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Q1. What are the functions of Statistics? Distinguish between Primary data and Secondary data.

Statistics as a discipline is considered indispensable in almost all spheres of human knowledge. There is hardly any branch of study which does not use statistics. Scientific, social and economic studies use statistics in one form or another. These disciplines make-use of observations, facts and figures, enquiries and experiments etc. using statistics and statistical methods. Statistics studies almost all aspects in an enquiry. It mainly aims at simplifying the complexity of information collected in an enquiry. It presents data in a simplified form as to make them intelligible. It analyses data and facilitates drawal of conclusions. Now let us briefly discuss some of the important functions of statistics.

Presents facts in. simple form: Statistics presents facts and figures in a definite form. That makes the statement logical and convincing than mere description. It condenses the whole mass of figures into a single figure. This makes the problem intelligible.

Reduces the Complexity of data: Statistics simplifies the complexity of data. The raw data are unintelligible. We make them simple and intelligible by using different statistical measures. Some such commonly used measures are graphs, averages, dispersions, skewness, kurtosis, correlation and regression etc. These measures help in interpretation and drawing inferences. Therefore, statistics enables to enlarge the horizon of one's knowledge.

Facilitates comparison: Comparison between different sets of observation is an important function of statistics. Comparison is necessary to draw conclusions as Professor Boddington rightly points out.” the object of statistics is to enable comparison between past and present results to ascertain the reasons for changes, which have taken place and the effect of such changes in future. So to determine the efficiency of any measure comparison is necessary. Statistical devices like averages, ratios, coefficients etc. are used for the purpose of comparison.

Testing hypothesis: Formulating and testing of hypothesis is an important function of statistics. This helps in developing new theories. So statistics examines the truth and helps in innovating new ideas.

Formulation of Policies : Statistics helps in formulating plans and policies in different fields. Statistical analysis of data forms the beginning of policy formulations. Hence, statistics is essential for planners, economists, scientists and administrators to prepare different plans and programmes.

Forecasting : The future is uncertain. Statistics helps in forecasting the trend and tendencies. Statistical techniques are used for predicting the future values of a variable. For example a producer forecasts his future production on the basis of the present demand conditions and his past experiences. Similarly, the planners can forecast the future population etc. considering the present population trends.

Derives valid inferences :

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Statistical methods mainly aim at deriving inferences from an enquiry. Statistical techniques are often used by scholars’ planners and scientists to evaluate different projects. These techniques are also used to draw inferences regarding population parameters on the basis of sample information. Statistics is very helpful in the field of business, research, Education etc., some of the uses of Statistics are:

Statistics helps in providing a better understanding and exact description of a phenomenon of nature.

Statistics helps in proper and efficient planning of a statistical inquiry in any field of study.

Statistical helps in collecting an appropriate quantitative data. Statistics helps in presenting complex data in a suitable tabular, diagrammatic and

graphic form for any easy and comprehension of the data. Statistics helps in understanding the nature and pattern of variability of a

phenomenon through quantitative observations. Statistics helps in drawing valid inference, along with a measure of their reliability

about the population parameters from the sample data

Any statistical data can be classified under two categories depending upon the sources utilized. These categories are,

1. Primary data 2. Secondary data

Primary Data:

Primary data is the one, which is collected by the investigator himself for the purpose of a specific inquiry or study. Such data is original in character and is generated by survey conducted by individuals or research institution or any organisation.

1. The collection of data by the method of personal survey is possible only if the area covered by the investigator is small. Collection of data by sending the enumerator is bound to be expensive. Care should be taken twice that the enumerator record correct information provided by the informants.

2. Collection of primary data by framing a schedules or distributing and collecting questionnaires by post is less expensive and can be completed in shorter time.

