The following marks have been obtained by a class of students in 2 papers of mathematics. Paper I: 45, 55, 56, 58, 60, 65, 68, 70, 75, 80, 85; Paper II: 56, 50, 48, 60, 62, 64, 65, 70, 74, 82, 90. Calculate the coefficient of correlation for the above data.

Find the quartile deviation and coefficient of quartile deviation of the following frequency distribution. Marks: <10, 10-20, 20-30, 30-40, 40-50, 50-60; No. of Students: 10, 20, 30, 50, 40, 30

Determine the eqns of regression lines for the following data. Also find the value of (i) y for x = 4.5 (ii) x when y = 13. x: 2, 3, 5, 7, 9, 10, 12, 15; y: 2, 5, 8, 10, 12, 14, 15, 16

The first four moments of four distribution about the value 4 are 2, 20, 40 and 100 respectively. i) Obtain the first central moments ii) Find mean, standard deviation iii) Find coefficients of skewness and kurtosis

In a certain company install 2000 LED bulbs on each floor. If LED bulbs have average life of 1000 burning hours with standard deviation of 200 hours. Using normal distribution find what number of LED bulbs might be expected to Fail in 700 hours. (Given : P(0 < z < 1.5) = 0.4332)

Between 2 pm to 4 pm the average no of phone calls per minute coming into a switch board of a company is 2.5. Find the probability that during a particular minute there will be i) no phone call ii) exactly 3 phone calls

A dice is thrown 10 times. If getting an odd number is a success. What is the probability of i) 8 sucess ii) At least 6 sucess

Weights of 4000 students are found to be normally distributed with mean 50 kg and standard deviation 5 kgs. Find the number of students with weights i) less than 45 kgs ii) between 45 to 60 kgs (for standard normal distribution z, area under the curve between z = 0 to z = 1 is 0.3413 and that between z = 0 to z = 2 is 0.4772)

If 10% bolts produced by a machine are defective. Determine the probability that out of 10 bolts choosen at random. i) two will be defective ii) at most two will be defective.

In a continuous distribution density function f(x) = kx(2 – x), 0 < x < 2 Find the value of k, mean and variance.

Random sample of 400 men and 600 women were asked whether they would have a school near their residence 200 men and 325 women were in favour of proposal. Test the hypothesis that the proportion of men and women in front of proposal is same at 5% level of significance. (Given Zalpha = 1.96 at 5% l.o.s)

The values given below are i) Observed frequencies of a distribution ii) The frequencies of a normal distribution having same mean, standard deviation and the total frequency as in a) apply 2 test of godness of fit. a) 1, 5, 20, 28, 42, 22, 15, 5, 2; b) 1, 6, 18, 25, 40, 25, 18, 6, 1. (Given 2 = 12.592 at 5% l.o.s.)

Fertilizers A and B are tried respectively on 10 and 8 randomly choosen experimental plots. The yields in the plots were as given below. Test using t-test whether in effects of the fertilizer as reflected in the mean yields. Fertilizers: A (8.0, 7.6, 8.2, 7.8, 8.3, 8.4, 8.2, 7.8, 7.1, 8.0); B (7.4, 8.1, 7.6, 8.1, 7.5, 7.6, 7.3, 7.2). (Given t0.05 = 2.201 at d.o.f 16)

The average marks in mathematics of a sample of 100 students was 51 with S.D. of 6 marks. Could this have a random sample from the population with average marks 50? (Given zalpha = 1.96 at 5% l.o.s.)

A coin is tossed 160 times and following are expected and observed frequencies for number of heads. No of heads: 0, 1, 2, 3, 4; Expected frequency: 17, 52, 54, 31, 6; Observed frequency: 10, 40, 60, 40, 10. Find the 2 value.

In two independent samples of size 8 and 10 the sum of squares deviations of the values form the respective sample means were 84.4 and 102.6. Test whether the difference of variances of the population is significant or not. (Given F0.05 = 3.29 at degrees of freedom (7,9))

State and prove Neyman-pearson Fundamental lemma.

Let p is the probability that a given die shows even number. To test H0: P=1/2 Vs H1: P=1/3 following procedure is adopted. Toss the die twice and accept H0 if both times. It shows even number. Find the probabilities of Type I and Type II error.

