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- University IGNOU (Indira Gandhi National Open University)
- Title Statistical techniques
- Language(s)
- Code AST-01
- Subject Mathematics
- Degree(s) BA, B.COM, B.Sc.
- Course Application-Oriented Courses (AOC)

- Unit 1 - Descriptive Statistics
- Unit 2 - Probability Theory
- Unit 3 - Probability Distributions

- Unit 1 - Sampling Distributions
- Unit 2 - Estimation
- Unit 3 - Tests of Significance
- Unit 4 - Applications of Chi-Square in Problems with Categorical Data

- Unit 1 - Analysis of Variance One-Way Classification
- Unit 2 - Regression Analysis
- Unit 3 - Forecasting and Time Series Analysis
- Unit 4 - Statistical Quality Control

- Unit 1 - Simple Random Sampling and Systematic Sampling
- Unit 2 - Stratified Sampling
- Unit 3 - Cluster Sampling and Multistage Sampling

1. a) The distribution of a discrete random variable X is given as :
Find :
(i) The value of K
(ii) P (X >2)
(iii) E (X)
(iv) Draw a rough sketch of the distribution.
b) For overall quality improvement of cloth, a textile manufacturer decides to monitor the number of defects in each bolt of cloth. The data from 10 inspections is reported as follows :
Draw an appropriate control chart to check whether the process is under statistical control.
2. a) Three dice are rolled. If no two dice show the same number on the top face, what is the probability that one dice shows ‘1’ on the top face?
b) If 2% of the books bound at a certain binary have defective bindings, determine the probability that 5 of 400 books bound by this binary will have defective bindings.
[Given that e-8 = 0.00034]
3. a) Out of 20,000 customers’ leader accounts, a sample of 600 accounts was taken to test the accuracy of posting and balancing wherein 45 mistakes were found. Determine the 95% confidence interval for number of defective cases.
b) In 16 one-hour test runs, the gasoline consumption of an engine on an average is found to be 16⋅ 4 gallons with a standard deviation of 2⋅1 gallons. At 5% level of significance, test the claim that the average gasoline consumption of this engine is 12⋅0 gallons per hour
c) A sample of 100 employees is to be drawn from a population of collages A and B. The population means and population mean squares of their monthly wages are given below:
Determine the sample size to be taken using (i) proportional allocation, and (ii) Neyman allocation methods.
4. a) A company has sales (y) which shows profit on the cost of production (x) of furniture when sold to government agencies. The regression lines are y = x + 5 and 16x = 9y + 95.
Find :
(i) Mean of sale and production cost.
(ii) Correlation coefficient between sales and production cost.
(iii) If σy = 4, find σx.
(iv) Predict the production cost when sale is 2,000.
b) An oil company claims that less than 20% of all the car owners have not tried its oil. Test the claim at the 0⋅01 level of significance if a random check reveals that 22 of 200 car owners not tried its oil.
5. a) The odds that a research monograph will be accepted by 3 independent referees are 3 to 2, 4 to 3 and 2 to 3, respectively. Find the probability that of the three reports :
(i) all will be accepted.
(ii) more than 1 of the reports will be favorable.
b) A company installed 10000 electric bulls in a metro. If these bulbs have an average life of 1000 hours with S. D of 200 hours. Assuming normality, what number of bulls might be expected to fail (i) in the first 800 hours (iii) between 800 and 1200 hours.
c) Cite two situations where systematic sampling is appropriate. Explain, how it is different from stratified sampling. Justify.
6. a) A random sample is selected from each of three types of ropes and their breaking strength (in pounds) are measured with the following results :
Test whether the breaking strength of the ropes differs significantly at 5% level of significance.
b) Works out the 5-yearly moving average for the data of a number of commercial industrial failures in a country during 7985 to 2000:
7. a) A cigarette company interested in the effect of gender on the type of cigarettes smoked and has collected the following data from a random sample of 150 persons :
At 95% level of significance, test whether the type of cigarette smoked and gender are independent.
b) The lifetime (in ‘000 hours) of five LED bulbs of 10 watts are 46, 40 48, 50 and 42. Draw all possible random samples of size 3 without replacement and estimate the average lifetime of the LED bulbs.
8. a) Following frequency distribution presents the lifetimes of 200 incandescent lamps :
Draw the histogram, ‘more than’ and ‘less than’ ogive. Also, find the median from the ogives.
b) Draw all the possible samples of sizes 2 without replacement from the population {8, 12, 20} and verify that the sample mean is the unbiased estimator of population mean. Find the variance of the estimate the population mean.
9. a) Suppose that a testing procedure A results in 20 unacceptable transistors out of 100 produced whereas another testing procedure B results in 12 unacceptable transistors out of 100 produced. Can we conclude at 5% level of significance that the two methods are equivalent?
b) Two different sampling techniques were adopted while investigating the same group of students to find the number of students falling in different intelligence level. The results are tabulated as follows :
At 5% of level of significance, test whether the sampling techniques adopted significantly different ?
10. Which of the following statements are True and False? Justify your answers.
a) If bxy = 0⋅9 and byx = −0⋅9, then r = 0⋅81.
b) If a pair of dice is thrown, then the probability that the sum of the number on the top faces is greater than 12 is 0.
c) If the probability of being left-handed is 0⋅1, then the probability that none of the 3 persons selected randomly is left-handed is 0⋅729.
d) Analysis of variance is a technique to test the equality of variation in several populations.
e) If the level of significance is the same, the area of the rejection in a 2-tailed test is less than in a 1-tailed test.