3. Suppose the questions are embarrassing or of complicated nature or the questions probe into personnel affairs of individuals, then the schedules may not be filled with accurate and correct information and hence this method is unsuitable

4. The information collected for primary data is mere reliable than those collected from the secondary data.

Importance of Primary data cannot be neglected. A research can be conducted without secondary data but a research based on only secondary data is least reliable and may have biases because secondary data has already been manipulated by human beings. In statistical surveys it is

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necessary to get information from primary sources and work on primary data: for example, the statistical records of female population in a country cannot be based on newspaper, magazine and other printed sources. One such sources are old and secondly they contain limited information as well as they can be misleading and biased.

Secondary Data:

Secondary data are those data which have been already collected and analysed by some earlier agency for its own use; and later the same data are used by a different agency. According to W.A.Neiswanger, ‘ A primary source is a publication in which the data are published by the same authority which gathered and analysed them. A secondary source is a publication, reporting the data which have been gathered by other authorities and for which others are responsible’.

1. Secondary data is cheap to obtain. Many government publications are relatively cheap and libraries stock quantities of secondary data produced by the government, by companies and other organizations.

2. Large quantities of secondary data can be got through internet.3. Much of the secondary data available has been collected for many years and therefore it

can be used to plot trends.4. Secondary data is of value to: - The government – help in making decisions and planning

future policy. Business and industry – in areas such as marketing, and sales in order to appreciate the general economic and social conditions and to provide information on competitors. Research organizations – by providing social, economical and industrial information.

Secondary data can be less valid but its importance is still there. Sometimes it is difficult to obtain primary data; in these cases getting information from secondary sources is easier and possible. Sometimes primary data does not exist in such situation one has to confine the research on secondary data. Sometimes primary data is present but the respondents are not willing to reveal it in such case too secondary data can suffice: for example, if the research is on the psychology of transsexuals first it is difficult to find out transsexuals and second they may not be willing to give information you want for your research, so you can collect data from books or other published sources.

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Q2. Draw a histogram for the following distribution:

Age 0-10 10-20 20-30 30-40 40-50

No. of people

5 10 15 8 2

Frequency0

2

4

6

8

10

12

14

16

0-10 10-20 20-30 30-40 40-50

Q3. Find the median value of the following set of values:

45, 32, 31, 46, 40, 28, 27, 37, 36, 41

Arranging in ascending order

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27, 28, 31, 32, 36, 37, 40, 41, 45, 46

Median= 36+37/2

= 36.5

Q4. Calculate the standard deviation of the following data:

Marks 78-80 80-82 82-84 84-86 86-88 88-90

No. of students

3 15 26 23 9 14

Arithmetic mean= 3+15+26+23+9+4

6

= 15

X M (X-M) (X-M)2

3 15 -12 14415 15 0 026 15 11 12123 15 8 649 15 -6 364 15 -11 121

S=√∑❑ ( x−M )2/n-1

= √❑81

= 9

Q5. An unbiased coin is tossed six times. What is the probability that the tosses will result in: (i) exactly two heads (ii) at least five heads

Q5. Explain briefly the types of sampling.

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Two categories: probability samples or non-probability samples.

PROBABILITY SAMPLES

The idea behind this type is random selection. More specifically, each sample from the population of

interest has a known probability of selection under a given sampling scheme. There are four

categories of probability samples described below.

SIMPLE RANDOM SAMPLING

The most widely known type of a random sample is the simple random sample (SRS). This is

characterized by the fact that the probability of selection is the same for every case in the population.

Simple random sampling is a method of selecting n units from a population of size N such that every

possible sample of size and has equal chance of being drawn.

An example may make this easier to understand. Imagine you want to carry out a survey of 100

voters in a small town with a population of 1,000 eligible voters. With a town this size, there are "old-

fashioned" ways to draw a sample. For example, we could write the names of all voters on a piece of

paper, put all pieces of paper into a box and draw 100 tickets at random. You shake the box, draw a

piece of paper and set it aside, shake again, draw another, set it aside, etc. until we had 100 slips of

paper. These 100 form our sample. And this sample would be drawn through a simple random

sampling procedure - at each draw, every name in the box had the same probability of being

chosen.