For distribution f(x, beta, gamma) = beta*gamma*x^(gamma-1)*exp(-beta*x^gamma), x>=0. Show that for H0 : beta = beta0 = gamma = gamma0 and H1 : beta = beta1 = gamma = gamma1 is the best critical region is given by (1/n) * sum(log(x)) * (beta1*gamma1 - beta0*gamma0) + log(beta1/beta0) + log(gamma1/gamma0) >= k.

Write short notes on : i) Critical region and Most powerful critical region. ii) Level of significance and power of Test.

For the following distribution, find (1) first 4 moments about any arbitrary point (2) Four Central Moments. X: 2, 2.5, 3, 3.5, 4, 4.5, 5 F: 5, 38, 65, 92, 70, 40, 10

The regression equations are 8x – 10y + 66 = 0 and 40x – 18y = 214. The value of variance of x is 9. Find : i) The mean values of x and y. ii) The correlation x and y and iii) The standard deviation of y.

The first four moments about the working mean 30.2 of distribution are 0.255, 6.222, 30.211 and 400.25. Calculate the first four moments about the mean. Also evaluate 1, 2 and comment upon the skewness and kurtosis of the distribution.

Obtain the regression lines y on x and x on y for the data : x: 5, 1, 10, 3, 9 y: 10, 11, 5, 10, 6

A Dice is thrown 10 times. If getting an odd number is a success. What is the probability of getting (i) 8 successes (ii) at least 6 success.

If the probability that an individual suffers a bad reaction from certain injection is 0.001. Determine the probability out of 2000 people, by using Poisson’s distribution (i) Exactly 3 (ii) More than 1 will suffer a bad reaction.

For a normal distribution when mean = 2, standard deviation = 4, find the probabilities of the following intervals. i) 4.43 < x < 7.29 ii) –0.43 < x < 5.39 [Given : A (z = 0.61) = 0.2291, A (z = 1.32) = 0.4066, A (z = 0.85) = 0.3023]

A Random variable X with following probability distribution X: 0, 1, 2, 3, 4 P(X): 0.1, k, 2k, 2k, k Find : i) k ii) P(x < 2) iii) P(x > 3) iv) P(1 < x < 3)

In a continuous distribution density function. f (x) = kx (2 –x), 0 < x < 2 Find the value of i) k ii) Mean iii) Variance

MNC company conducted 1000 candidates aptitude test. The average score is 45 and the standard deviation of score is 25. Assuming normal distribution for the result. Find : i) The number of candidates whose score exceed 60. ii) The number of candidates whose score lies between 30 & 60. [Given : A (z = 0.6) = 0.2257]

The table below gives the number of customers visit the certain company on various days of week. Days: Sun, Mon, Tue, Wed, Thurs, Fri, Sat Number of Customers: 6, 4, 9, 7, 8, 10, 12 Test at 5% of level of significance whether customer visits are uniformly distributed over the days. [Given 2 6,0.05 = 15.592]

In a Batch of 500 articles, produced by a machine, 16 articles are found defective. After overhauling the machine, it is found that 3 articles are defective in a batch of 100.Has the machine improved? [Given Z = 1.96]

In two independent samples of size 8 and 10 the sum of squares deviations of the sample values from the respective sample means were 84.4 and 102.6. Test whether the difference of variances of the population is significance or not. [Given Ft,0.05 = 3.29]

In an experiment on pea breeding, the following frequencies of seeds were obtained. Round and Green: 222, Wrinkled and Green: 120, Round and Yellow: 32, Wrinkled and Yellow: 150 Total: 524 Theory Predicts that the frequencies should be in Proportion 8:2:2:1. Examine the correspondence between theory and experiment. [Given 2 3,0.05 = 7.815]

For sample I : n1=1000, x1=49000, (x1-xbar1)2=7,84,000, For sample II : n2=1500, x2=70500, (x2-xbar2)2=24,00,000. Discuss the significance difference between mean score. [Given Z = 1.96]

Samples of size 10 and 14 were taken from two normal populations with standard deviation 3.5 and 5.2. The sample means were found to be 20.3 and 18.6. The whether the means of the two populations are at the same level. [Given t22,0.05 = 2.07]

State & prove Neyman-Pearson Fundamental Lemma.

