The following table shows the information as:

. Then find

(i) the expected advertising expenditure of the company if sale is Rs. 125 lakhs, and

(ii) the expected sales of the company if the advertising expenditure is Rs 32 lakhs.

Calculate Karl Pearson's coefficient of correlation between X and Y for the following data:
 and . 

The value of Spearman's rank correlation coefficient of a set of non-repeating values was found to be 2/3. The sum of the squares of difference between the corresponding ranks was 55. Find the number of pairs.

The following is the distribution of age (in years) of 800 workers:

Find (i) Median, (ii) Quartile Deviation, and (iii) Coefficient of Quartile Deviation

The numbers 3.2, 5.8, 7.9 and 4.5 have frequencies Y, (Y + 2), (Y - 3) and (Y + 6), respectively. If the arithmetic mean is 4.876, find the value of Y and write the whole series. 

State whether the following statements are true or false and also give the reason in support of your answer: 

(a) If  and  are the variate values of two variables X and Y, and their geometric means are G₁ and G₂, respectively, then geometric mean of  will be (G₁/G₂).
(b) If X and Y are two independent variables and the variables  and , then the 
(c) If each value of X is divided by 2 and of Y is multiplied by 2, then b'_YX will be same as b.
(d) The mean and standard deviation of a set of values are 25 and 5, respectively. If a constant value 5 is added to each value, the coefficient of variation of the new set of values is equal to 10%.
(e) If  and  then .

Four coins were tossed and number of heads noted. The experiment is repeated 200 times. The number of tosses showing 0, 1, 2, 3 and 4 heads were found distributed as under. Fit a binomial distribution to these observed results assuming that the nature of the coins is not known.

 Write any 10 characteristics of normal distribution.

 Three unbiased coins are tossed simultaneously. In which of the following cases are the events A and B independent?

(i) A be the event of getting exactly one tail

B be the event of getting exactly one head

(ii) A be the event that first coin shows tail

B be the event that third coin shows head

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the expected value for the number of kings.

Find the probability that a third child in a family is the family's second daughter, assuming the male and female are equally probable.

A factory produces certain type of output by 3 machines. The respective daily production figures are- machine X : 4500 units, machine Y: 2000 units and machine Z: 3500 units. Past experience shows that 2% of the output produced by machine X is defective. The corresponding fractions of defectives for the other two machines are 5 and 10 percent respectively. An item is drawn from the day's production. If the drawn item is found to be defective, what is the probability that it has been produced by machine Y?

If a problem is randomly selected from a particular book, then probability that students A, B and C can solve the problem are 1/2, 1/3 and 1/4 respectively. If a problem is selected randomly from this book and given to the students A, B and C then what is the probability that problem is solved. They solve the problem independently.

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

(a) If events A, B and C are three mutually exclusive and exhaustive events, then it is possible that ,  and .

(b) If X is a discrete random variable, then X cannot take countably infinite values.

(c) If X is a standard normal variate, then .

(d) If random variables X and Y follow Bernoulli distributions with parameters 1/2 and 1/3 respectively then the random variable X + Y also follows a Bernoulli distribution with parameter 5/6.

Describe the following:

(i) t-distribution

(ii) General procedure of Testing of hypothesis 

6. Let  be a random sample from the Exponential distribution with pdf

(i) Obtain the maximum likelihood estimator (MLE) of .

(ii) Hence, find the maximum likelihood estimate of  for the observed sample: 

2.1, 1.6, 3.4, 0.9, 2.5

5. A researcher wants to compare the average test scores of students taught using Method A and Method B. A random sample of students was selected from each group, and their scores are given below:

Assuming that the populations are normally distributed with equal variances. For testing whether there is a significant difference between the mean scores of the two teaching methods at the 5% level of significance

(i) State the null and alternative hypotheses.

(ii) Name the appropriate and justify your choice.

(iii) Compute the test statistic and critical value.

(iv) State your conclusion. 

A survey was conducted to study the preference for online learning among students from urban and rural areas. Out of 300 urban students, 210 preferred online learning, while out of 250 rural students, 150 preferred online learning. Construct a 95% confidence interval for the difference between the population proportions of students preferring online learning in urban and rural areas. 
 

3. Let  be a random sample from a population with mean  and variance . Consider the estimator

(i) Find the bias of the estimator .

(ii) Find the variance of .
(iii) Check whether it is more efficient than sample mean. 

A school caretaker looks after 6 children. The ages (in years) of the children are given below:

(i) What is the nature (type) of the population of the ages of children?
(ii) Construct the sampling distribution of the sample mean for all possible samples of size .
(iii) Comment on whether the sampling distribution is approximately normal.
(iv) Find the mean and standard error of the sampling distribution.