mb0040 solved fall drive assignment 2011

8/3/2019 MB0040 Solved Fall Drive Assignment 2011

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1. (a) Statistics is the backbone of decision-making.Comment.

(b) Statistics is as good as the user. Comment.

(a) Statistics is the backbone of decision-makingDue to advanced communication network, rapid changes in consumerbehaviour, varied expectations of variety of consumers and newmarket openings, modern managers have a difficult task of makingquick and appropriate decisions. Therefore, there is a need for themto depend more upon quantitative techniques like mathematicalmodels, statistics, operations research and econometrics.

Decision making is a key part of our day-to-day life. Even when we

wish to purchase a television, we like to know the price, quality,durability, and maintainability of various brands and models beforebuying one. As you can see, in this scenario we are collecting dataand making an optimum decision. In other words, we are usingStatistics.

Again, suppose a company wishes to introduce a new product, it hasto collect data on market potential, consumer likings, availability ofraw materials, feasibility of producing the product. Hence, datacollection is the back-bone of any decision making process.Many organisations find themselves data-rich but poor in drawinginformation from it. Therefore, it is important to develop the ability toextract meaningful information from raw data to make betterdecisions. Statistics play an important role in this aspect.Statistics is broadly divided into two main categories. The twocategories of Statistics are descriptive statistics and inferentialstatistics.

Descriptive Statistics: Descriptive statistics is used to presentthe general description of data which is summarised quantitatively.This is mostly useful in clinical research, when communicating the

results of experiments.

Inferential Statistics: Inferential statistics is used to make validinferences from the data which are helpful in effective decisionmaking for managers or professionals.Statistical methods such as estimation, prediction and hypothesistesting belong to inferential statistics. The researchers makedeductions or conclusions from the collected data samples regardingthe characteristics of large population from which the samples aretaken. So, we can say Statistics is the backbone of decision-making.

(b) Statistics is as good as the user:


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Statistics is used for various purposes. It is used to simplify mass dataand to make comparisons easier. It is also used to bring out trendsand tendencies in the data as well as the hidden relations betweenvariables. All this helps to make decision making much easier. Let uslook at each function of Statistics in detail.

1. Statistics simplifies mass data

The use of statistical concepts helps in simplification of complex data.Using statistical concepts, the managers can make decisions moreeasily. The statistical methods help in reducing the complexity of thedata and consequently in the understanding of any huge mass ofdata.

2. Statistics makes comparison easier

Without using statistical methods and concepts, collection of data andcomparison cannot be done easily. Statistics helps us to comparedata collected from different sources. Grand totals, measures ofcentral tendency, measures of dispersion, graphs and diagrams,coefficient of correlation all provide ample scopes for comparison.

3. Statistics brings out trends and tendencies in the dataAfter data is collected, it is easy to analyse the trend and tendencies

in the data by using the various concepts of Statistics.

4. Statistics brings out the hidden relations between variablesStatistical analysis helps in drawing inferences on data. Statisticalanalysis brings out the hidden relations between variables.

5. Decision making power becomes easierWith the proper application of Statistics and statistical software

packages on the collected data, managers can take effectivedecisions, which can increase the profits in a business.

Seeing all these functionality we can say Statistics is as good as the user.


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2. Distinguish between the following with example.(a) Inclusive and Exclusive limits.(b) Continuous and discrete data.(c) Qualitative and Quantitative data(d) Class limits and class intervals.

Answer :

a) Inclusive and Exclusive limits.Inclusive and exclusive limits are relevant from data tabulation andclass intervals point of view.

Inclusive series is the one which doesn't consider the upper limit, forexample,00-1010-2020-3030-4040-50

In the first one (00-10), we will consider numbers from 00 to 9.99only. And 10 will be considered in 10-20. So this is known as inclusive

series.Exclusive series is the one which has both the limits included, forexample,00-0910-1920-2930-3940-49

Here, both 00 and 09 will come under the first one (00-09). And 10will come under the next one.

b) Continuous and discrete data.All data that are the result of counting are called quantitative

discrete data. These data take on only certain numerical values. Ifyou count the number of phone calls you receive for each day of theweek, you might get 0, 1, 2, 3, etc.

