**Statistics for Management assignment sample question-answer**

**Q1. A statistical survey is a scientific process of collection and analysis of numerical data. Explain the stages of statistical survey. Describe the various methods for collecting data in a statistical survey.**

**Ans: **

__Meaning of statistical survey__

__Meaning of statistical survey__

A Statistical survey is a scientific process of collection and analysis of numerical data. Statistical surveys are used to collect numerical information about units in a population. Surveys involve asking questions to individuals. Surveys of human populations are common in government, health, social science and marketing sectors.

__Stages of statistical survey (Listing and Explanation)__

Statistical surveys are categorised into two stages:

**Planning****Execution**

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**Planning a Statistical Survey**: The relevance and accuracy of data obtained in a survey depends upon the care exercised in planning. A properly planned investigation can lead to best results with least cost and time. The figure 2.2 gives the explanation of steps involved in the planning stage.

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**Execution of Statistical survey**: Control methods should be adopted at every stage of carrying out the investigation to check the accuracy, coverage, methods of measurements, analysis and interpretation. The collected data should be edited, classified, tabulated and presented in diagrams and graphs. The data should be carefully and systematically analysed and interpreted.

__Methods for collecting data__** **

__Methods for collecting data__

**Direct personal observation:**In the direct personal observation method, the investigator collects data by having direct contact with units of investigation. The accuracy of data depends upon the ability, training and attitude of the investigator.

**Indirect oral interview:**Indirect oral interview is used when the area to be covered is large. The investigator collects the data from a third party or witness or head of institution. This method is generally used by police department in cases related to enquiries on causes of fires, thefts or murders.

**Information through agencies:**Methods of collecting information through local agencies or correspondents are generally adopted by newspaper and television channels. Local agents are appointed in different parts of the area under investigation. They send the desired information at regular intervals.

**Information through mailed questionnaires:**Often, information is collected through questionnaires. The questionnaires are filled with questions pertaining to the investigation. They are sent to the respondents with a covering letter soliciting cooperation from the respondents. The respondents are asked to give correct information and to mail the questionnaire back.

**Information through schedule filled by investigators:**Information can be collected through schedules filled by investigators through personal contact. In order to get reliable information, the investigator should be well trained, tactful, unbiased and hard working.

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**Q2. Explain the approaches to define probability.**

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**Ans: **There are four approaches to probability. The figure 5.3. shows the four approaches to probability. They are:

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**Classical / Mathematical / Priori approach**

Under this approach the probability of an event is known before conducting the experiment.

The following are some of the examples of classical approach.

a) Getting a head in tossing a coin

b) Drawing a king from well shuffled pack

c) Getting a ‘6’ in throwing a die.

**Statistical / Relative Frequency / Empirical / Posteriori approach**

Under this approach the probability of an event is arrived at after conducting an experiment. If we want to know the probability that a particular household in an area will have two earning members, then we have to gather data on all household in that area and arrive at the probability. The greater number of households surveyed, the more accurate will be the probability arrived.

**Subjective approach**

Under this approach the investigator or researcher assigns probability to the events either from his experience or from past records. It is more suitable when the sample size is ten or less than ten. The investigator has full knowledge about the characteristics of each and every individual. However, there is a chance of personal bias being introduced in such probability.

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**Axiomatic approach**

This approach is based on set theory. The probability of an event is defined as:

P(A) = n(A) / n(S); Such that

- 0 ≤ P(Ai) ≤ 1
**b.**∑ P(Ai) = 1 for I =I 1 to n

where, Ai is ‘n’ mutually exclusive and exhaustive events.

**Q3.**

**a) The procedure of testing hypothesis requires a researcher to adopt several steps. Describe in brief all such steps.**

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**Ans: **There are five steps involved in testing on hypothesis while are as follows:

**Formulate a Hypothesis:** The first step is to set up two hypothesis instead of one in such a way that if one hypothesis is true, the other is false. Alternatively; if one hypothesis is false or rejected then the other is true or accepted.

**Set up a suitable significance level: **After formulating the hypothesis, the next steps is to test its validity at a certain level of significance. The confidence with which a null hypothesis is rejected or accepted depends on the significance level used for the purpose.

**Select test criterion: **The next steps in hypothesis testing in the selection of an appropriate statistical technique as a test criterion. There are many techniques from which one is to be chosen. For example, when the hypothesis partners to a large of more than 30, the Z test implying normal distribution is used for population mean. If the sample is small (n<30) the t test will be more appropriate. The test criteria that are frequently used in hypothesis testing are Z, t, f and x^{2}.

**Compute: **After selecting the sampling technique to less the hypotheses, the next step includes various computations necessary for the application of that particular test. These computations include the testing statistic as also its standard error.

**Making decision:** The final step in hypothesis testing is to draw a statistical decreases, involving the acceptance or rejection of the null hypothesis.

