a: reducing prices in seasons b: introducing different products having different seasons
c: both the above d: none of the above
24 What is the sequence of the following phases in business (i) depression, ii) prosperity, iii) recovery, iv) recession):
a: ii, iv, i, iii b: iv, c: iv, i, ii, iii d: iii, ii, iv, i
25 Which period give more tangible trend:
a: shorter period b: long period c: medium period
26 The seasonal changes are always:
a: periodic but not regular b: not periodic c: periodic and regular
27 Which is the feature of prosperity:
a: inflation b; optimism c: high profits
Answers
01 a 02 b 03 d 04 b 05 c 06 a 07 a
08 c 09 b 10 c 11 a 12 b 13 a 14 d
15 c 16 a 17 a 18 d 19 a 20 d 21 a
22 d 23 d 24 a 25 b 26 c 27 d
Bond Valuation - Objective Type Questions
1.Bonds have a ____ maturity andvalue at maturity:
a.fixed, variable b. fixed, fixed c. variable, fixed d.variable, d.variable
2.Regular repayment in the form of interest on a bond is called:
a.discount, b.interest c.coupon d. dividend
3.On 8%, 5 year bond of Rs.10000, the investors gets annually, as
a. 80.00, interest, b. 800.130, discount, c. 80.00, coupon d. 800.00, coupon
4.A bond the value of which is changed into a share is called:
a.convertible bond, b.zero coupon bond, c.negative bond, d.floating rate bond
5.A bond where the interest rate can be reset is: a. convertible bond, b.zero coupon bond c. negative bond,
d.floating rate bond
6. Abond is issued at a price below its face value while on payment, the face value is paid:
a. convertible bond, b.zero coupon bond, c. negative bond d. floating rate bond
7.In which of the following bonds, no interest payment Is made:
a.convertible bond b. zero coupon bond, c. negative bond d. floating rate bond
8.The income that a bond earns from year to year Is called: a. coupon, b.yield to maturity, c. current yield, d.effective yield
9.The actual yield to an investor, instead of nominal interest, is called: a. coupon, b.yield to maturity, c. current yield, d.effective yield
10.The discount rate that makes the present value of bond's payment equal to its price is called:
a.coupon b. yield to maturity c. current yield d. effective yield
11. A risk that arises on account of change in the current market price of the bond, is known as:
a. interest risk b.yield risk c. return risk d. price risk
12.The return to an investor in case of zero coupon bond is different between the price and value:
a: sale, face
b: face, purchase
c: purchase, face
d: face, sale
13 The face value ofbond changes with current interest rates:
a: convertible bond
b: zero coupon bond
c: negative bond
d: floating rate bond
14 Coupon / (1+r), where ris compounded return =
a: future value
b: discount rate
c: present value
d: maturity value
15 To calculate present value under annuity formula, we use the following:
a: PV of investment + PV of face value
b: PV of coupon + PV of face value
c: PV of coupon + PV of investment
d: PV of interest + PV of investment
16 For a 3 year 6% bond of Rs.1000, the present value is if the compounding is annual:
a: Rs.1010.77
b: Rs.1010.91
c: Rs.1100.77
d: Rs.1017.11
17 For a 3 year 6% bond of Rs.1000, the present value is , if the compounding is semi-annual (half yearly):
a: Rs.1010.77 , b. 1010.91 , c.Rs.1100.77 , d. Rs.1017.11
18 Where the current market interest are lower that the fixed coupon rate of a bond:
a: bond is sold at a discount to face value
b: bond earns premium on face value
c: bond sells at par d. there is no effect
19 Where the current market interest rate is higher than the fixed coupon rate of a bond:
a: bond is sold at a discount to face value
b: bond earns premium on face value
c: bond sells at par
d: there is no effect
20 Where the current market rate of interest is equal to coupon rate of the bond:
a: bond is sold at a discount to face value
b: bond earns premium on face value
c: bond is traded at par d. there is no effect
Answers
01 b 02 c 03 d
04 a 05 d 06 b
07 b 08 c 09 d
10 b 11 d 12 c
13 c 14 c 15 b
16 a 17 b 18 b
19 a 20 c
Objective Type Questions on linear programming
1 The word 'linear' represents which of the following:
a: drawing lines for solving management problem b: representation of relationships by straight line.
c: solution for production problem through statistical methods d: representing the solution on straight lines.
