SMU MBA (Assignment Semester – I) – Statistics for Management (MB0040)

[su_note note_color=”#64ef94″ radius=”6″]Semester – MBA 1st Semester
Subject code – MB0040
Subject Name – Statistics for Management[/su_note]
Q. No 1 Statistics plays a vital role in almost every facet of human life. Describe the functions of Statistics. Explain the applications of statistics?

Ans.1

Definition of Statistics

“Statistics is a science which deals with the method of collecting, classifying, presenting, comparing and interpreting the numerical data to throw light on enquiry”. – Seligman

According to Horace Secrist, “Statistics may be defined, as the aggregate of facts affected to a marked extent by multiplicity of causes, numerically expressed, enumerated or estimated according to a reasonable standard of accuracy, collected in a systematic manner for a predetermined purpose and placed in relation to each other”. This definition is both comprehensive and exhaustive.

Functions of Statistics

Statistics is used for various purposes. It is used to simplify mass data and to make comparisons easier. It is also used to bring out trends and tendencies in the data, and the hidden relations between variables. All these help in easy decision making. Let us look 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 more easily.

2. Statistics makes comparison easier

Without using statistical methods and concepts, collection of data and comparison would be difficult. Statistics helps us to compare data collected from various sources.

3. Statistics brings out trends and tendencies in the data

After 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 variables

Statistical analysis helps in drawing inferences on the data. Statistical analysis brings out the hidden relations between variables.

Application of Statistics

Statistical methods are applied to specific problems in various fields such as Biology, Medicine, Agriculture, Commerce, Business, Economics, Industry, Insurance, Sociology and Psychology.

In the field of agriculture, an important concept of statistics such as analysis of variance (ANOVA) is used in experiments related to agriculture, to test the significance between two sample means.

In Biology, Medicine and Agriculture, Statistical methods are applied in the following:

• Study of the growth of plants
• Movement of fish population in the ocean
• Migration pattern of birds
• Analysis of the effect of newly invented medicines
• Theories of heredity
• Estimation of yield of crop
• Study of the effect of fertilizers on yield
• Birth rate
• Death rate
• Population growth
• Growth of bacteria

Insurance companies decide on the insurance premiums based on the age composition of the population and the mortality rates.

Q. No 2 a) Explain the approaches to define probability.
b) State the addition and multiplication rules of probability giving an example of each case.

Ans.2

(a) Approaches to define probability

There are four approaches to define probability. They are as follows:

1) Classical / mathematical / priori approach
2) Statistical / relative frequency / empirical / posteriori approach
3) Subjective approach
4) Axiomatic approach

1) Classical / mathematical / priori approach

Under this approach the probability of an event is known before conducting the experiment. In this case, each of possible outcomes is associated with equal probability of occurrence and number of outcomes favourable to the concerned event is known.

2) 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 households in that area and then arrive at the probability.

3) 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.

4) Axiomatic approach

Let S be a sample space consisting of all events of a random experiment and , then the probability of an event A is a set function satisfying the following axioms:

i) Axioms of positivity,
ii) Axiom of certainty,

Addition and multiplication rules of probability giving an example of each – 

Example

A sales manager may like to know the probability that he will exceed the target for product A or product B. Sometimes, he would like to know the probability that the sales of product A and B will exceed the target. The first type of probability is answered by addition rule. The second type of probability is answered by multiplication rule.

1. Addition rule

The addition rule of probability states that:

i) If ‘A’ and ‘B’ are any two events then the probability of the occurrence of either ‘A’ or ‘B’ is given by:

P(A ∪ B) = P(A) + P(B) – P (A ∪ B)

ii) If ‘A’ and ‘B’ are two mutually exclusive events then the probability of occurrence of either ‘A’ or ‘B’ is given by:

P(A ∪ B) = P(A) + P(B)

iii) If ‘A’, ‘B’ and ‘C’ are any three events then the probability of occurrence of either ‘A’ or ‘B’ or ‘C’ is given by:

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P(A ∩ C) + P (A ∪ B ∪ C)

(b) Multiplication rule

If ‘A’ and ‘B’ are two independent events then the probability of occurrence of ‘A’ and ‘B’ is given by:

P(A ∩ B) = P(A) * P(B)

Example

• Show that P(A) = 1 – P(A’)
• Show that probability is a value between 0 and 1.
• Show that P(Ф) = 0 where Ф is null event.