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**b) A sample of 400 items is taken from a normal population whose mean as well as variance is 4. If the sample mean is 4.5, can the sample be regarded as a truly random sample?**

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**Solution:**

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**Step1:**

H_{0}: µ = 4, H_{1}: µ ≠ 4

**Step2:**

z = 4.5 – 4 = 4.5 – 4 = 5

∂ 2

√n 20

Note: Since the sample size is large, normal test is applicable.

**Step2:**

Since the value of calculated z is greater than even 1% value of tabulated z i.e. 2.58, the null hypothesis is rejected. The sample cannot be regarded as a truly random sample.

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Q4. a) What is a Chi-square test? Point out its applications. Under what conditions is this test applicable?

b) What are the components of time series? Enumerate the methods of determining trend in time series.

Q5. What do you mean by cost of living index? Discuss the methods of construction of cost of living index with an example for each.

Q6. a. What is analysis of variance? What are the assumptions of the technique?

- Three samples below have been obtained from normal populations with equal variances. Test the hypothesis at 5% level that the population means are equal.

A | B | C |

8 | 7 | 12 |

10 | 5 | 9 |

7 | 10 | 13 |

14 | 9 | 12 |

11 | 9 | 14 |

[The table value of F at 5% level of significance for 1 = 2 and 2 = 12 is 3.88]

Q7. Analysis of daily wages of workers in two organisations A and B yielded the following results:

Organisation

A B

No. of workers | 10 | 20 |

Average daily wages (Rs) | 30 | 15 |

Variance | 25 | 100 |

Obtain the average daily wages and the standard deviation of wages of all workers in the two organisations taken together. Which organisation is more equitable in regard to wages?

#### Q8. a. State the addition and multiplication rules of probability giving an example of each case.

b. In a bolt factory machines A, B, C manufacture 25, 35 and 40 percent of the total output. Of their total output 5, 4 and 2 percent are defective respectively. A bolt is drawn at random and is found to be defective. What are the probabilities that it was manufactured by machines A, B and C?

Q9. Discuss the types of measurement scales with examples.

Q10. Explain the Components of Time series.

## Q11. Statistics plays a vital role in almost every facet of human life. Describe the functions of Statistics. Explain the applications of statistics.

Q12. a. Explain the various measures of Dispersion.

b. Obtain the values of the median and the two Quartiles.

391 | 384 | 591 | 407 | 672 | 522 | 777 | 733 | 2488 | 1490 |

Q13. a. What is correlation? Distinguish between positive and negative correlation.

Q13. b. Calculate coefficient of correlation from the following data.

X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Y | 9 | 8 | 10 | 12 | 11 | 13 | 14 | 16 | 15 |

Q14. Index number acts as a barometer for measuring the value of money. What are the characteristics of an index number? State its utility.

Q15. Business forecasting acquires an important place in every field of the economy. Explain the objectives and theories of Business forecasting.

Q16. The weekly wages of 1000 workers are normally distributed around a mean of Rs. 70 and a standard deviation of Rs. 5. Estimate the number of workers whose weekly wages will be:

- Between 70 and 72
- Between 69 and 72
- More than 75
- Less than 63

Q17.

## Explain the characteristics of Statistics.

- What are the components of Statistics? Give a brief description of each of the components.

Q18. Explain the objectives of Statistical Average. What are the requisites of a good average?

Q19. a. Mention the Characteristics of Chi-square test.

b. Two research workers classified some people in income groups on the basis of sampling studies. Their results are as follow:

Investigators | Income groups | Total | ||

Poor | Middle | Rich | ||

A | 160 | 30 | 10 | 200 |

B | 140 | 120 | 40 | 300 |

Total | 300 | 150 | 50 | 500 |

Show that the sampling technique of atleast one research worker is defective.

Q20. Define trend. Enumerate the methods of determining trend in time series

Q21. Explain Chi-square test and the conditions for applying chi-square test

Q22. The following data represent the number of units of production per day turned out by 5 different workmen using different types of machines. 10 marks

Workmen | Machine Type | |||

A | B | C | D | |

1 | 44 | 38 | 47 | 36 |

2 | 46 | 40 | 52 | 43 |

3 | 34 | 36 | 44 | 32 |

4 | 43 | 38 | 46 | 33 |

5 | 38 | 42 | 49 | 39 |

i) Test whether the mean productivity is the same for the four different machine types.

ii) Test whether 5 men differ with respect to mean productivity.

Q23. a) The procedure of testing hypothesis requires a researcher to adopt several steps. Describe in brief all such steps.

b) Explain the Components of time series.

Q24. Distinguish between Classification and Tabulation. Explain the structure and components of a Table with an example.