In real-world social research, designs that employ simple random sampling are difficult to come by.

We can imagine some situations where it might be possible - you want to interview a sample of

doctors in a hospital about work conditions. So you get a list of all the physicians that work in the

hospital, write their names on a piece of paper, put those pieces of paper in the box, shake and

draw. But in most real-world instances it is impossible to list everything on a piece of paper and put it

in a box, then randomly draw numbers until desired sample size is reached.

STRATIFIED RANDOM SAMPLING

In this form of sampling, the population is first divided into two or more mutually exclusive segments

based on some categories of variables of interest in the research. It is designed to organize the

population into homogenous subsets before sampling, then drawing a random sample within each

subset. With stratified random sampling the population of N units is divided into subpopulations of

units respectively. These subpopulations, called strata, are non-overlapping and together they

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comprise the whole of the population. When these have been determined, a sample is drawn from

each, with a separate draw for each of the different strata. The sample sizes within the strata are

denoted by respectively. If a SRS is taken within each stratum, then the whole sampling procedure is

described as stratified random sampling.

The primary benefit of this method is to ensure that cases from smaller strata of the population are included in sufficient numbers to allow comparison.

Stratification is a common technique. There are many reasons for this, such as:

1. If data of known precision are wanted for certain subpopulations, than each of

these should be treated as a population in its own right.

2. Administrative convenience may dictate the use of stratification, for example, if

an agency administering a survey may have regional offices, which can

supervise the survey for a part of the population.

3. Sampling problems may be inherent with certain sub populations, such as

people living in institutions (e.g. hotels, hospitals, prisons).

4. Stratification may improve the estimates of characteristics of the whole

population. It may be possible to divide a heterogeneous population into sub-

populations, each of which is internally homogenous. If these strata are

homogenous, i.e., the measurements vary little from one unit to another; a

precise estimate of any stratum mean can be obtained from a small sample in

that stratum. The estimate can then be combined into a precise estimate for the

whole population.

5. There is also a statistical advantage in the method, as a stratified random

sample nearly always results in a smaller variance for the estimated mean or

other population parameters of interest.

SYSTEMATIC SAMPLING

This method of sampling is at first glance very different from SRS. In practice, it is a variant of simple

random sampling that involves some listing of elements - every nth element of list is then drawn for

inclusion in the sample. Say you have a list of 10,000 people and you want a sample of 1,000.

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Creating such a sample includes three steps:

1. Divide number of cases in the population by the desired sample size. In this

example, dividing 10,000 by 1,000 gives a value of 10.

2. Select a random number between one and the value attained in Step 1. In this

example, we choose a number between 1 and 10 - say we pick 7.

3. Starting with case number chosen in Step 2, take every tenth record (7, 17, 27,

etc.).

More generally, suppose that the N units in the population are ranked 1 to N in some order (e.g.,

alphabetic). To select a sample of n units, we take a unit at random, from the 1st k units and take

every k-th unit thereafter.

The advantages of systematic sampling method over simple random sampling include:

1. It is easier to draw a sample and often easier to execute without mistakes. This

is a particular advantage when the drawing is done in the field.

2. Intuitively, you might think that systematic sampling might be more precise than

SRS. In effect it stratifies the population into n strata, consisting of the 1st k

units, the 2nd k units, and so on. Thus, we might expect the systematic sample

to be as precise as a stratified random sample with one unit per stratum. The

difference is that with the systematic one the units occur at the same relative

position in the stratum whereas with the stratified, the position in the stratum is

determined separately by randomization within each stratum.

CLUSTER SAMPLING

In some instances the sampling unit consists of a group or cluster of smaller units that we call

elements or subunits (these are the units of analysis for your study). There are two main reasons for

the widespread application of cluster sampling. Although the first intention may be to use the

elements as sampling units, it is found in many surveys that no reliable list of elements in the

population is available and that it would be prohibitively expensive to construct such a list. In many

countries there are no complete and updated lists of the people, the houses or the farms in any large

geographical region.

