MBA SEMESTER 1

MB0040 – STATISTICS FOR MANAGEMENT

Assignment Set- 1

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

Ans:- Due to advanced communication network, rapid changes in consumer behaviour,

varied expectations of variety of consumers and new market openings, modern

managers have a difficult task of making quick and appropriate decisions. Therefore,

there is a need for them to depend more upon quantitative techniques like

mathematical models, statistics, operations research and econometrics.

As you can see, what the General Manager is doing here is using Statistics to solve a

problem and to increase profits.

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 before buying one. As you can see, in this scenario we are collecting

data and making an optimum decision. In other words, we are using Statistics.

Again, suppose a company wishes to introduce a new product, it has to collect data on

market potential, consumer likings, availability of raw materials, feasibility of producing

the product. Hence, data collection is the back-bone of any decision making process.

Many organisations find themselves data-rich but poor in drawing information from it.

Therefore, it is important to develop the ability to extract meaningful information from

raw data to make better decisions. Statistics play an important role in this aspect.

Statistics is broadly divided into two main categories. Figure 1.1 illustrates the two

categories. The two categories of Statistics are descriptive statistics and inferential

statistics.

Divisions in Statistics

Descriptive Statistics: Descriptive statistics is used to present the 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 valid inferences from the

data which are helpful in effective decision making for managers or professionals.

Statistical methods such as estimation, prediction and hypothesis testing belong to

inferential statistics. The researchers make deductions or conclusions from the collected

data samples regarding the characteristics of large population from which the samples

are taken. So, we can say ‘Statistics is the backbone of decision-making’.

Q.1. (b) Give plural meaning of the word Statistics?

Ans:- Plural of Word “Statistic”:

The word statistics is used as the plural of the word “Statistic” which refers to a

numerical quantity like mean, median, variance etc…, calculated from sample value.

In plural sense, the word statistics refer to numerical facts and figures collected in a

systematic manner with a definite purpose in any field of study. In this sense, statistics

are also aggregates of facts which are expressed in numerical form. For example,

Statistics on industrial production, statistics or population growth of a country in

different years etc.

For Example: If we select 15 student from a class of 80 students, measure their

heights and find the average height. This average would be a statistic.

Q 2. a. In a bivariate data on ‘x’ and ‘y’, variance of ‘x’ = 49, variance of ‘y’ = 9 and covariance (x,y) = -17.5. Find coefficient of correlation between ‘x’ and ‘y’.

Ans:- We know that:

Given

Hence, there is a highly negative correlation.

Q 2. b. Enumerate the factors which should be kept in mind for proper planning.

Ans:- 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. Steps involved in the planning stage.

Q 3. The percentage sugar content of Tobacco in two samples was represented in table 11.11. Test whether their population variances are same.

Table 1. Percentage sugar content of Tobacco in two samples

Sample A 2.4 2.7 2.6 2.1 2.5

Sample B 2.7 3 2.8 3.1 2.2 3.6

Ans:-

Required values of the method I to calculate sample mean

X d = X - 2.5 d2

2.4 0.1 0.01

2.7 -0.2 0.04

2.6 -0.1 0.01

2.1 0.4 0.16

2.5 0 0

Total 0.2 0.22

Required values of the method II to calculate sample mean

X d = X – 3 d2 3

2.7 0.3 0.09

3 0 0

2.8 0.2 0.04

3.1 -0.1 0.1

2.2 0.8 0.64

3.6 -0.6 0.36

Total 0.6 1.23

S

2

=

1

[ ∑d2 -

(∑d)2

]1

n1

-1n1

=1

[0.22

-0.04

/ 5 ]4

= 0.053

S2

=1

[ ∑d2 -(∑d)2

]2 n2 -1 n2

=1

[1.23-0.053

]5 6

= 0.244 not significant

Q 4. a. Explain the characteristics of business forecasting.

Ans:- Characteristics of business forecasting

Based on past and present conditions

Business forecasting is based on past and present economic condition of the business. To forecast the future, various data, information and facts concerning to economic condition of business for past and present are analysed.

Based on mathematical and statistical methods

The process of forecasting includes the use of statistical and mathematical methods. By using these methods, the actual trend which may take place in future can be forecasted.

Period

The forecasting can be made for long term, short term, medium term or any specific

Estimation of future

The business forecasting is to forecast the future regarding probable economic conditions.