2 Linear programming is a _______technique:
a: economics b: mathematical c: algebraic d: any
of the above
3 Linear expression in linear programming is known as:
a: linear function b: algebraic function c: objective function d: main function
4 Linear inequalities in linear programming are known as:
a: linear constraints b: linear functions c: linear relationships d: none of the above
5 The value of decision variables can be:
a: negative or positive b: zero or negative c: positive or zero d: a and b
6 The decision variable in LP that should not assume negative values are called because they represent impossible situation.
a: negative restrictions b: positive restrictions c: Non-negative restrictions d: non-negative and positive restrictions
7 The relationships to be expressed either as equations or inequalities should be:
a: positive b: straight c: cervical d: linear
8 The non-zero variables in a basic feasible solution are called:
a: surplus variables b: slack variables c: artificial variables d: basic variables
9 Positive variables that are added to the left hand side to the constraints to convert them to equalities are called:
a: surplus variables b: slack variables c: artificial variables d: basic
variables
10 Positive variables that are subtracted to the left hand side to the constraints to convert them to equalities are called:
a: surplus variables b: slack variables c: artificial variables d: basic variables
11 The original linear programming problem is called:
a: original problem b: primary problem c: dual problem d: primal problem
12 The problem that is related to original linear programming problem is called:
a: original problem b: primary problem c: dual problem d: primal problem
13 A linear programming problem shows degeneracy:
a: when a basic variable acquires a zero value
b: in the final solution either the no. of basic variables is not equal to the no. of constraints
c: the no. of fern variable does not mita! the nn. of derision variahle.s
d: any of the above
14 For formulation of a linear programming problem, which of the following is an important steps:
a: determination of unknown variables
b: identification of constraints and expressing them in linear equations
c: identification of objectives and represent them as a linear function of decision variables
d: all the above
15 Which of the following combinations are the methods employed in linear programming problems:
a: geometrical, graphical methods b: simplex method, graphical methods
c: geometrical, simplex method d: b and c both
16 Geometrical method can be used effectively when the no. of variable is :
a: minimum 2 b: maximum 2 c: maximum 5 d:
no restrictions on minimum and maximum
17 Which of the following is most effective methods under LPP:
a: graphical b: geometrical c: simplex d: all the above
18 An arithmetical step by step procedure is called:
a: linear programming b: algorithm c: iterative procedure d: none of the above
19 Simplex method is basically and essentially procedure that evaluates corner points in search of the optimal solution:
a: iterative b: continuous c: regular d: minimum
20 In Simplex method, the value of objection function is increased at each step of iteration till:
a: further possibility of increase has ended b: maximum benefit has been obtained
c: no further improvement is possible d: none of the above
21 When Simplex method is used for problems such as maximization of production, profit or contribution, such problem is called:
a: Simplex problem b: minimization problem c: maximization problem
d : a o r c
22 When Simplex method is used for problems such as minimisation of cost or time, such problem is called:
a: Simplex problem b: minimization problem c: maximization problem
d: a or c
23 In which of the following areas, the application of LP techniques can be done:
a: portfolio selection b: staffing problem c: profit planning d:all the above
24 In which of the following areas, the application of LP techniques can not be done:
a: trimming of loss b: agriculture and contract awarding c: financial mix strategy, d. none of the above
Answer
01 b 02 b 03 c 04 a 05 c 06 c 07
d 08 a 09 b 10 a
11 d 12 c 13 d 14 d 15 d 16 d 17
c 18 b 19 a 20 c
21 c 22 b 23 d 24 d
1 Where items are included in a sample, based on the judgement of the person who is selecting the sample, it is called non-random.
(True/False)
2 The statistic is a characteristic of a population (True/False)
3 A sampling plan that selects members from a population at uniform intervals in time order or space is called stratified sampling.