All data that are the result of measuring are quantitative continuousdata assuming that we can measure accurately. Measuring angles inradians might result in the numbers /6, /3, /2, /, 3/4, etc. If youand your friends carry backpacks with books in them to school, thenumbers of books in the backpacks are discrete data and the weightsof the backpacks are continuous data.


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c) Qualitative and Quantitative dataData may come from a population or from a sample. Small letters

like x or y generally are used to represent data values. Most datacan be put into the following categories:

Qualitative

Quantitative

Qualitative dataQualitative data are the result of categorizing or describing attributesof a population. Hair color, blood type, ethnic group, the car a persondrives, and the street a person lives on are examples of qualitativedata. Qualitative data are generally described by words or letters. Forinstance, hair color might be black, dark brown, light brown, blonde,gray, or red. Blood type might be AB+, O-, or B+. Qualitative data arenot as widely used as quantitative data because many numericaltechniques do not apply to the qualitative data. For example, it doesnot make sense to find an average hair color or blood type.

Quantitative dataQuantitative data are always numbers and are usually the data ofchoice because there are many methods available for analyzing thedata. Quantitative data are the result of counting or measuringattributes of a population. Amount of money, pulse rate, weight,number of people living in your town, and the number of students whotake statistics are examples of quantitative data. Quantitative data

may be either discrete or continuous.

All data that are the result of counting are called quantitative discretedata. These data take on only certain numerical values. If you countthe number of phone calls you receive for each day of the week, youmight get 0, 1, 2, 3, etc.

Example 2: Data Sample of Quantitative Continuous Data

The data are the weights of the backpacks with the books in it. Yousample the same five students. The weights (in pounds) of their

backpacks are 6.2, 7, 6.8, 9.1, 4.3. Notice that backpacks carryingthree books can have different weights. Weights are quantitativecontinuous data because weights are measured.

Example 3: Data Sample of Qualitative Data

The data are the colors of backpacks. Again, you sample the same fivestudents. One student has a red backpack, two students have blackbackpacks, one student has a green backpack, and one student has agray backpack. The colors red, black, black, green, and gray arequalitative data.


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Inclusive and Exclusive Class Intervals

Inclusive Class Interval:When the lower and the upper class limit is included, then it is aninclusive class interval. For example - 220 - 234, 235 - 249 ..... etc.are inclusive type of class intervals. Usually in the case of discretevariate, inclusive type of class intervals are used.

Exclusive Class Interval:When the lower limit is included, but the upper limit is excluded, thenit is an exclusive class interval. For example - 150 - 153, 153 -156.....etc are exclusive type of class intervals. In the class interval150 - 153, 150 is included but 153 is excluded.

Usually in the case of continuous variate, exclusive type of classintervals are used. Consider the frequency table shown below

Note: While analysing a frequency distribution, if there areinclusive type of class intervals they must be converted into

exclusive type.

This can be done by extending the class intervals from both the ends. Thus the class intervals 220 - 234, 235 - 249, ....... should beconverted into exclusive type 219.5 - 234.5, 234.5 - 249.5.... etc.

After the conversion the frequency table would look like this


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3. In a management class of 100 students three languages areoffered as an additional subject viz. Hindi, English andKannada. There are 28 students taking Kannada, 26 taking Hindiand 16 taking English. There are 12 students taking bothKannada and English, 4 taking Hindi and English and 6 that aretaking Hindi and Kannada. In addition, we know that 2 studentsare taking all the three languages.

i) If a student is chosen randomly, what is the probability that he/sheis not taking any of these three languages?

ii) If a student is chosen randomly, what is the probability that he/ sheis taking exactly one language?