- a) Following is the distribution of marks (out of 25) obtained by 10 students in Physics and Mathematics. Draw a scatter diagram for all the 10 students and calculate the correlation between marks of Physics and Mathematics. b) Two cards are drawn simultaneously or successively without replacement from a well-shuffled deck of 52 cards. Find the probability distribution of number of aces (x). Give a graphical representation of probability distribution.
- a) There are two samples of 1200 and 900 people drawn from populations respectively, which have 30% and 25% of fair-haired people. Test whether the samples drawn from this population maintain difference or not. (Use α = 05.0) b) A bicycle shop sells the following number of bicycles from 1990 to 2000. Compute the first three moving averages of length 3 for the bicycle sales data and place them in line with the corresponding year.
- a) Two samples of 9 and 8 sizes give the sum of squares of deviations from respective means equal to 160 and 91 inches squares. Test whether these samples have been drawn from same normal population or not? (Use α =0.05 ) b) In a locality of 18,000 families, a random sample of 840 families was taken. Of these 840 families, 206 families were found to have a monthly income of Rs. 500 or less. Give the confidence interval for the families having income Rs. 500 or less.
- a) Plating of gold on wrist watches requires a chemical called AiCI3. Concentration of the chemical that is added to the solution is an important factor. The objective is to choose that concentration for which plating is uniform. Three concentration of AiCI3 chosen were 5%, 15% and 20%. These were added in the solution and plating was done and thickness (in units) measured on 5 samples is gives below. Use ANOVA to comment on whether the concentration of AiCI3 gives same result or 3 not. (Use α = 0.05 ) b) Consider a population of 11 schools from which a sample of size 3 is to be selected. List all the possible samples by circular systematic sampling.
- a) Consider the case of tossing two dices. Let A: be the event of getting an odd total B: the event of getting on the first dice, and C: the event of getting a total of seven on two dices. Then (i) Find P (A∩C),P (A∩B) and P(B∩C). Check whether A,B and C are independent or not. Give reasons for your answers. b) It is desired to estimate average annual wool yield per sheep for a herd of 150 at a certain farm house using stratified simple random sampling. Sheeps in the herd are to be grouped into three strata on the basis of first shaving in days. Optimum method sample allocation is said to be used for selecting the overall sample of 25 sheeps from the three strata. Determine approximately optimum strata boundaries using the information on first shaving given below.
- a) Find the probability that at most 5 defective fuses will be found in a box of 200, if experience shows that 20% of such fuses are defective. b) The following table gives the hours spent by 50 students in the playground in a week: Calculate the mean median and standard deviation of the hours required by students. c) In a television appearance, an MLA from a state asks voters in that state to indicate whether the MLA should vote against or for a particular piece of legislation. The MLA office receives 4100 replies. Since voters from western part tends to be democrats while those from eastern parts tend to e communists, the MLA divides the replies on the basis of their geographical origin. 68% of the replies from the western part are against the legislation and 36% of the replies from the eastern part are against it. (i) Did the MLA carry out a stratified sampling? (ii) Did the MLA carry out a stratified random sampling? (iii) How would you justify the sample design? (iv) What improvements would you suggest?
- a) At a call centre, callers have to wait till an operator is ready to take their call. To monitor this process, 5 calls were recorded every hour for the 8-hour working day. The data below shows the waiting time in weeks: Compute the control limits for x and R -charts. Draw the control charts for x and ,R and comments. b) Write three advantage and three disadvantage of using a sampling approach instead of a census approach for studying a characteristic.
- a) A shop manufacturer is trying to determine whether or not to market a new type of shop. It chooses a random samples of equivalent size in England and France. It asks each of the people in each sample to try the new soap, and see whether he or she likes it better than other soaps. The results are as follows: Test the hypothesis that there are international difference in the proportion of people who prefer the new soap, using 5% level of significance. b) Suppose from a total of 120 guava trees, 5 clusters of 4 trees each selected and the yield (in kg) is recorded below: Estimate the average yield per tree and its standard error.
- a) A manufacture of rayon wants to compare that the yield strength of 5 11⋅ kg/ 2 mm is met or not at 5% level of significance. The manufacturer draws a sample and calculates the mean to be 8 12⋅ kg/ 2 mm and the standard derivation is known to be 2⋅0kg/mm2. Carry out the statistical test appropriate for this. b) A box contains 10 screws, 3 of which are defective. Two screws are drawn at random. Find the probability that none of them is defective, if the sample is drawn (i) with replacement, (ii) without replacement. c) A researcher measures the temperature of a solution 5 times. The observations are 25, 32, 27, 29, 30 o C. Determine an unbiased estimate of the temperature.
- Which of the following statements are true? Give reasons for your answer. (i) F-distribution is always used in goodness of fit. (ii) Measure of central tendency in a data set refers to the extent to which the observations are scattered. (iii) A process is said to be under assignable cause when the points are above the VCL link of chart. (X,R) (iv) Simple random sampling is done by using random number tables where the probability of drawing a digit is 0.1. (v) All time series have a trend.

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