(True/False)
4 As a general rule, it is not necessary to include a finite population multiplier in a computation for standard error of the mean when
the size of the samples is greater than 50. (True/False)
5 The probability distribution of all the possible means of samples is known as the sample distribution of the mean. (True/False)
6 The principles of simple random sampling are the theoretical foundation of statistical inference. (True/False)
Z The standard error of the mean is the standard deviation of the distribution of sample means (True/False)
8 A sampling plan that divides the population into well defined groups from which random samples are drawn is called as cluster
sampling. (True/False)
9 With increasing sample size, the sampling distribution of the mean approaches normality, regardless of the distribution of the
population (True/False)
10 The standard error of the means decreases in direct proportion to sample size. (True/False)
11 To perform a complete enumeration, one would need to examine every item in a population (True/False)
12 In everyday life, we see many examples of infinite populations of physical objects. (True/False)
13 To obtain a theoretical sampling distribution, we consider all the samples of a given size. (True/False)
14 large samples are always a good idea because they decrease the standard error. (True/False)
15 if the mean for a certain population were 15, it is likely that most of the samples we could take from that population would
have means of 15 (True/False)
16 the precision of a sample is determined by the no. of items in the sample and not the proportion of the total population that is
sampled. (True/False)
17 the standard error of a sample statistic is the standard deviation of its sampling distribution. (True/False)
18 judgement sampling has the disadvantage that it may lose some representativeness of a sample (True/False)
19 the sampling fraction compares the size of a sample to the size of the population. (True/False)
20 Any sampling distribution can be totally described by its mean and standard deviation. (True/False)
21 the precision with which the sample mean can be used to estimate the population mean decreases as the standard error
increases. (True/False)
22 Which of the following is a method of selecting samples from a population
a: judgement sampling b: random sampling c: probability sampling d: a and b but not c
23 Choose the pair of symbols that best completes this sentence : is a parameter, whereas is a statistics:
a: N, p b: a, S c: N, n d: all the above
24 In random sampling, we can describe mathematically how objective our estimates are. Why is this ?
a: we always know the chance that any population element will be included in the sample
b: every sample always has an equal chance of being selected
c: all the samples are exactly the same size and can be counted
d: none of the above e: a and b but not c
25 Suppose you are performing stratified sampling on a particular population and have divided it into strata of different sizes. How
· can you make your sample selection?
a: • select at random an equal no. of elements from each stratum
b: draw equal no. of elements from each stratum and weigh the results
c: draw no. of elements from each stratum proportional to their weights in the population
d: a and b only e: b and c only
26 In which of the following situations would a g=cr/Vn be the correct formula to use for computing:
a: sampling is from an infinite population
b: sampling is from a finite population with replacement
c: sampling is from a finite population without replacement
d: a and b only e: b and c only.
27 The dispersion among sample means is less than the dispersion among the sampled items themselves because:
a: each sample is smaller than the population from which it is drawn.
b: very large values are averaged down and very small values are averaged up
c: the sampled items are all drawn from the same population
d: none of the above e: b and c and not a.
28 Suppose a population with N = 144 has p = 24. What is the mean of the sampling distribution of the mean for samples of size 25:
a: 24 b: 2 c: 4.8 d: cannot be determined from the information given
29 The central limit theorem assures us that the sampling distribution of the mean :
a: is always normal
b: is always normal for large sample sizes
c: approaches normality as sample size increases
d: appears normal only when Nis greater than 1000
30 Suppose that for a certain population a it is calculated as 20 when samples of size 25 are taken and as 10 when samples of size 100 are
taken. A quadrupling of sample size, then, only halved a R. We can conclude that increasing sample size is:
a: always cost-effective b: sometimes cost•ef€ective
c: never cost-effective d: none of the above
31 For the above questions, what must the value of a for this infinite population
a: 1000 b: 500 c: 377.5 d: 100
32 The finite population multiplier does not have to be used when the sampling fraction is :
a: greater than 0.05 b: greater than 0.50 c: less than 0.50
d: greater than 0.90 e: none of the above
33 The standard error of the mean for a sample size of two or more is:
a: always greater than the standard deviation of the population
b: generally greater than the standard deviation of the population
c: usually less than the standard deviation of the population
d: none of the above
34 A broader patrol check point that stops every passenger van is using:
a: simple random sampling b: systematic sampling c: stratified sampling d: complete enumeration
35 In a normally distributed population, the sampling distribution of the mean:
a: is normally distributed
a: has a mean equal to the population mean
b: has a standard deviation equal to the population standard deviation divided by the square root of the sample size
c: both a and b
36 The central limit theorem:
a: requires some knowledge of the frequency distribution
b: permits us to use sample statistics to make inferences about population parameters
c: relates the shape of a sampling distribution of the mean to the mean of the sample
d: requires a sample to contain fewer than 30 observations.