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Even when a list of individual houses is available, economic considerations may point to the choice

of a larger cluster unit. For a given size of sample, a small unit usually gives more precise results

than a large unit. For example a SRS of 600 houses covers a town more evenly than 20 city blocks

containing an average of 30 houses apiece. But greater field costs are incurred in locating 600

houses and in traveling between them than in covering 20 city blocks. When cost is balanced

against precision, the larger unit may prove superior.

Important things about cluster sampling:

1. Most large scale surveys are done using cluster sampling;

2. Clustering may be combined with stratification, typically by clustering within

strata;

3. In general, for a given sample size n cluster samples are less accurate than the

other types of sampling in the sense that the parameters you estimate will have

greater variability than an SRS, stratified random or systematic sample.

NONPROBABILITY SAMPLING

Social research is often conducted in situations where a researcher cannot select the kinds of

probability samples used in large-scale social surveys. For example, say you wanted to study

homelessness - there is no list of homeless individuals nor are you likely to create such a list.

However, you need to get some kind of a sample of respondents in order to conduct your research.

To gather such a sample, you would likely use some form of non-probability sampling.

To reiterate, the primary difference between probability methods of sampling and non-probability

methods is that in the latter you do not know the likelihood that any element of a population will be

selected for study.

There are four primary types of non-probability sampling methods:

AVAILABILITY SAMPLING

Availability sampling is a method of choosing subjects who are available or easy to find. This method

is also sometimes referred to as haphazard, accidental, or convenience sampling. The primary

advantage of the method is that it is very easy to carry out, relative to other methods. A researcher

can merely stand out on his/her favorite street corner or in his/her favorite tavern and hand out

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surveys. One place this used to show up often is in university courses. Years ago, researchers often

would conduct surveys of students in their large lecture courses. For example, all students taking

introductory sociology courses would have been given a survey and compelled to fill it out. There are

some advantages to this design - it is easy to do, particularly with a captive audience, and in some

schools you can attain a large number of interviews through this method.

The primary problem with availability sampling is that you can never be certain what population the

participants in the study represent. The population is unknown, the method for selecting cases is

haphazard, and the cases studied probably don't represent any population you could come up with.

However, there are some situations in which this kind of design has advantages - for example,

survey designers often want to have some people respond to their survey before it is given out in the

"real" research setting as a way of making certain the questions make sense to respondents. For

this purpose, availability sampling is not a bad way to get a group to take a survey, though in this

case researchers care less about the specific responses given than whether the instrument is

confusing or makes people feel bad.

Despite the known flaws with this design, it's remarkably common. Ask a provocative question, give

telephone number and web site address ("Vote now at CNN.com), announce results of poll. This

method provides some form of statistical data on a current issue, but it is entirely unknown what

population the results of such polls represents. At best, a researcher could make some conditional

statement about people who are watching CNN at a particular point in time who cared enough about

the issue in question to log on or call in.

QUOTA SAMPLING

Quota sampling is designed to overcome the most obvious flaw of availability sampling. Rather than

taking just anyone, you set quotas to ensure that the sample you get represents certain

characteristics in proportion to their prevalence in the population. Note that for this method, you have

to know something about the characteristics of the population ahead of time. Say you want to make

sure you have a sample proportional to the population in terms of gender - you have to know what

percentage of the population is male and female, then collect sample until yours matches. Marketing

studies are particularly fond of this form of research design.

The primary problem with this form of sampling is that even when we know that a quota sample is

representative of the particular characteristics for which quotas have been set, we have no way of

knowing if sample is representative in terms of any other characteristics. If we set quotas for gender

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and age, we are likely to attain a sample with good representativeness on age and gender, but one

that may not be very representative in terms of income and education or other factors.

