State whether the following statements are True or False. Give reasons in support of your answers.

(a) The solution of a transportation problem with 5 rows (supplies) and 4 columns (destinations) is feasible if number of possible allocations are 8.

(b) The moving averages of suitable period in a time-series are free from the influences of seasonal and cyclic variations.

(c) If the basic solutions for a system of equations are (-2, 0, 1), (0, 1, 3), (-2, 3, 0), then only (0, 1, 3) is feasible.

(d) In the stepwise selection method of multiple regression model, once a variable enters in the model then it always remains in the model.

(e) An enterprise requires 1000 units per month. The ordering cost is estimated to be 50 per order. The purchase price is 20 per unit and the carrying cost per unit is 10% of it. Then the economic lot size to be ordered is 775.

A psychology class performed an experiment to determine whether a recall score in which instructions to form images of 25 words were given differs from an initial recall score for which no imagery instructions were given. Twenty students participated in the experiment with the results listed in the table:

(a) What two testing procedures can be used to test for differences in the distribution of recall scores with and without imagery? What assumptions are required for the parametric procedure? Do these data satisfy these assumptions?

(b) Use both the parametric and non-parametric tests for differences in the distributions of recall scores under these two conditions.

(c) Compare the results of the tests in part b. Are the conclusions for same? If not, why not?

A company plans to promote a new product by using one of three advertising campaigns. To investigate the extent of product recognition from these three campaigns, 15 market areas were selected and five were randomly assigned to each advertising plan. At the end of the ad campaigns, random samples of 400 adults were selected in each area and the proportions who were familiar with new product were recorded. The responses were not approximately normal. Is there a significant difference among the three population distributions from which these samples came? Use an appropriate nonparametric method to answer this question at 5% level of significance. then show that the estimators of the parameter u using maximum likelihood and method of moments are same.

If magnitude of earthquakes recorded in a region of a country follows a distribution with parameter a whose pdf is given below:

then show that the estimators of the parameter u using maximum likelihood and method of moments are same.

The following data relate to the number of items produced per shift by two workers for a number of days:

A bond proposal for school construction will be submitted to the voters at the next municipal election. A major portion of the money derived form this bond issue will be used to build schools in a rapidly developing selection of the city, and the remainder will be used to renovate and update school buildings in the rest of the city. To assess the viability of the bond proposal, a random sample of-50 residents in the developing section and n100 residents from the other parts of the city were asked whether they plan to vote for the proposal. The results are tabulated below

Sample Values for Opinion on Bond Proposal

(a) Estimate the difference in the true proportions favoring the bond proposal with a 99% confidence interval.

(b) If both samples were pooled into one sample of size n-150, with 103 in favor of the proposal, provide a point estimate of the proportion of city residents who will vote for the bond proposal.

A study was conducted to compare the mean numbers of police emergency calls per 8-hour shift in two districts of a large city. Samples of 100 8-hour shifts were randomly selected from the police records for each of the two regions and the number of emergency calls was recorded for each shift. The sample statistics are listed here:

Find a 90% confidence interval for the difference in the mean numbers of police emergency calls per shift between the two districts of the city. Interpret the interval.

If a finite population has four elements: 6, 1, 3, 2.

(a) How many different samples of size n=2 can be selected from this population if you sample without replacement?

(b) List all possible samples of size n=2.

(c) Compute the sample mean for each of the samples given in part b.

(d) Find the sampling distribution off and draw the histogram.

(e) Compute standard error.

(f) If all four population values are equally likely, calculate the value of the population mean ? Do any of the samples listed in part (b) produce a value off exactly equal to

tate whether the following statements are True or False. Give reason in support of your answer:

(a) If the probability of non rejection of Ho when H, is true is 0.4 then power of the test will be 0.6.

(b) If T₁ and T₂ are two estimators of the parameter & such that Var(T₁)-1/n and Var(T₂)n then T, is more efficient than T₂.

(c) A 95% confidence interval is smaller than 99% confidence interval.

(d) If the level of significance is the same, the area of the rejection region in a two-tailed test is less than that in a one-tailed test.

(e) Non parametric tests are more powerful than the parametric tests.

State Monty Hall problem and solve it.

Consider the joint PDF for the type of customer service X (0- telephonic hotline, 1- Email) and of satisfaction score Y (1 unsatisfied, 2 satisfied, 3 very satisfied):

(a) Determine and interpret the marginal distributions of both X and Y.

(b) Calculate the 75% quantile for the marginal distribution of Y.

(c) Determine and interpret the conditional distribution of satisfaction level for X=1.

(d) Are the two variables independent?

(e) Calculate and interpret the covariance of X and Y.

Random variable X follows Beta distribution with parameters a 3, b-2 and has pdf f(x12x(1x),0x1 10,otherwise Find (1) CDF of X (ii) P[0

Random variable X follows Beta distribution with parameters a=3, b = 2 and has pdf

Find (1) CDF of X (ii) P(0<X<1/2] (iii) mean and variance of X without using direct formula for mean and variance.

A group of 100 people are comparing their birthdays (as usual, assume their birthdays are independent and not on February 29, etc.). Find the expected number of pairs of people with the same birthday, and the expected number of days in the year on which at least two of these people were born.

Random variable X follows Beta distribution with parameters a 3, b-2 and has pdf f(x12x(1x),0x1 10,otherwise Find (1) CDF of X (ii) Pdf 

is a valid PMF for a discrete random variable. Also find out its CDF.

A group of 100 people are comparing their birthdays (as usual, assume their birthdays are independent and not on February 29, etc.). Find the expected number of pairs of people with the same birthday, and the expected number of days in the year on which at least two of these people were born.

Show that

f(x1,2, 1,2,3,4,5, is a valid PMF for a discrete random variable. Also find out its CDF.

random 5-card poker hand is dealt from a standard deck of cards. Find the probability (in terms of binomial coefficients) of getting a flush (all 5 cards being of the same suit: do not count a royal flush, which is a flush with an ace, king, queen, jack and

There are 4 black, 3 blue and 8 red balls in an urn. Three balls are selected one by one without replacement. What is the probability that:

(i) First ball drawn is black, second one is red and third one is blue

(ii) All the three balls are of the same colour.

State whether the following statements are True or False and also give the reason in support of your answer.

(a) Sample space of a (i) random experiment tossing two coins simultaneously and (ii) One coin two times is the same.

(b) Standard deviation of a random variable X may take any real value, i.e. its value lies in the interval(ゆす。,

c) If events  are mutually exclusive and exhaustive then will be greater than 1/2 but less than 1.

(d) If S is sample space of a random experiment and E is an event defined on this sample space then P(SE)-1.

(e) If X is a random variable having range set (0, 1, 2, 3) then the set (xS:X(x)0 is an event having at least one outcome of the random experiment.

In the given data, two frequencies are missing and its mean is found to be 1.46.

Find the missing frequencies.