Let P be the probability that a coin will fall head in a single toss in order to test H0: P=1/2 against H1: P=3/4. The coin is tossed 5 times and H0 is rejected if more than 3 heads are obtained. Find the probability of type I error and power of the test.

Show that the likelihood ratio test for testing the equality of variances of two normal distribution is the usual F-test.

Write short note on : i) Population and sample ii) Type I and Type II Error iii) Critical Region iv) Power of test

Calculate: i) Quartile deviation (Q.D.), ii) Mean Deviation (M.D.) from mean, for the following data: Marks: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60, 60-70; No. of students: 6, 5, 8, 15, 7, 6, 8

The variables X and Y are connected by the equation aX + bY + c = 0. Show that the correlation between them is -1 if the signs of a and b are alike and +1 if they are different.

An analysis of monthly wages paid to the workers of two firms A and B belonging to the same industry give the following results: Firm A (Number of workers: 500, Average daily wage: Rs. 186.00, Variance: 81); Firm B (Number of workers: 600, Average daily wage: Rs. 175.00, Variance: 100). i) Which firm, A or B, has a larger wage bill? ii) In which firm, A or B, is there greater variability in individual wages? iii) Calculate (a) the average daily wage, and (b) the variance of the distribution of wages of all the workers in the firm A and B taken together.

In a partially destroyed laboratory, record of an analysis of correlation data, the following results only are legible: Variance of X=9. Regression equations: 8X–10Y+66=0, 40X–18Y=214. what are: i) the mean values X and Y, ii) the correlation coefficient between X and Y, iii) the standard deviation of Y?

A Dice is thrown 10 times. If getting an odd number is a success. What is the probability of getting i) 8 successes ii) at least 6 success?

Fit Poisson’s distribution to following data and calculate theoretical frequencies. x: 0, 1, 2, 3, 4; f: 122, 60, 15, 2, 1

In a Sample of 1000 cases the means of a certain test is 14 and standard deviation is 2.5 assuming the distribution to be normal find i) How many students scored between 12 & 15. ii) How many scored below 8. [Given: A(z = 0.8) = 0.2881), A(z = 0.4) = 0.1554), A (z = 2.4) = 0.4918]

A Random variable X with following probability distribution X: 1, 2, 3, 4, 5, 6, 7; P(X): k, 2k, 3k, k2, k2+k, 2k2, 4k2. Find: i) k ii) P(x > 5) iii) P(1 <= x <= 5)

In a continuous distribution density function f(x) = kx(1-x), 0 <= x <= 1. Find the value of i) k ii) Mean iii) Variance

MNC company conducted 1000 candidates’ aptitude test. The average score is 45 and the standard deviation of score is 25. Assuming normal distribution for the result. Find: i) The number of candidate whose score exceed 60. ii) The number of candidates whose score lies between 30 & 60. [Given: A(z = 0.6) = 0.2257)]

In an experiment of pea breeding, the following frequencies of seeds were obtained. (Round and green: 222, Wrinkle and green: 120, Round and yellow: 32, Wrinkle and yellow: 150). Total: 524. Theory predicts that the frequencies should be in the proportion 8:2:2:1. Examine the correspondence between theory and experiment. Given chi-square (0.05,3) = 7.815

The average marks in mathematics of a sample of 100 students was 51 with standard deviation of 6 marks. Could this have a random sample from the population with average marks 50? Given Z at 5% level of significance = 1.96

A random sample of 16 newcomers gave a mean of 1.67 m and standard deviation of 0.16 m. Is the mean height of newcomers significantly different from the mean height of students’ population of the previous year? Given t0.05, 15 = 2.13

Following table shows number of books issued on the various days of week from a certain library At 5% level of significance test the null hypothesis that number of books issued in department of the day. Day: Mon, Tue, Wed, Thurs, Fri, Sat; No. of books issued: 120, 130, 110, 115, 135, 110. Given: Chi-square value at 5% level of significance for degrees of freedom 5 is 11.071.