Moreover, because researchers can set quotas for only a small fraction of the characteristics

relevant to a study quota sampling is really not much better than availability sampling. To reiterate,

you must know the characteristics of the entire population to set quotas; otherwise there's not much

point to setting up quotas. Finally, interviewers often introduce bias when allowed to self-select

respondents, which is usually the case in this form of research. In choosing males 18-25,

interviewers are more likely to choose those that are better-dressed, seem more approachable or

less threatening. That may be understandable from a practical point of view, but it introduces bias

into research findings.

PURPOSIVE SAMPLING

Purposive sampling is a sampling method in which elements are chosen based on purpose of the

study. Purposive sampling may involve studying the entire population of some limited group

(sociology faculty at Columbia) or a subset of a population (Columbia faculty who have won Nobel

Prizes). As with other non-probability sampling methods, purposive sampling does not produce a

sample that is representative of a larger population, but it can be exactly what is needed in some

cases - study of organization, community, or some other clearly defined and relatively limited group.

SNOWBALL SAMPLING

Snowball sampling is a method in which a researcher identifies one member of some population of

interest, speaks to him/her, then asks that person to identify others in the population that the

researcher might speak to. This person is then asked to refer the researcher to yet another person,

and so on.

Snowball sampling is very good for cases where members of a special population are difficult to

locate. For example, several studies of Mexican migrants in Los Angeles have used snowball

sampling to get respondents.

The method also has an interesting application to group membership - if you want to look at pattern

of recruitment to a community organization over time, you might begin by interviewing fairly recent

recruits, asking them who introduced them to the group. Then interview the people named, asking

them who recruited them to the group.

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The method creates a sample with questionable representativeness. A researcher is not sure who is

in the sample. In effect snowball sampling often leads the researcher into a realm he/she knows little

about. It can be difficult to determine how a sample compares to a larger population. Also, there's an

issue of who respondents refer you to - friends refer to friends, less likely to refer to ones they don't

like, fear, etc.

Q1. Explain the following terms with respect to statistics:(i) Sample

In statistics, a sample is a subset of a population. Typically, the population is very large, making a census or a complete enumeration of all the values in the population impractical or impossible. The sample represents a subset of manageable size. Samples are collected and statistics are calculated from the samples so that one can make inferences or extrapolations from the sample to the population. This process of collecting information from a sample is referred to as sampling.

A complete sample is a set of objects from a parent population that includes ALL such objects that satisfy a set of well-defined selection criteria. For example, a complete sample of Australian

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men taller than 2m would consist of a list of every Australian male taller than 2m. But it wouldn't include German males, or tall Australian females, or people shorter than 2m. So to compile such a complete sample requires a complete list of the parent population, including data on height, gender, and nationality for each member of that parent population. In the case of human populations, such a complete list is unlikely to exist, but such complete samples are often available in other disciplines, such as complete magnitude-limited samples of astronomical objects.

An unbiased sample is a set of objects chosen from a complete sample using a selection process that does not depend on the properties of the objects. For example, an unbiased sample of Australian men taller than 2m might consist of a randomly sampled subset of 1% of Australian males taller than 2m. But one chosen from the electoral register might not be unbiased since, for example, males aged under 18 will not be on the electoral register. In an astronomical context, an unbiased sample might consist of that fraction of a complete sample for which data are available, provided the data availability is not biased by individual source properties.

The best way to avoid a biased or unrepresentative sample is to select a random sample, also known as a probability sample. A random sample is defined as a sample where each individual member of the population has a known, non-zero chance of being selected as part of the sample. Several types of random samples are simple random samples, systematic samples, stratified random samples, and cluster random samples.

(ii) Variable

A variable is a characteristic that may assume more than one set of values to which a numerical measure can be assigned.

Height, age, amount of income, province or country of birth, grades obtained at school and type of housing are all examples of variables. Variables may be classified into various categories, some of which are outlined in this section.

Categorical variables: A categorical variable (also called qualitative variable) is one for which each response can be put into a specific category. These categories must be mutually exclusive and exhaustive. Mutually exclusive means that each possible survey response should belong to only one category, whereas, exhaustive requires that the categories should cover the entire set of possibilities. Categorical variables can be either nominal or ordinal.