Fill-in the blanks:
37 A portion of the elements in a population chosen for direct examination or measurement is a
38 the proportion of the population contained in a sample is the
39 is the process by which inferences about a population are made from information about a sample
40 The is the distribution obtained by finding the sampling distribution of all samples of a given size of a population.
41 sampling should be used when each group considered has small variation within itself but there Is wide variation between
different groups.
42 A method of random sampling in which elements are selected from the population at uniform intervals is called sampling.
43 ________is the degree of accuracy with which the sample mean can estimate the population mean.
44 Within a population, groups that are similar to each other (although the groups themselves have wide internal variation) are called
45 A sampling distribution of the proportion is a probability distribution of the
State whether the following is true or false
46: A rupee to be received in future has less value than a rupee available today (T/F)
47: Money has been invested in capital of a firm instead of investment in bonds. The return on bonds is called opportunity cost
of capital (T/F)
48: When there are inflationary conditions the discount rates are low (T/F)
49: Present value of a perpetuity is expressed as = 1/r (T/F)
46: The difference between cash outflows and present value of inflows is called net present value
(T/F) Answers
01 T 02 F 03 F 04 F 05 T
06 T 07 T 08 F 09 T 10 F
11 T 12 F 13 T 14 F 15 F
16 T 17 T 18 T 19 T 20 F
21 T 22 d 23 b 24 e 25 e
26 d 27 b 28 a 29 c 30 b
31 d 32 u 33 d Si d 35 d
36 b 37 sample 38 sample fraction 39 statistical
40 theoretical sampling distribution inference
41 stratified 42. systematic 43 precision 44 cluster 45 sample
46 T 47 T 48 F 49 T 50 T proportion
MCQs ON CORRELATION & REGRESSION
1 Coefficient of correlation is always , according to Karl Pearson:
a: plus or minus 1 b: plus 1
c: minus 1 d: 0
2 r = { xy /N ax ay refers to:
a: price index b: standard deviation
c: coefficient of correlation d: mean deviation
3 Where there is no correlation, the coefficient of correlation will be:
a: +1 b: -1
c: both d: none of the above
4 If two variables changes in the same ratio and in the upward direction, the correlation is said to be:
a: perfectly negative
a: perfectly positive
a: zero
b: negative
5 +1 refers to the coefficient of correlation being:
a: no correlation
b: negative correlation
a: positive correlation
c: perfect positive correlation
6 Correlation means which of the following:
a: measuring relationship between two variables
b: measures changes in prices
c: measuring average correlation between the variables
d: prediction
7 If r M -1, this means that there is:
e: positive correlation
b: negative correlation
b: perfectly positive correlation
c: perfectly negative correlation
8 If correlation is between + 0.9 and + 1, it means there is:
a: high degree of positive correlation
b: high degree of negative correlation
c: perfect degree of positive correlation
d: perfect degree of negative correlation
9 If the value of two series move in the opposite direction, there is:
a: perfect correlation b: positive correlation
c: negative correlation d: lack of correlation
10 Scatter diagram is used to :
a: determine the extent of correlation
b: know the direction of correlation
c: estimate the extent of rank correlation
d: none of the above
11 What is the correct value of coefficient of correlation:
a: there is nn limit of it
a: it is between plus or minus 1
b: it is greater than 1
c: it is less than 1
12 There is positive correlation between X and Y when:
a: X and Y both increase
b: X and Y both decrease
c: X increases and Y falls
d: Y increases and X falls
13 The probable error of correlation is:
a: 0.6745 of standard error
b: 0.6745 of correlation co-efficient
c: 0.6745 of standard deviation
d: none of the above
14 If two lines intersect each other at 90 degree angle, the co-efficient of correlation will be:
a: 0 b: -1 c: +1 d: none of the above
15 The relationship between price and supply is generally:
a: negative b: neutral c: positive d: linear
16 Between income and consumption there is generally:
a: no correlation b: positive correlation c: negative correlation d: partial correlation