Scope

The forecasting can be physical as well as financial.

Q 4. b. Differentiate between prediction, projection and forecasting.

Ans:- Prediction, projection and forecasting

A great amount of confusion seem to have grown up in the use of words ‘forecast’, ‘prediction’ and ‘projection’.

Forecasts are made by estimating future values of the external factors by means of prediction, projection or forecast and from these values calculating the estimate of the dependent variable.

Q 5. What are the components of time series? Bring out the significance of moving average in analysing a time series and point out its limitations.

Ans:- Components of Time Series

The behaviour of a time series over periods of time is called the movement of the time series. The time series is classified into the following four components:

i) Long term trend or secular trend

ii) Seasonal variations

iii) Cyclic variations

iv) Random variations

Method of moving averages

Moving averages method is used for smoothing the time series. That is, it smoothes the fluctuations of the data by the method of moving averages.

When period of moving average is odd

To determine the trend by this method, the procedure is described in

Procedure for determining the trend when moving average is odd

By plotting these trend values (if desired) you can obtain the trend curve with the help of which you can determine the trend whether it is increasing or decreasing. If needed, you can also compute short-term fluctuations by subtracting the trend values from the actual values.

When period of moving averages is even

When period of moving average is even (such as 4 years), we compute the moving averages by using the steps described in below

Procedure for determining the trend when moving average is even

Merits and demerits of moving averages method

Merits Demerits

This is a simple method. No functional relationship between the values and the time. Thus, this method is not helpful in forecasting and predicting the values on the basis of time.

This method is objective in the sense that anybody working on a problem with this method will get the same results.

No trend values for some years in the beginning and some in the end. For example, for 5 – yearly moving average, there will be no trend values for the first two years and the last three years.

This method is used for determining seasonal, cyclic and irregular variations besides the trend values.

In case of non–linear trend, the values obtained by this method are biased in one or the other direction.

This method is flexible enough to add more figures to the data because the entire calculations are not changed.

The period selection of moving average is a difficult task. Hence, great care has to be taken in period selection, particularly when there is no business cycle during that time.

If the period of moving averages coincides with the period of cyclic fluctuations in the data, such fluctuations are automatically eliminated.

Q 6. List down various measures of central tendency and explain the difference between them?

Ans:- Measures of Central Tendency

Several different measures of central tendency are defined below.

1 Arithmetic Mean

The 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 is used 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 sample and N is the

number of numbers in the sample. As an example, the mean of the numbers 1 + 2 + 3

+ 6 + 8 = 20/5 = 4 regardless of whether the numbers constitute the entire population

or just a sample from the population.

The table, Number of touchdown passes (Table 1: Number of touchdown passes),

shows the number of touchdown (TD) passes thrown by each of the 31 teams in the

National Football League in the 2000 season.

The mean number of touchdown passes thrown is 20.4516 as shown below.

Number of touchdown passes

Although the arithmetic mean is not the only "mean" (there is also a geometric

mean), it is by far the most commonly used. Therefore, if the term "mean" is used

without specifying whether it is the arithmetic mean, the geometric mean, or some

other mean, it is assumed to refer to the arithmetic mean.

2 Median

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

midpoint of a distribution: the same number of scores are above the median as below

it. For the data in the table, Number of touchdown passes (Table 1: Number of

touchdown passes), there are 31 scores. The 16th highest score (which equals 20) is

the median because there are 15 scores below the 16th score and 15 scores above

the 16th score. The median can also be thought of as the 50th percentile3. Let's

return to the made up example of the quiz on which you made a three discussed

previously in the module Introduction to Central Tendency4 and shown in Table 2:

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

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

For Dataset 1, the median is three, the same as your score. For Dataset 2, the

median is 4. Therefore, your score is below the median. This means you are in the

lower half of the class. Finally for Dataset 3, the median is 2. For this dataset, your

score is above the median and therefore 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 is the mean of the two middle numbers.

Thus, the median of the numbers 2, 4, 7, 12 is 4+7/2 = 5:5.