A random sample of 900 members has mean 3.4 cms. Can it be reasonable regarded as a sample from a large population of mean 3.2 cms and standard deviation 2.3 cms.

Find the F-statistics form the following data: Sample 1 (n=8, Sum of observations=9.6, Sum of squares of observations=61.52), Sample 2 (n=11, Sum of observations=16.5, Sum of squares of observations=73.26)

State & Prove Neyman-Pearson Fundamental Lemma.

Given the frequency function f(x, theta) = 1/theta, 0 <= x <= theta, and that you are testing the null hypothesis H0: theta = 1 vs H1: theta = 2 by means of a single observed value of x. what would be the size of Type I and Type II error. If you choose the interval i) 0.5 <= x ii) 1 <= x <= 1.5. Also obtain the power function of the test.

Write short notes on i) Most powerful test ii) Uniformly most powerful test iii) Advantages and disadvantages of non-parametric tests iv) Level of significance

Explain in detail about test for the Equality of means of several normal populations.

The first four moments of a distribution about the value 5 are 2,20,40 and 50. From the given information obtain the first four central moments, coefficient of skewness and kurtosis.

Obtain the regression lines y on x and x on y for the data. x: 5, 1, 10, 3, 9 y: 10, 11, 5, 10, 6

Calculate standard deviation for the following frequency distribution. Decide whether Arithmetic mean is good or not. Wages in rupees earned per day: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60 No. of laborer’s: 5, 9, 15, 12, 10, 3

Following are the values of import of raw material and export of finished product in suitable units. Export: 10, 11, 14, 14, 20, 22, 16, 12, 15, 13 Import: 12, 14, 15, 16, 21, 26, 21, 15, 16, 14 Calculate the coefficient of Correlation between the import values and export values.

If the two lines of regression are 9x + y – λ = 0 and 4x + y = μ and the means of x and y are 2 and –3 respectively, find the values of λ, μ and the coefficient of correlation between x and y.

Compute the first four moments about arbitrary mean A = 25 for the following frequencies. No of Jobs: 0-10, 10-20, 20-30, 30-40, 40-50 No of Workers: 6, 26, 47, 15, 6

20% of bolts produced by a machine are defective. Determine the probability that out of 4 bolts chosen at a random. i) 1 is defective ii) Zero are defective iii) At most 2 bolts are defective

The average number of misprints per page of a book 1.5. Assuming the distribution of number of misprints to be Poisson, Find i) The Probability that a particular book is free from misprints. ii) Number of pages containing more than one misprint if the book contains 900 pages.

For a normal distribution When mean = 2, standard deviation σ = 4, find the probabilities of the following intervals. i) 4.43 ≤ x ≤ 7.29 ii) –0.43 ≤ x ≤ 5.39 [Given : A(z = 0.61) = 0.2291, A(z = 1.32) = 0.4066, A(z = 0.85) = 0.3023]

A Random variable X with following probability distribution. X: 0, 1, 2, 3, 4 P(X): 0.1, k, 2k, 2k, k Find i) k ii) P (x < 2) iii) P (x ≥ 3) iv) P (1 ≤ x ≤ 3)

Fit a Poisson Distribution to the following data and calculate theoretical frequencies x: 0, 1, 2, 3, 4, Total f: 109, 65, 22, 3, 1, 200

The lifetime of an article has a normal distribution with mean 400 hours and standard deviation 50 hours. Find the expected number of articles out of 2000 whose lifetime lies between 335 hours to 465 hours. (Given : A(z = 1.3) = 0.4032)

The Table below gives the number of customers visit the certain company on various days of week Days: Sun, Mon, Tue, Wed, Thurs, Fir, Sat Number of Customers: 6, 4, 9, 7, 8, 10, 12 Test at 5% of level of significance whether customer visits are uniformaly distributed over the days. [Given χ² 6,0.05 = 15.592]

In a Batch of 500 articles, produced by a machine, 16 articles are found defective. After overhauling the machine, it is found that 3 articles are defective in a batch of 100. Has the machine improved? (Given Zα = 1.96)