17 Between ages of husbands and wives, there is generally correlation:
a: negative b: neutral c: zero d: positive
18 If two series move in the same direction, the correlation is said to be:
a: positive b: negative c: zero d: none of these
19 Study of relationship between three or more variables is known as:
a: multiple correlation b: simple correlation c: neither of the two
20 When we consider more than two variables, but consider only two variables to be influencing each other, the effect of other
influencing variable being constant, it is called:
a: multiple correlation b: partial correlation c: simple correlationd: none of the above
21 If the amount of change in one variable tends to bear constant ratio to the amount of change in other variable then the
correlation is said to be: a: linear b: non-linear c: none of the above
22 Correlation analysis deals with measurement of:
a: association between two ore more variables
b: degree of relationship between variables
c: co-variation between two or more variables
a: all the above
23 Correlation analysis tells:
a: degree of relationship between variables
b: cause and effect relationship between variables
c: none of the above
24 Correlation observed between variables that cannot conceivably be casually related is called:
a: spurious correlations
a: nonsense correlation
b: both a and b
c: none of the above
25 The study of tea and coffee is known as :
a: partial correlation b: multiple correlation c: simple correlation d: none of the above
26 When we study the relationship between the yield of rice per acre and both the amount of rainfall and fertilizers used, it
is a problem of:
a: multiple correlation b: simple correlation c: none of the above
27 The coefficient of correlation:
a: cannot be negative b: cannot be positive
c: can be positive
d: can be both positive or negative
28 Negative correlation implies that on an average as one variable is decreasing the other is:
a: decreasing b: increasing
c: neutral d: none of the above
29 If the amount of change in one variable does not bear a constant ratio to the amount of change in other variable, this
type of correlation is known as: a: linear b: non-linear c: scattered d: none of the above
30 Which is the non-mathematical method of studying the correlation between the variables:
a: scatter diagram method
b: graphic method
c: both the above
d: none of the above
31 If 'r' is more than 6 times the probable error, the correlation is:
a: negative b: positive c: significant d: none of the above
32 The coefficient of correlation describes:
a: magnitude of correlation
a: direction of correlation
b: both the above
c: none of the above
33 If sum of the product of deviations of X and Y series from their mean is Zero, the co-efficient of correlation shall be:
a: +1 b: 0 c: 1 d: none of the above
Answers
01 a 02 c 03 c 04 b 05 d
06 a 07 d 08 a 09 c 10 b
11 b 12 a 13 a 14 a 15 c
16 b 17 d 18 a 19 a 20 b
21 a 22 d 23 a 24 c 25 c
26 a 27 d 28 b 29 b 30 c
31 c 32 c 33 b
QUESTIONS ON REGRESSION ANALYSIS
1 Greater the value of r
a: better are the estimates obtained through regression analysis
b: worst are the estimates
c: really makes no difference
d: none of the above
2 When one regression co-efficient is negative, the other would be:
a: negative b: positive c: zero d: none of the above
3 If the variables are independent of each other:
a: r is zero
a: lines of regression are at right angles
b: both a and b
c: none of the above
4 Nearer the two regression lines from each other:
a: higher the degree of correlation
b: lesser the degree of correlation
c: neither a nor b
5 The more and more distance between two regression lines means:
a: higher degree of correlation
b: lesser the degree of correlation
c: none of the above
d: both the above
6 Value of correlation coefficient is:
a: 4bxy and byx
b: + or - bxy and by:c
c: none of the above
7 The regression lines cut each other at the point of:
a: average of X and Y b: Average of X only
c: Average of ? only d: none of the above
8 The regression line of Y on X:
a: minimize total of squares of the horizontal deviations b: total of the squares of the vertical deviations
c: both vertical and horizontal deviations d: none of the above
9 There will be only one regression line in case of two variables if:
a: r = 0 b: r = +1
c: r = -1 d: r is either +1 or -1.