Let students taking Kannada as language be S(K) = 28Let students taking Hindi as language be S(H) = 28Let students taking English as language be S(E) = 28

Let students taking Kannada and English be S (K E ) = 12

Let students taking Hindi and English be S ( H E ) = 4


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Let students taking Hindi and Kannada be S ( H K ) = 6

Let students taking all the three subjects be S (K H E ) = 2

If a student is chosen randomly, probability that he/she is not taking anyof these three languages is P ( not taking any language)

= [ 1 { S(K)+ S(H)+S(E)+ S (K E )+ S ( H E )+ S ( H K )+ S (K HE} / 100 ]

= [1 {28+26+26+6+4+12+2} / 100]= 1 (94 / 100)= 1 0.94 = 0.06

If a student is chosen randomly, probability that he/ she is taking exactlyone language is

P (taking exactly one language) = [ { S(K)+ S(H)+S(E) } ] / 100= [28+26+16] = 70

= 70= {70/100}= 0.7

4. List down various measures of central tendency and explainthe difference between them?

Measures of Central Tendency

Several different measures of central tendency are defined below.

1 Arithmetic MeanThe arithmetic mean is the most common measure of central tendency.It simply the sum of the numbers divided by the number of numbers.The symbol m is used for the mean of a population. The symbol M isused for the mean of a sample. The formula for m is shown below:

Where X is the sum of all the numbers in the numbers in the sampleand N is the number of numbers in the sample. As an example, themean of the numbers 1 + 2 + 3 + 6 + 8 = 20/5 = 4 regardless of

whether the numbers constitute the entire population or just a samplefrom the population.

The table, Number of touchdown passes (Table 1: Number of touchdownpasses), shows the number of touchdown (TD) passes thrown by each ofthe 31 teams in the National Football League in the 2000 season.

The mean number of touchdown passes thrown is 20.4516 as shownbelow.

Number of touchdown passes


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Although the arithmetic mean is not the only "mean" (there is also ageometric mean), it is by far the most commonly used. Therefore, if theterm "mean" is used without specifying whether it is the arithmeticmean, the geometric mean, or some other mean, it is assumed to referto the arithmetic mean.

2 MedianThe median is also a frequently used measure of central tendency. The

median is the midpoint of a distribution: the same number of scores areabove the median as below it. For the data in the table, Number oftouchdown passes (Table 1: Number of touchdown passes), there are 31scores. The 16th highest score (which equals 20) is the median becausethere are 15 scores below the 16th score and 15 scores above the 16thscore. The median can also be thought of as the 50th percentile3. Let'sreturn to the made up example of the quiz on which you made a threediscussed previously in the module Introduction to Central Tendency4and shown in Table 2: Three possible datasets for the 5-point make-upquiz.

Three possible datasets for the 5-point make-up quiz

For Dataset 1, the median is three, the same as your score. For Dataset2, the median is 4. Therefore, your score is below the median. Thismeans you are in the lower half of the class. Finally for Dataset 3, themedian is 2. For this dataset, your score is above the median andtherefore in the upper half of the distribution.

Computation of the Median: When there is an odd number of numbers,the median is simply the middle number. For example, the median of 2,4, and 7 is 4. When there is an even number of numbers, the median isthe mean of the two middle numbers. Thus, the median of the numbers2, 4, 7, 12 is 4+7/2 = 5:5.

3 mode


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The mode is the most frequently occuring value. For the data in thetable, Number of touchdown passes (Table 1: Number of touchdownpasses), the mode is 18 since more teams (4) had 18 touchdown passesthan any other number of touchdown passes. With continuous data suchas response time measured to many decimals, the frequency of eachvalue is one since no two scores will be exactly the same (see discussion

of continuous variables5). Therefore the mode of continuous data isnormally computed from a grouped frequency distribution. The Groupedfrequency distribution (Table 3: Grouped frequency distribution) tableshows a grouped frequency distribution for the target response timedata. Since the interval with the highest frequency is 600-700, the modeis the middle of that interval (650).