### Q25. a) Describe the characteristics of Normal probability distribution.

b) In a sample of 120 workers in a factory, the mean and standard deviation of wages were Rs. 11.35 and Rs.3.03 respectively. Find the percentage of workers getting wages between Rs.9 and Rs.17 in the whole factory assuming that the wages are normally distributed.

Q26. b) Distinguish between:

- Stratified random sampling and Systematic sampling
- Judgement sampling and Convenience sampling
__Judgement sampling and Convenience sampling__

Q27. a) What is regression analysis? How does it differ from correlation analysis?

b) Calculate Karl Pearson’s coefficient of correlation between X series and Y series.

X | 110 | 120 | 130 | 120 | 140 | 135 | 155 | 160 | 165 | 155 |

Y | 12 | 18 | 20 | 15 | 25 | 30 | 35 | 20 | 25 | 10 |

Q28. Briefly explain the methods and theories of Business forecasting.

Q29. Construct Fisher’s Ideal Index for the given information and check whether Fisher’s formula satisfies Time Reversal and Factor Reversal Tests.

Items | P_{0} | Q_{0} | P_{1} | Q_{1} |

A | 16 | 5 | 20 | 6 |

B | 12 | 10 | 18 | 12 |

C | 14 | 8 | 16 | 10 |

D | 20 | 6 | 22 | 10 |

E | 80 | 3 | 90 | 5 |

F | 40 | 2 | 50 | 5 |

Q30. Define “Statistics” What are the functions of Statistics? Distinguish between Primary data and Secondary data

Q31. Find the (i) arithmetic mean and (ii) the median value of the following set of values: 40, 32, 24, 36, 42, 18, 10.

Q32. Explain the following terms with respect to Statistics: (i) Sample, (ii) Variable,

(iii) Population.

Q33. An unbiased coin is tossed six times. What is the probability that the tosses will result in: (i) exactly two heads and (ii) at least five heads.

Q34. Find the median value of the following set of values: 45, 32, 31, 46, 40, 28, 27, 37, 36, 41.

## Q35. Explain briefly the types of sampling

Q36. What are the types of classification of data?

Q37. Find the (i) arithmetic mean and (ii) range of the following data: 15, 17, 22, 21, 19, 26, 20.

Q38. Suppose two houses in a thousand catch fire in a year and there are 2000 houses in a village. What is the probability that: (i) none of the houses catch fire and (ii) At least one house catch fire?

Q39. Find Karl Pearson’s correlation coefficient between the sales and expenses from the data given below:

Sales (Rs. Lakhs) | 50 | 50 | 55 | 60 | 65 | 65 | 65 | 60 | 60 | 50 |

Expenses (Rs. Lakhs) | 11 | 13 | 14 | 16 | 16 | 15 | 15 | 14 | 13 | 13 |

Q40. The incidence of occupational disease in an industry is such that the workers have a 20% chance of suffering from it. What is the probability that out of six workers 4 or more will contract the disease?

Q41. Construct index numbers of price for the following data by applying:

i) Laspeyre’s method

ii) Paasche’s method

iii) Fisher’s Ideal Index number

Commodity | Base year | Current year | ||

Price | Quantity | Price | Quantity | |

A | 2 | 8 | 4 | 6 |

B | 5 | 10 | 6 | 5 |

C | 4 | 14 | 5 | 10 |

D | 2 | 19 | 2 | 13 |

Q42. Find Karl Pearson’s correlation co-efficient for the data given in the below table:

X | 18 | 16 | 12 | 8 | 4 |

Y | 22 | 14 | 12 | 10 | 8 |

Q43. Find the (i) arithmetic mean (ii) range and (iii) median of the following data: 15, 17, 22, 21, 19, 26, 20.

# Q44. (a) Explain Arithmetic mean.

(b) The mean wage is Rs. 75 per day, SD wage is Rs. 5 per day for a group of 1000 workers and the same is Rs. 60 and Rs. 4.5 for the other group of 1500 workers. Find the mean and standard deviation for the entire group.

Q45. Mr. Arun and Mr. Bhandari play a game. If Mr. Arun picks up an even number from 1 to 6, Mr. Bhandari will pay him double the amount equal to picked up number. If Mr. Arun picks up an odd number then he has to pay amount equal to double the picked up number. What is Mr. Arun’s expectation?

Q46. The probability that an employee will get an occupational disease is 20%. In a firm having five employees, what is the probability that:

i) None of the employees get the disease

ii) Exactly two will get the disease

iii) More than four will contract the disease

Q47. Microsoft estimated that out of 10,000 potential software buyers, 35% wait to purchase the new OS Windows Vista, until an upgrade has been released. After an advertising campaign to reassure the public was released, Microsoft surveyed 3000 buyers and found 950 who are still skeptical. At 5% level of significance, can the company conclude that the population of skeptical people had decreased?

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