10 The regression line of Y on X gives:
a: the most probable values of X for given value of Y b: the most probable value of Y for given value of X
c: either a or b d: none of the above
11 X on Y gives:
a: the most probable values of X for given value of Y b: the most probable values of Y for given value of X
c: any of the above d: none of the above
12 If byx = 0.6 andbxy= 1,3 then r+
a: 0.75 b: 0.73 c: 0.88 d: none of the above
13 Regression is:
a: measure of average relationship between 2 ore more variables
b: to find a relation between 2 or more variables that are related casually
c: any of the above
d: none of the above
14 Coefficient of correlation is under-root of two:
a: standard deviations b: mean deviations a regression coefficients d: none of the above
15 The farther the two regression lines cut each other:
a: the greater the degree of correlation , b. the lesser the degree of correlation c: does not matter d: none of the
above
16 Where r = 0, the regression line cut each other at:
a: an angle of 45 degree b: an angle of 60 degree c: an angle of 90 degree d: none of the above
17 The regression equation X or Y is shown as:
a: X = a + by b: Y = a + bx c: Ex --- a +.b Ey d: EXY=aN-EbEy2
18 The regression equation Y or X is shown as:
a: Y = a + bx b: Y = a + by c: Y = Na + b EX d: none of the above
19 If two regression lines are nearer to each other thEn there would be:
a: low degree of correlation
b: higher degree of correlation
c: no correlation
d: none of the above
20 If coefficient of correlation between two variables is zero, the regression lines will be:
a: coinciding each other b: parallel to each other c: perpendicular to each other d: none of the above
21 If bxy = 1.3 and byx = 1.5, r would be:
a: 1.256 b: 1.296 c: 1.315 d: 1.396
Answers
01 a 02 a 03 c 04 a 05 b
06 a 07 a 08 a 09 d 10 b
11 a 12 c 13 b 14 c 15 b
16 c 17 a 18 a 19 b 20 c 21 d
OBJECTIVE TYPE QUESTIONS ON TIME SERIES
1 Ups and downs in the economy due to trade cycles are called:
a: cyclical variations b: seasonal variations c: secular tends d: none of the above
2 Which one of the following is the indication of seasonal variation:
a: improvement in techniques b: demand of woolen clothes in winter
c: purchase of books during the beginning of the academic sessions d: none of the above
3 Y = ? in series:
a: TxS+C+1 b: T-S+C-1 c: T-SxCx1 d: TxSxCx1
4 The fluctuations which complete the whole sequence of change within the span of a year or has some pattern year after year is
known as:
a: secular trend b: seasonal variation c: cyclical fluctuation d: none of the above
5 Cyclical fluctuations are also known as ;
e: oscillatory variations b: business cycle fluctuations c: either a or bd: none of the above
6 The period of moving average is to be decided on the basis of :
a: length of the cycle b: policies of the company c: both the above d: none of the above
7 Method of [east squares is:
a: flexible in nature b: non-flexible in nature c: semi-flexible in nature d: none of the above
8 Method of least squares cannot be used to fit:
a: Gompertz curve b: logistic curve c: both the above d: none of the above
9 In shifting the trend origin, the value of 'b':
a: changes b: remains unchanged c: badly fluctuates d: either a or b
10 The moving average may constitute a satisfactory trend for a series that is of:
a: linear duration b: whose cycles are regular in duration c: both the above d: none of the
above
11 To convert an annual trend equation to a monthly basis, when original data are given as totals:
a: 'a' is divided by 12 and 'b' by 144 b: 'a' is divided by 12 and 'b' by 12
c: 'a' is divided by 144 and 'b' by 12 d: 'a' is divided by 144 and 'b' by 144
12 Changes that have taken place as a result of such forces that could not be predicted like floods etc. are known as:
a: seasonal variations b: erratic variation c: cyclical variation d: none of the above
13 Seasonal variations repeat during a period of:
a: 12 months b: 24 months c: 36 months d: 48 months
14 The most important factors covering seasonal variations are:
a: growth of population b: technological improvements c: change in fashions d: weather
15 Seasonal moments are:
a: repetitive b: predictable around the trend time c: trend value d: none of the above
16 The general movement persisting over long period is called:
a: secular trend b: seasonal trend c: trend value d: none of the above
17 The most widely used method of measuring seasonal variations is :
a: ratio to moving averages method b: ratio to trend method
c: link relative method d: method of simple averages
18 Irregular variations are caused by:
a: floods b: strikes c: wars d: all the above
19 A time series is a set of observations taken at specified time, usually at:
a: equal intervals b: unequal intervals c: only at short-term intervals d: none of the above
20 Which one of the following is a component of time series:
a: secular trend b: seasonal variations c: cyclical variations
21 In Yc = a + bx, if b is positive, it shows:
a: rising trend b: seasonal variations c: cyclical variations d: all the above
22 We find irregular variations due to:
a: lock outs b: transport bottlenecks c: floods d: all the above
23 How can be remove seasonal variations:
a: reducing prices in seasons b: introducing different products having different seasons
c: both the above d: none of the above
24 What is the sequence of the following phases in business (i) depression, ii) prosperity, iii) recovery, iv) recession):
a: ii, iv, i, iii b: iv, c: iv, i, ii, iii d: iii, ii, iv, i
25 Which period give more tangible trend:
a: shorter period b: long period c: medium period
26 The seasonal changes are always:
a: periodic but not regular b: not periodic c: periodic and regular
27 Which is the feature of prosperity:
a: inflation b; optimism c: high profits
Answers
01 a 02 b 03 d 04 b 05 c 06 a 07 a
08 c 09 b 10 c 11 a 12 b 13 a 14 d
15 c 16 a 17 a 18 d 19 a 20 d 21 a
22 d 23 d 24 a 25 b 26 c 27 d
Bond Valuation - Objective Type Questions
1.Bonds have a ____ maturity andvalue at maturity:
a.fixed, variable b. fixed, fixed c. variable, fixed d.variable, d.variable
2.Regular repayment in the form of interest on a bond is called:
a.discount, b.interest c.coupon d. dividend
3.On 8%, 5 year bond of Rs.10000, the investors gets annually, as
a. 80.00, interest, b. 800.130, discount, c. 80.00, coupon d. 800.00, coupon
4.A bond the value of which is changed into a share is called:
a.convertible bond, b.zero coupon bond, c.negative bond, d.floating rate bond
5.A bond where the interest rate can be reset is: a. convertible bond, b.zero coupon bond c. negative bond,
d.floating rate bond
6. Abond is issued at a price below its face value while on payment, the face value is paid:
a. convertible bond, b.zero coupon bond, c. negative bond d. floating rate bond
7.In which of the following bonds, no interest payment Is made:
a.convertible bond b. zero coupon bond, c. negative bond d. floating rate bond
8.The income that a bond earns from year to year Is called: a. coupon, b.yield to maturity, c. current yield, d.effective yield
9.The actual yield to an investor, instead of nominal interest, is called: a. coupon, b.yield to maturity, c. current yield, d.effective yield
10.The discount rate that makes the present value of bond's payment equal to its price is called:
a.coupon b. yield to maturity c. current yield d. effective yield
11. A risk that arises on account of change in the current market price of the bond, is known as:
a. interest risk b.yield risk c. return risk d. price risk
12.The return to an investor in case of zero coupon bond is different between the price and value:
a: sale, face
b: face, purchase
c: purchase, face
d: face, sale
13 The face value ofbond changes with current interest rates:
a: convertible bond
b: zero coupon bond
c: negative bond
d: floating rate bond
14 Coupon / (1+r), where ris compounded return =
a: future value
b: discount rate
c: present value
d: maturity value
15 To calculate present value under annuity formula, we use the following:
a: PV of investment + PV of face value
b: PV of coupon + PV of face value
c: PV of coupon + PV of investment
d: PV of interest + PV of investment
16 For a 3 year 6% bond of Rs.1000, the present value is if the compounding is annual:
a: Rs.1010.77
b: Rs.1010.91
c: Rs.1100.77
d: Rs.1017.11
17 For a 3 year 6% bond of Rs.1000, the present value is , if the compounding is semi-annual (half yearly):
a: Rs.1010.77 , b. 1010.91 , c.Rs.1100.77 , d. Rs.1017.11
18 Where the current market interest are lower that the fixed coupon rate of a bond:
a: bond is sold at a discount to face value
b: bond earns premium on face value
c: bond sells at par d. there is no effect
19 Where the current market interest rate is higher than the fixed coupon rate of a bond:
a: bond is sold at a discount to face value
b: bond earns premium on face value
c: bond sells at par
d: there is no effect
20 Where the current market rate of interest is equal to coupon rate of the bond:
a: bond is sold at a discount to face value
b: bond earns premium on face value
c: bond is traded at par d. there is no effect
Answers
01 b 02 c 03 d
04 a 05 d 06 b
07 b 08 c 09 d
10 b 11 d 12 c
13 c 14 c 15 b
16 a 17 b 18 b
19 a 20 c
Objective Type Questions on linear programming
1 The word 'linear' represents which of the following:
a: drawing lines for solving management problem b: representation of relationships by straight line.