(i) If A and A’ are complementary events, A A’ = S

By the axiom 2, P(S) = 1. And so, P(A ∪ A’) =1 …. Result 1

But A and A’ are mutually exclusive events. Therefore, by the axiom 3,

P(A ∪ A’) = P(A) + P(A’) …Result 2

By the results 1 and 2, P(A) + P(A’) = 1

That is, P(A) = 1-(A’)

(ii) Let A be an event. Then, by the axiom 1,

P(A)≥0 ….Result 1

If A’ is the complementary event of A,

P(A’) = 1 – P(A)

But, by axioml, P(A’) ≥0

Therefore, 1 – P(A) ≥ 0 …Result 2

Hence, P(A)≤ By the results 1 and 2,

0 ≤ P(A) ≤ 1, i.e., the probability is a value between 0 and 1.

(iii) If A is an event and if Φ is a null event, A Φ = A

∴ P (A ∪ φ) = P(A) ….. Result 1

But, A and Φ are mutually exclusive. Therefore,

P (A ∪ φ) = P(A) + P(φ) ….. Result 2

By the result 1 and 2

P(A) + P(Φ) = P(A)

That is, P(Φ) = P(A) – P(A) = 0

Q. No 3 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.

Ans.3

(a) Hypothesis testing procedure

Having calculated appropriate z-statistic or t-statistic, to reject or accept the null hypothesis, it is necessary to identify the rejection region with reference to the given level of significance. If the calculated statistic is in the rejection region, we accept the alternative hypothesis against the null hypothesis at that level of significance. Otherwise, we accept null hypothesis at given level of significance. Table 9.4 depicts the rejection region, normally denoted by ‘R’.

Table 9.4: Kinds of Tests

Kind of test	z- statistic	t- statistic
Two tail test	R: |z| > |ztable|	R: |t| > |ttable|
Lower tail test	R: z < z table	R: t < t table
Upper tail test	R: t > t table	R: t > t table

Figure 9.4 depicts the hypothesis testing procedure.

(b) Components of time series

Any time series can contain some or all of the following components:

1. Trend (T)
2. Cyclical (C)
3. Seasonal (S)
4. Irregular (I)

These components may be combined in different ways. It is usually assumed that they are multiplied or added, i.e.,

yt = T × C × S × I

yt = T + C + S + I

To correct for the trend in the first case one divides the first expression by the trend (T). In the second case it is subtracted.

Trend component

The trend is the long term pattern of a time series. A trend can be positive or negative depending on whether the time series exhibits an increasing long term pattern or a decreasing long term pattern. if a time series does not show an increasing or decreasing pattern then the series is stationary in the mean.

Cyclical component

Any pattern showing an up and down movement around a given trend is identified as a cyclical pattern. The duration of a cycle depends on the type of business or industry being analyzed.

Seasonal component

Seasonality occurs when the time series exhibits regular fluctuations during the same month (or months) every year, or during the same quarter every year. For instance, retail sales peak during the month of December.

Irregular component

This component is unpredictable. Every time series has some unpredictable component that makes it a random variable. In prediction, the objective is to “model” all the components to the point that the only component that remains unexplained is the random component

Q. No 4 a) What is a Chi-square test? Point out its applications. Under what conditions is this test applicable?
b) Discuss the types of measurement scales with examples ?

Ans.4

(a) Chi-square test

The data used in calculating a chi square statistic must be random, raw, mutually exclusive, drawn from independent variables and be drawn from a large enough sample. For example, the results of tossing a coin 100 times would meet these criteria.

Application of Chi-square test test

1. Tests for independence of attributes

In the test for independence, the null hypothesis is that the row and column variables are independent of each other. We have studied earlier, that the hypothesis testing is done under the assumption that the null hypothesis is true.

2. Test of goodness of fit

The test of goodness of fit of a statistical model measures how accurately the test fits a set of observations

Conditions for applying the Chi-Square test

The following are the conditions for using the Chi-Square test:

1. The frequencies used in Chi-Square test must be absolute and not in relative terms.
2. The total number of observations collected for this test must be large.
3. Each of the observations which make up the sample of this test must be independent of each other.