Nominal variables: A nominal variable is one that describes a name or category. Contrary to ordinal variables, there is no 'natural ordering' of the set of possible names or categories.

Ordinal variables: An ordinal variable is a categorical variable for which the possible categories can be placed in a specific order or in some 'natural' way.

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Numeric variables: A numeric variable, also known as a quantitative variable, is one that can assume a number of real values—such as age or number of people in a household. However, not all variables described by numbers are considered numeric. For example, when you are asked to assign a value from 1 to 5 to express your level of satisfaction, you use numbers, but the variable (satisfaction) is really an ordinal variable. Numeric variables may be either continuous or discrete.

Continuous variables: A variable is said to be continuous if it can assume an infinite number of real values. Examples of a continuous variable are distance, age and temperature.

The measurement of a continuous variable is restricted by the methods used, or by the accuracy of the measuring instruments. For example, the height of a student is a continuous variable because a student may be 1.6321748755... metres tall.

Discrete variables: As opposed to a continuous variable, a discrete variable can only take a finite number of real values. An example of a discrete variable would be the score given by a judge to a gymnast in competition: the range is 0 to 10 and the score is always given to one decimal (e.g., a score of 8.5).

(iii) Population

A statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest. Notice that if we choose a population like all crows, we will be limited to observing crows that exist now or will exist in the future. Probably, geography will also constitute a limitation in that our resources for studying crows are also limited.

Population is also used to refer to a set of potential measurements or values, including not only cases actually observed but those that are potentially observable. Suppose, for example, we are interested in the set of all adult crows now alive in the county of Cambridge shire, and we want to know the mean weight of these birds. For each bird in the population of crows there is a weight, and the set of these weights is called the population of weights.

A subset of a population is called a subpopulation. If different subpopulations have different properties, the properties and response of the overall population can often be better understood if it is first separated into distinct subpopulations.

For instance, a particular medicine may have different effects on different subpopulations, and these effects may be obscured or dismissed if such special subpopulations are not identified and examined in isolation.

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Similarly, one can often estimate parameters more accurately if one separates out subpopulations: distribution of heights among people is better modeled by considering men and women as separate subpopulations, for instance.

Populations consisting of subpopulations can be modeled by mixture models, which combine the distributions within subpopulations into an overall population distribution.

Q2. What are the types of classification of Data?

According to Nature1. Quantitative data- information obtained from numeral variables(e.g. age, bills, etc) 2. Qualitative Data- information obtained from variables in the form of categories, characteristics names or labels or alphanumeric variables (e.g. birthdays, gender etc.)

According to Source1. Primary data- first- hand information (e.g. autobiography, financial statement) 2. Secondary data- second-hand information (e.g. biography, weather forecast from news papers)

According to Measurement1. Discrete data- countable numerical observation.-Whole numbers only has an equal whole number interval obtained through counting(e.g. corporate stocks, etc.) 2. Continuous data-measurable observations. -decimals or fractions obtained through measuring(e.g. bank deposits, volume of liquid etc.)

QUALITATIVE DATA

Qualitative data is a categorical measurement expressed not in terms of numbers, but rather by means of a natural language description. In statistics, it is often used interchangeably with "categorical" data.

For example: favorite color = "yellow"

height = "tall"

Although we may have categories, the categories may have a structure to them. When there is not a natural ordering of the categories, we call these nominal categories. Examples might be gender, race, religion, or sport.

When the categories may be ordered, these are called ordinal variables. Categorical variables that judge size (small, medium, large, etc.) are ordinal variables. Attitudes (strongly disagree, disagree, neutral, agree, strongly agree) are also ordinal variables, however we may not know which value is the best or worst of these issues. Note that the distance between these categories is not something we can measure.

QUANTITATIVE DATA

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Quantitative data is a numerical measurement expressed not by means of a natural language description, but rather in terms of numbers. However, not all numbers are continuous and measurable. For example, the social security number is a number, but not something that one can add or subtract.

