State whether the following statements are True or False. Give reason in support of your answer:
(a) The Wilcoxon rank-sum test is a non-parametric alternative to the two-sample t-test when normality is not assumed.
(b) The power of a statistical test is the probability of accepting the null hypothesis when it is true.
(c) If the sample size tends to infinity, the variance of a consistent estimator must tend to zero.
(d) In the method of moments, population moments are equated to corresponding sample moments.
(e) The chi-square distribution is symmetric for all degrees of freedom.
See Answer →Times between successive crashes of a computer system were generated for a 6-month period and are given in increasing order as follows (time in hours):
1 10 20 30 40 52 63 70 80 90 100 102
130 140 190 210 266 310 530 590 640 1340
The parameter , mean
.
Use the Kolmogorov-Smirnov test to examine the goodness of fit of exponential distribution.
Generate a complete cycle for the LCG given below: , with
. A man tosses an unbiased coin ten times. Using the first ten random numbers generated above, obtain a sequence of heads and tails by taking Head (H) as
.
The distribution function of Pareto distribution is given by ,
.
Given a , generate a random number from the above distribution, when
and
.
Suppose , then find x.
Identify the design given in the following table and then carry out the analysis:
| Column | Row | |||
|---|---|---|---|---|
| I | II | III | IV | |
| I | A 8 | C 18 | B 11 | D 8 |
| II | C 16 | B 10 | D 7 | A 4 |
| III | B 12 | D 10 | A 6 | C 20 |
| IV | D 10 | A 9 | C 28 | B 16 |
In the following data, two values are missing. Estimate these values by Yates method and analyse the data by suitable technique.
| Treatments | Blocks | ||
|---|---|---|---|
| I | II | III | |
| A | 12 | 14 | 12 |
| B | 10 | y | 8 |
| C | x | 15 | 10 |
A researcher wants to test four diets A, B, C, D on growth rate in mice. These animals are divided into 3 groups according to their weights. Heaviest 4, next 4 and lightest 4 are put in Block I, Block II, and Block III, respectively. Within each block, one of the diets is given at random to the animals. After 15 days, increase in weight is noted, which is given in the following table:
| Blocks | Treatments/Diets | |||
|---|---|---|---|---|
| A | B | C | D | |
| I | 12 | 8 | 6 | 5 |
| II | 15 | 12 | 9 | 6 |
| III | 14 | 10 | 8 | 5 |
Perform a two-way ANOVA to test whether the data indicates any significant difference between' the four diets due to different blocks.
See Answer →The following data relate to production in kg of three varieties P, Q, R of wheat:
| P : | 14 | 16 | 18 | ||
| Q: | 14 | 13 | 15 | 22 | |
| R : | 18 | 16 | 19 | 15 | 20 |
Is there any significant difference among the three varieties at 5% level of significance?
See Answer →To determine the yield rate of wheat in a district of Punjab, 6 groups of 6 plots each were constructed. The data are given in the following table:
| Plot No. | Group 1 | Group 2 | Group 3 | Group 4 | Group 5 | Group 6 |
|---|---|---|---|---|---|---|
| 1 | 8 | 6 | 18 | 13 | 17 | 12 |
| 2 | 13 | 5 | 8 | 7 | 15 | 15 |
| 3 | 11 | 16 | 6 | 13 | 10 | 11 |
| 4 | 26 | 5 | 10 | 6 | 21 | 17 |
| 5 | 13 | 16 | 16 | 7 | 20 | 8 |
| 6 | 31 | 5 | 20 | 2 | 25 | 10 |
Select a cluster sample of 3 clusters from the above data and find its sample mean. Further, explain the procedure of two-stage sampling if we want to draw a sample of 6 plots. Which are the 6 plots in your sample?
See Answer →A sample of 60 students is to be drawn from a population consisting of 600 students belonging to two villages, A and B. The means and standard deviations of their marks are give below:
| Villages | Stratum sizes ($N_i$) | Means ($x_i$) | Standard deviations |
|---|---|---|---|
| Village A | 400 | 60 | 20 |
| Village B | 200 | 120 | 80 |
What are the sample sizes for the two villages using proportional allocation technique?
See Answer → Draw all possible samples of size 2 from the population [2, 3, 4] and verify that . Also find variance of
.
State whether the following statements are true or false and also give the reason in support of your answer:
(a) The total number of all possible samples of size 2 without replacement from a population of size 7 is 21.
(b) Consecutive 3 random numbers starting from 8937 by 'middle square method' are 8937, 8699, 6726.