Grouped frequency distribution

Proportions and Percentages

When the focus is on the degree to which a population possesses aparticular attribute, the measure of interest is a percentage or aproportion.

A proportion refers to the fraction of the total that possesses acertain attribute. For example, we might ask what proportion ofwomen in our sample weigh less than 135 pounds. Since 3 womenweigh less than 135 pounds, the proportion would be 3/5 or 0.60.

A percentage is another way of expressing a proportion. Apercentage is equal to the proportion times 100. In our example of

the five women, the percent of the total who weigh less than 135pounds would be 100 * (3/5) or 60 percent.

NotationOf the various measures, the mean and the proportion are mostimportant. The notation used to describe these measures appears below:

X: Refers to a population mean.

x: Refers to a sample mean.

P: The proportion of elements in the population that has a particularattribute.

p: The proportion of elements in the sample that has a particularattribute.


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Q: The proportion of elements in the population that does not havea specified attribute. Note that Q = 1 - P.

q: The proportion of elements in the sample that does not have aspecified attribute. Note that q = 1 - p.

5. Define population and sampling unit for selecting a randomsample in each of the following cases.

a) Hundred voters from a constituencyb) Twenty stocks of National Stock Exchangec) Fifty account holders of State Bank of Indiad) Twenty employees of Tata motors.

Statistical survey or enquiries deal with studying various characteristics ofunit belonging to a group. The group consisting of all the units is calledUniverse or Population

Sample is a finite subset of a population. A sample is drawn from a

population to estimate the characteristics of the population. Sampling is atool which enables us to draw conclusions about the characteristics of thepopulation.

In sampling there are two types namely discrete and the other is thecontinuous. Discrete sampling is that the data given are of the finite andtheir calculations are made easy. Continuous sampling is one where thedata are of infinite form. Its intervals are indicated by , greater thanbut lesser than, lesser than and greater than..

The finite number of items in a sample is size. in practice samples greaterthan 30 are large samples and if less it is small samples.


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A measure associated with the entire population is called as populationparameter. Or just an parameter.

Given a population, suppose we consider all possible samples of a certainsize N that can be drawn from the population. For each sample supposewe compute a statistic such as mean, standard deviation etc. These

sample vary from sample to sample. We group these different statisticsaccording to their frequencies which is called as frequency distribution toformed so called as sampling distribution., standard deviation of asampling distribution is called its standard error.

Suppose we draw all possible samples of a certain size N from apopulation and find the mean of X bar of each of these samples.Frequency distribution of these means is called as sampling distribution

of means. If the population is infinite , then , , be the standarddeviation and mean respectively then the standard deviation denoted by

is given by

= / sqrt of N

is used to calculate the standard normal variate for the population whereits size is more than 30.

6. What is a confidence interval, and why it is useful? What is aconfidence level?

Under a given hypothesis H the sampling distribution of a statistic S is a

normal distribution with the mean and the standard deviation

then Z = is the standard normal variate associated with S sothat for the distribution of z the mean is zero and the standard deviation

is 1. Accordingly for z the Z% confidence level is ( -z c , zc) this meansthat we can be Z% confident that if the hypothesis H is true than thevalue of z lie between zc and zc. This is equivalent saying that when H istrue there is (100 Z ) %chance that the value of z lies outside theinterval (-zc . zc)if we reject a true hypothesis H on the grounds that thevalue of z lies outside the interval (-z, zc) we would be making a type 1error and the probability of making this error is (100-Z)% the level ofsignificance.


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Confidence level is very much useful as we can predict any assumptionscan be made so that it will not lead us to the wrong way even if it doesntbe so great. As explained the confidence level is between zc to z and thepeak is at 100% which is the best.

In some cases we predict but do not consider it , and sometimes we willnot predict but hypothesis need it so this is called as the TYPE 1 errorsand TYPE 2 errors.

According to the levels of the Z the confidence is assured.. in the abovethe field shaded portion is the critical region.

mb0040 solved fall drive assignment 2011

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