For example: favorite color = "450 nm"

height = "1.8 m"

Quantitative data always are associated with a scale measure.

Probably the most common scale type is the ratio-scale. Observations of this type are on a scale that has a meaningful zero value but also have an equidistant measure (i.e., the difference between 10 and 20 is the same as the difference between 100 and 110). For example, a 10 year-old girl is twice as old as a 5 year-old girl. Since you can measure zero years, time is a ratio-scale variable. Money is another common ratio-scale quantitative measure. Observations that you count are usually ratio-scale (e.g., number of widgets).

A more general quantitative measure is the interval scale. Interval scales also have a equidistant measure. However, the doubling principle breaks down in this scale. A temperature of 50 degrees Celsius is not "half as hot" as a temperature of 100, but a difference of 10 degrees indicates the same difference in temperature anywhere along the scale. The Kelvin temperature scale, however, constitutes a ratio scale because on the Kelvin scale zero indicates absolute zero in temperature, the complete absence of heat. So one can say, for example, that 200 degrees Kelvin is twice as hot as 100 degrees Kelvin.

PRIMARY DATA

Primary data means original data that has been collected specially for the purpose in mind. It means when an authorized organization, investigator or an enumerator collects the data for the first time from the original source. Data collected this way is called primary data.

SECONDARY DATA

Secondary data is data that has been collected for another purpose. When we use Statistical Method with Primary Data from another purpose for our purpose we refer to it as Secondary Data. It means that one purpose's Primary Data is another purpose's Secondary Data. Secondary data is data that is being reused. Usually in a different context.

Q3. Find the (i) arithmetic mean and (ii) range of the following data 15, 77, 22, 21, 19, 26, 20

Arithmetic mean= (15+77+22+21+19+26+20)/7

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=140/7

=20

Range = highest number- lowest number/2

= 58/2

=29

Q4. 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 (ii) Atleast one house catch fire

Q5. (i) What are the characteristics of Chi-square test?

1. It is not symmetric2. The shape of the chi-square distribution depends upon the degrees of freedom,

just like Student’s t-distribution. 3. As the number of degrees of freedom increases, the chi-square distribution

becomes more symmetric as is illustrated in Figure 1. 4. The values are non-negative. That is, the values of are greater than or equal to 0.5. This is not a test, but a distribution. The Chi-square distribution, is derived from

the Normal distribution. It is the distribution of a sum of squared Normal distributed variables. That is, if all Xi are independent and all have an identical, standard Normal distribution then X^2 = X1*X1 + X2*X2 + X3*X3 + ... + Xv*Xv is Chi-square distributed with v degrees of freedom with mean = v and variance = 2*v. The importance of the Chi-square distribution stems from the fact that it describes the distribution of the Variance of a sample taken from a Normal distributed population.

6. Chi-square is non-negative. Is the ratio of two non-negative values, therefore must be non-negative itself

7. There are many different chi-square distributions, one for each degree of freedom8. The degrees of freedom when working with a single population variance is n-1.

since the chi-square distribution isn't symmetric, the method for looking up left-tail values is different from the method for looking up right tail values.

Area to the right - just use the area given. Area to the left - the table requires the area to the right, so subtract the given area from

one and look this area up in the table.

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Area in both tails - divide the area by two. Look up this area for the right critical value and one minus this area for the left critical value.

(ii) The data given in the below table shows the production n three shifts and the number of defective goods that turned out in three weeks. Test at 5% level of significance whether the weeks and shifts are independent.

Shift 1st week 2ndweek 3rd week TotalI 15 5 20 40II 20 10 20 50III 25 15 20 60Total 60 30 60 150

Q6. Find Karl Pearson’s correlation co-efficient for the data given below table:

X 20 16 12 8 4Y 22 14 4 12 8

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MB0040-STATISTICS FOR MANAGEMENT

MB0040 Page 19