3 mode

The mode is the most frequently occuring value. For the data in the table, Number of

touchdown passes (Table 1: Number of touchdown passes), the mode is 18 since

more teams (4) had 18 touchdown passes than any other number of touchdown

passes. With continuous data such as response time measured to many decimals, the

frequency of each value is one since no two scores will be exactly the same (see

discussion of continuous variables5). Therefore the mode of continuous data is

normally computed from a grouped frequency distribution. The Grouped frequency

distribution (Table 3: Grouped frequency distribution) table shows a grouped

frequency distribution for the target response time data. Since the interval with the

highest frequency is 600-700, the mode is the middle of that interval (650).

Grouped frequency distribution

Proportions and Percentages

When the focus is on the degree to which a population possesses a particular

attribute, the measure of interest is a percentage or a proportion.

• A proportion refers to the fraction of the total that possesses a certain

attribute. For example, we might ask what proportion of women in our sample

weigh less than 135 pounds. Since 3 women weigh less than 135 pounds, the

proportion would be 3/5 or 0.60.

• A percentage is another way of expressing a proportion. A percentage is equal

to the proportion times 100. In our example of the five women, the percent of

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

Notation

Of the various measures, the mean and the proportion are most important. 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 particular attribute.

• p: The proportion of elements in the sample that has a particular attribute.

• Q: The proportion of elements in the population that does not have a specified

attribute. Note that Q = 1 - P.

• q: The proportion of elements in the sample that does not have a specified

attribute. Note that q = 1 - p.

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

Ans;-

Confidence Intervals

In statistics, a confidence interval (CI) is a particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval (i.e. it is calculated from the observations), in principle different from sample to sample, that frequently includes the parameter of interest, if the experiment is repeated. How frequently the observed interval contains the parameter is determined by the confidence level or confidence coefficient.

A confidence interval with a particular confidence level is intended to give the assurance that, if the statistical model is correct, then taken over all the data that might have been obtained, the procedure for constructing the interval would deliver a confidence interval that included the true value of the parameter the proportion of the time set by the confidence level. More specifically, the meaning of the term "confidence level" is that, if confidence intervals are constructed across many separate data analyses of repeated (and possibly different) experiments, the proportion of such intervals that contain the true value of the parameter will approximately match the confidence level; this is guaranteed by the reasoning underlying the construction of confidence intervals.

A confidence interval does not predict that the true value of the parameter has a particular probability of being in the confidence interval given the data actually obtained. (An interval intended to have such a property, called a credible interval, can be estimated using Bayesian methods; but such methods bring with them their own distinct strengths and weaknesses).

The confidence level sets the boundaries of a confidence interval, this is conventionally set at 95% to coincide with the 5% convention of statistical significance in hypothesis testing. In some studies wider (e.g. 90%) or narrower (e.g. 99%) confidence intervals will be required. This rather depends upon the nature of your study. You should consult a statistician before using CI's other than 95%.

You will hear the terms confidence interval and confidence limit used. The confidence interval is the range Q-X to Q+Y where Q is the value that is central to the study question, Q-X is he lower confidence limit and Q+Y is the upper confidence limit.

Familiarise yourself with alternative CI interpretations:

Common

A 95% CI is the interval that you are 95% certain contains the true population value as it might be estimated from a much larger study.

The value in question can be a mean, difference between two means, a proportion etc. The CI is usually, but not necessarily, symmetrical about this value.

Pure Bayesian

The Bayesian concept of a credible interval is sometimes put forward as a more practical concept than the confidence interval. For a 95% credible interval, the value of interest (e.g. size of treatment effect) lies with a 95% probability in the interval. This interval is then open to subjective moulding of interpretation. Furthermore, the credible interval can only correspond exactly to the confidence interval if prior probability is so called "uninformative".

Pure frequentist

Most pure frequentists say that it is not possible to make probability statements, such CI interpretation, about the study values of interest in hypothesis tests.

Neymanian

A 95% CI is the interval which will contain the true value on 95% of occasions if a study were repeated many times using samples from the same population.

Neyman originated the concept of CI as follows: If we test a large number of different null hypotheses at one critical level, say 5%, then we can collect all of the rejected null hypotheses into one set. This set usually forms a continuous interval that can be derived mathematically and Neyman described the limits of this set as confidence limits that bound a confidence interval. If the critical level (probability of incorrectly rejecting the null hypothesis) is 5% then the interval is 95%. Any values of the treatment effect that lie outside the confidence interval are regarded as "unreasonable" in terms of hypothesis testing at the critical level.