c: solution for production problem through statistical methods d: representing the solution on straight lines.
2 Linear programming is a _______technique:
a: economics b: mathematical c: algebraic d: any
of the above
3 Linear expression in linear programming is known as:
a: linear function b: algebraic function c: objective function d: main function
4 Linear inequalities in linear programming are known as:
a: linear constraints b: linear functions c: linear relationships d: none of the above
5 The value of decision variables can be:
a: negative or positive b: zero or negative c: positive or zero d: a and b
6 The decision variable in LP that should not assume negative values are called because they represent impossible situation.
a: negative restrictions b: positive restrictions c: Non-negative restrictions d: non-negative and positive restrictions
7 The relationships to be expressed either as equations or inequalities should be:
a: positive b: straight c: cervical d: linear
8 The non-zero variables in a basic feasible solution are called:
a: surplus variables b: slack variables c: artificial variables d: basic variables
9 Positive variables that are added to the left hand side to the constraints to convert them to equalities are called:
a: surplus variables b: slack variables c: artificial variables d: basic
variables
10 Positive variables that are subtracted to the left hand side to the constraints to convert them to equalities are called:
a: surplus variables b: slack variables c: artificial variables d: basic variables
11 The original linear programming problem is called:
a: original problem b: primary problem c: dual problem d: primal problem
12 The problem that is related to original linear programming problem is called:
a: original problem b: primary problem c: dual problem d: primal problem
13 A linear programming problem shows degeneracy:
a: when a basic variable acquires a zero value
b: in the final solution either the no. of basic variables is not equal to the no. of constraints
c: the no. of fern variable does not mita! the nn. of derision variahle.s
d: any of the above
14 For formulation of a linear programming problem, which of the following is an important steps:
a: determination of unknown variables
b: identification of constraints and expressing them in linear equations
c: identification of objectives and represent them as a linear function of decision variables
d: all the above
15 Which of the following combinations are the methods employed in linear programming problems:
a: geometrical, graphical methods b: simplex method, graphical methods
c: geometrical, simplex method d: b and c both
16 Geometrical method can be used effectively when the no. of variable is :
a: minimum 2 b: maximum 2 c: maximum 5 d:
no restrictions on minimum and maximum
17 Which of the following is most effective methods under LPP:
a: graphical b: geometrical c: simplex d: all the above
18 An arithmetical step by step procedure is called:
a: linear programming b: algorithm c: iterative procedure d: none of the above
19 Simplex method is basically and essentially procedure that evaluates corner points in search of the optimal solution:
a: iterative b: continuous c: regular d: minimum
20 In Simplex method, the value of objection function is increased at each step of iteration till:
a: further possibility of increase has ended b: maximum benefit has been obtained
c: no further improvement is possible d: none of the above
21 When Simplex method is used for problems such as maximization of production, profit or contribution, such problem is called:
a: Simplex problem b: minimization problem c: maximization problem
d : a o r c
22 When Simplex method is used for problems such as minimisation of cost or time, such problem is called:
a: Simplex problem b: minimization problem c: maximization problem
d: a or c
23 In which of the following areas, the application of LP techniques can be done:
a: portfolio selection b: staffing problem c: profit planning d:all the above
24 In which of the following areas, the application of LP techniques can not be done:
a: trimming of loss b: agriculture and contract awarding c: financial mix strategy, d. none of the above
Answer
01 b 02 b 03 c 04 a 05 c 06 c 07
d 08 a 09 b 10 a
11 d 12 c 13 d 14 d 15 d 16 d 17
c 18 b 19 a 20 c
21 c 22 b 23 d 24 d
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