4. As λ² test is based wholly on sample data, no assumption is made concerning the population distribution. In other words, it is a non parametric-test.

(b) Types of Measurement Scales

Statisticians often refer to the “levels of measurement” of a variable, a measure, or a scale to distinguish between measured variables that have different properties. There are four basic levels: nominal, ordinal, interval, and ratio.

Nominal

A variable measured on a “nominal” scale is a variable that does not really have any evaluative distinction. One value is really not any greater than another. A good example of a nominal variable is sex (or gender).

Ordinal

Something measured on an “ordinal” scale does have an evaluative connotation. One value is greater or larger or better than the other. Product A is preferred over product B, and therefore A receives a value of 1 and B receives a value of 2.

Interval

A variable measured on an interval scale gives information about more or betterness as ordinal scales do, but interval variables have an equal distance between each value. The distance between 1 and 2 is equal to the distance between 9 and 10.

Ratio

Something measured on a ratio scale has the same properties that an interval scale has except, with a ratio scaling, there is an absolute zero point. Temperature measured in Kelvin is an example.

Q. No 5 Business forecasting acquires an important place in every field of the economy. Explain the objectives and theories of Business forecasting.

Ans.5

Business Forecasting

Business forecasting refers to the analysis of past and present economic conditions with the object of drawing inferences about probable future business conditions. The process of making definite estimates of future course of events is referred to as forecasting and the figure or statements obtained from the process is known as ‘forecast’; future course of events is rarely known.

Objectives of forecasting in business

Forecasting is a part of human nature. Businessmen also need to look to the future. Success in business depends on correct predictions. In fact when a man enters business, he automatically takes with it the responsibility for attempting to forecast the future.

To a very large extent, success or failure would depend upon the ability to successfully forecast the future course of events. Without some element of continuity between past, present and future, there would be little possibility of successful prediction. But history is not likely to repeat itself and we would hardly expect economic conditions next year or over the next 10 years to follow a clear cut prediction. Yet, past patterns prevail sufficiently to justify using the past as a basis for predicting the future.

Theories of Business Forecasting

There are a few theories that are followed while making business forecasts.

Some of them are:

1. Sequence or time-lag theory
2. Action and reaction theory
3. Economic rhythm theory
4. Specific historical analogy
5. Cross-cut analysis theory

Sequence or time-lag theory

This is the most important theory of business forecasting. It is based on the assumption that most of the business data have the lag and lead relationships.

Action and reaction theory

This theory is based on the following two assumptions.

• Every action has a reaction
• Magnitude of the original action influences the reaction

Economic Rhythm Theory

The basic assumption of this theory is that history repeats itself and hence assumes that all economic and business events behave in a rhythmic order.

Specific historical analogy

History repeats itself is the main foundation of this theory. If conditions are the same, whatever happened in the past under a set of circumstances is likely to happen in future also.

Cross-cut analysis theory

This theory proceeds on the analysis of interplay of current economic forces. In this method, the combined effects of various factors are not studied. The effect of each factor is studied independently.

Q. No 6 a) What is analysis of variance? What are the assumptions of this technique?
b) 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 v₁ = 2 and v₂ = 12 is 3.88]

Ans.6

(a) analysis of variance

Analysis of Variance (ANOVA) is useful in such situations as comparing the mileage achieved by five different brands of gasoline, testing which of four different training methods produce the fastest learning record, or comparing the first-year earnings of the graduates of half a dozen different business schools.

Technique of analysis of variance

Technique of analysis of variance is referred to as ANOVA. Initially, the technique was applied in the field of Zoology and Agriculture, but in a later stage, it was applied to other fields also. In ANOVA.

In fact, ANOVA is the classification and cross-classification of statistical data with the view of testing whether the means of specific classification differ significantly or whether they are homogeneous

(b) Solution

t = = 2.0544 for A & B , B & C Comparisons
t = = 4.10885 for A & C Comparisons

P-value = 2*P(t > 2.0544 when df = 4) = 2(.0546) = .1092

P-value = 2*P(t> 4.1089when df = 4) = 2(.0074) = = .0148.0148 < .05

A&C comparison significantly says populations mean are NOT equal.

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