(c) RBD is suitable in situations where it is not possible to divide the experimental material into a number of homogeneous blocks.
(d) As we increase the sample size, representativeness of the population by the sample decreases.
(e) In a big hall, there are 50 rows and each row has 60 students. A research scholar selects 10 rows randomly and then randomly selects 15 students from each selected row. It is an example of cluster sampling procedure.
A restaurant produces fresh burgers for its customers every day. The company is known for supplying fresh burgers and never uses burgers prepared on the previous day. Demand for burgers is uncertain, preparation capacity is limited, and the restaurant has the option of producing 0, 100, 200, 300 and 400 burgers every day. It has been estimated that the cost of producing each burgers pack is Rs.15. Each burger is sold for Rs. 20. Prepare a payoff matrix when 0, 100, 200, 300 or 400 demands of the burgers turn up on any given day. Prepare an opportunity loss table and hence find the optimum strategy.
See Answer →In a manufacturing process, 4 items are inspected every hour for 10 consecutive hours. The measured quality characteristic (in mm) is given below:
| Hour | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 50 | 52 | 51 | 49 |
| 2 | 51 | 50 | 53 | 52 |
| 3 | 49 | 50 | 48 | 51 |
| 4 | 52 | 54 | 53 | 51 |
| 5 | 50 | 49 | 51 | 50 |
| 6 | 53 | 52 | 54 | 55 |
| 7 | 48 | 49 | 50 | 47 |
| 8 | 51 | 52 | 50 | 51 |
| 9 | 54 | 55 | 53 | 54 |
| 10 | 50 | 51 | 49 | 50 |
Construct control chart for variability and mean and comment on the state of statistical control. If the process is out of control, obtain revised control limits.
See Answer →A system consists of six independent components arranged as follows:
• Two components with reliabilities 0.9 and 0.8 connected in series.
• This series combination is connected in parallel with a component of reliability 0.7.
• The resulting subsystem is connected in series with two components of reliabilities 0.85 and 0.95.
Draw the reliability block diagram and calculate the overall reliability of the system.
See Answer →A two-person zero-sum game having the following payoff matrix for player A
| Player B | ||||||
|---|---|---|---|---|---|---|
| I | II | III | IV | V | ||
| Player A | I | 2 | 4 | 3 | 8 | 4 |
| II | 5 | 6 | 3 | 7 | 8 | |
| III | 6 | 7 | 9 | 8 | 7 | |
| IV | 4 | 2 | 8 | 4 | 3 | |
(i) Check whether saddle point exit or not.
(ii) If saddle point does not exit then determine optimal strategies for both the manufacturers and value of the game.
See Answer →3. A company manufactures water pumps. The quality control inspector of the company takes a sample of 100 water pumps at regular intervals. The numbers of defective pumps for 15 samples are given below:
| Sample No. | Defective Pumps | Sample No. | Defective Pumps | Sample No. | Defective Pumps |
| 1 | 5 | 6 | 0 | 11 | 6 |
| 2 | 6 | 7 | 4 | 12 | 1 |
| 3 | 3 | 8 | 8 | 13 | 10 |
| 4 | 2 | 9 | 2 | 14 | 2 |
| 5 | 1 | 10 | 2 | 15 | 1 |
Use the data to construct a suitable chart. Observe the results and comment on the control of the process as indicated by the chart.
See Answer →A factory purchases bolts in lots of 800. Acceptance is decided using a single-sampling plan with sample size n = 20 and acceptance number c = 3. Assume that 2% of defective items are considered acceptable quality and 7% defective items are considered unacceptable quality.
Find:
(i) The probability of accepting a lot when the incoming quality level is 5% defective.
(ii) The Average Outgoing Quality (AOQ), assuming rejected lots are completely screened and defectives are replaced.
(iii)The Average Total Inspection (ATI)
See Answer →State whether the following statements are True or False. Give reason in support of your answer:
(a) The c-chart is suitable for monitoring to proportion of defective.
(b) In single sampling plan, if we increase acceptance number then the OC curve will be steeper.
(c) In a series system, improving the reliability of the weakest component gives the maximum improvement in system reliability.
(d) If the probabilities are not associated with the occurrence of different states of nature, then the situation is known as decision making under risk.
(e) In single sampling plan, if we increase acceptance number then the OC curve will be steeper.
See Answer →Seven successive observations on a stationary time-series are as follows:
12, 14, 13, 10, 15, 12, 15
(a) Calculate auto-covariances C0, C1, C2, C3 and C4.
(b) Calculate auto-correlation coefficients r1, r2, r3 and r4.
(c) Plot the correlogram.
See Answer →