Samples of Size 10 and 14 were taken from two normal populations with Standard deviation 3.5 and 5.2. The sample means were found to be 20.3 and 18.6. Test whether the means of the two populations are at the same level. (Given t0.05,22 = 2.07)

In an experiment on pea breeding, the following frequencies of seeds were obtained. Round and Green: 222 Wrinkled and Green: 120 Round and Yellow: 32 Wrinkled and Yellow: 150 Total: 524 Theory Predicts that the frequencies should be in Proportion 8:2:2:1. Examine the correspondence between theory and experiment [Given χ² 3,0.05 = 7.815]

For sample I : n1 = 1000, Σx = 49000, Σ(x – x̄)² = 7,84,000, For Sample II : n2 = 1500, Σx = 70500, Σ(x – x̄)² = 24,00,000. Discuss the significant difference between mean score. (Given Zα = 1.96)

Find the F statistics from the following data: Sample Size (n): 1 (8), 2 (11) Total observation Σx: 1 (9.6), 2 (16.5) Sum of squares of observations Σx²: 1 (61.52), 2 (73.26)

Let P be the probability that a coin will fall head in a single toss in order to test H0 : P = 1/2 against P = 3/4. The coin is tossed 5 times and H0 is rejected if more than 3 heads are obtained. Find the probability of type I error and power of the test.

Show that the likelihood ratio test for testing the equality of variances of two normal distribution is the usual F-test.

Write short notes on i) Most Powerful test ii) Level of significance iii) Advantages and disadvantages of non-parametric test

State and Prove Neyman - Pearson lemma for testing a simple hypothesis against a simple alternative hypothesis.

What are the limitations and importance of Statistics.

What are the methods of estimation? Give brief on testing of hypothesis.

Explain the Scope of Statistics in engineering & Technology.

What is population and sample? Explain the type of sampling in brief.

What are random sample? Explain lottery method and random numbers in detail.

What is Statistics? Explain the scope of statistics in medical and biological fields.

Draw a frequency polygon for the following data. Marks 0-20 20-40 40-60 60-80 80-100 No. of students 2 18 42 28 5 Also state the advantages of graphical representation of data (any four).

Calculate the mean for the following frequency distribution. Class 0-10 10-20 20-30 30-40 40-50 50-60 Frequency 12 18 27 20 17 6 Also state merits of Mean (any four).

Age distribution of hundred life insurance policy holders is as follows : Age 17-19 20-22 23-25 26-28 29-31 32-34 35-37 38-40 Number 9 16 12 26 14 12 6 5 Calculate mode

What is histogram? Draw the histogram for the following data. Age group 0-20 21-40 41-60 61-80 81-100 101-120 (in years) Population 500 2100 2200 2000 1600 400

Calculate median for the following distribution. Class 0-10 10-20 20-30 30-40 40-50 Frequency 5 14 29 21 25 Also write merits of median (any 2).

What are the merits and demerits of Harmonic mean (2 each). Also calculate Harmonic mean of the following series. Values 2 6 10 14 18 Frequency 4 12 20 9 5

Explain types of Sampling.

What is Statistics? Explain its importance.

Explain the importance of sampling and state its limitations.

What is distrust of Statistics? Explain the limitations of Statistics.

State the difference between Population and Sampling. Explain the process of sampling.

What is random sampling? Explain its types.

Calculate the median marks of students from the following distribution: Marks 0-10, 10-20, 20-30, 30-40, 40-50, 50-60, 60-70; No. of students 3, 6, 13, 15, 14, 5, 4

What is harmonic mean? Explain its merits and demerits.

Draw the frequency polygon for the following data: Class Interval 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90; Frequency 4, 6, 8, 10, 12, 14, 7, 5

Consider the following frequency distribution. Calculate the mean weight of students: Weight (in Kg) 31-35, 36-40, 41-45, 46-50, 51-55, 56-60, 61-65, 66-70, 71-75; No. of students 9, 6, 15, 3, 1, 2, 2, 1, 1

What is mode? How is it calculated? State its merits.

Draw the Histogram for the following data : Height in meters 0-3, 3-6, 6-9, 9-12, 12-15; No. of Trees 80, 100, 120, 90, 50