Assignment Code: MMPC-005/TMA/JULY/2022

Course Code: MMPC-005

Assignment Name: Quantitative Analysis for Managerial Applications

Year: 2022-2023

Verification Status: Verified by Professor

1. The income of a group of 10,000 persons was found to be normally distributed with mean Rs.750 per month and a standard deviation of Rs. 50, show that of this group about 95% has income exceeding Rs. 668 and only 5% had income exceeding Rs. 832. (area between 750 and 668 = 0.4495, area between 750 and 832 = 0.4495).

Ans) Given: Income of a group of 10,000 persons was found to be normally distributed with a mean of Rs.750 per month and a standard deviation of Rs. 50, showing that of this group about 95% have income exceeding Rs. 668 and only 5% had income exceeding Rs. 832.

To Find: Distribution of Deviation

Solution:

As we are given in the question,

Mean of Income of 10000 people = Rs.750

Standard Deviation of 10000 people = Rs.50

The Group has an income exceeding Rs.668:

P ((y - μ)/σ > - 1.6449) = 95% 

P (y - μ > - 1.6449σ) = 95% 

P (y > μ - 1.6449σ) = 95% 

P (y > 750 - 1.6449(50)) = 95% 

P (y > 667.755) = 95% 

P (y > 668) ≈ 95% 

Therefore, 95% of people in the group have an income exceeding Rs.668.

The Group has income exceeding Rs. 832

P ((y - μ)/σ > 1.6449) = 5% 

P (y - μ > + 1.6449σ) = 5% 

P (y > μ + 1.6449σ) = 5% 

P (y > 750 + 1.6449(50)) = 5% 

P (y > 832.245) = 5% 

P (y > 832) = 5%

Therefore, only 5% of people in the group have an income exceeding Rs.832.

2. Why is forecasting so important in business? Explain the application of forecasting for long term decisions.

Ans) Business forecasting involves making informed guesses about certain business metrics, regardless of whether they reflect the specifics of a business, such as sales growth, or predictions for the economy as a whole. Financial and operational decisions are made based on economic conditions and how the future looks, albeit uncertain.

1. Forecasting is valuable to businesses so that they can make informed business decisions.

2. Financial forecasts are fundamentally informed guesses, and there are risks involved in relying on past data and methods that cannot include certain variables.

Companies use forecasting to help them develop business strategies. Past data is collected and analysed so that patterns can be found. Today, big data and artificial intelligence has transformed business forecasting methods. There are several different methods by which a business forecast is made. All the methods fall into one of two overarching approaches: qualitative and quantitative.

The application of forecasting for long term decisions are:

Technological Forecasting: Technological growth is often haphazard, especially in developing countries like India. This is because Technology seldom evolves and there are frequent technology transfers -due to imports of knowhow resulting in a leap-frogging phenomenon. In spite of this, it is generally seen that logarithms of many technological variables show linear trends with time, showing exponential growth.

Some extrapolations reported by Rohatgi et al. are:

1. Passenger kms carried by Indian Airlines.

2. Fertilizer applied per hectare of cropped area.

3. Demand and supply of petroleum crude.

4. Installed capacity of electricity generation in millions of KW.

Delphi: This is a subjective method relying on the opinion of experts designed to minimise bias and error of judgment. A Delphi panel consists of a number of experts with an impartial leader or coordinator who organises the questions. Specific questions (rather than general opinions) with yes-no or multiple type answers or specific dates/events are sought from the experts.

For instance, questions could be of the following kind:

1. When do you think the petroleum reserves of the country would be exhausted?

2. When would the level of pollution in Delhi exceed danger limit?

3. What would the population of India be in 2020, 2040 and 2060?

4. When would fibre optics become a commercial viability for communication?

Opinion Polls: Opinion polls are a very common method of gaining knowledge about consumer tastes, responses to a new product, popularity of a person or leader, reactions to an election result or the likely future prime minister after the impending polls. In any opinion poll two things are of primary importance. First, the information that is sought and secondly the target population from whom the information is sought. Both these factors must be kept in mind while designing the appropriate mechanism for conducting the opinion poll.

Opinion polls may be conducted through

1. Personal interviews.

2. Circulation of questionnaires.

3. Meetings in groups.

4. Conferences, seminars and symposia.

3. What do you understand by Primary Data? What are the various methods of collecting primary data? Also, mention what points to be kept in mind while designing the questionnaire?

Ans) Primary data is a type of data that is collected by researchers directly from main sources through interviews, surveys, experiments, etc. Primary data are usually collected from the source—where the data originally originates from and are regarded as the best kind of data in research. The sources of primary data are usually chosen and tailored specifically to meet the demands or requirements of particular research. Also, before choosing a data collection source, things like the aim of the research and target population need to be identified.

The various methods of collecting primary data are:

In this method, the interviewer sits down with the respondent face-to-face and writes down what he says. This is a good way to get more accurate and reliable information because the interviewer can clear up any questions and double-check the answers. This method takes a lot of time and can be very expensive if there are a lot of respondents who live in different places.

Mail Questionnaire: In this method, a list of questions, called a "questionnaire," is made and mailed to the people who answered the survey. The investigator wants the respondents to fill out the questionnaire and send it back to them. Mail questionnaires are sometimes given to respondents in other ways, like by putting them on the products they buy or putting them in newspapers or magazines. This method is easy to use when the area being studied is large and the people being asked to take part are spread out over a large area. But this method can only be used with people who can read and write and can understand and answer written questions.

Phone: In this method, the investigator calls the people and asks them the right questions. This method is cheaper, but only people who have phones can be interviewed, and only a few questions can be asked over the phone.

The questionnaire method is a quick and easy way to gather information. But it has a major flaw in that it may be very hard to get information about sensitive things like income, age, or personal life details that the respondent may not want to share with the investigator. With other methods, too, different people may have different ideas about how to answer the questions, which could lead to mistakes and wrong information.

The points to be kept in mind while designing the questionnaire are:

1. A cover letter should be sent with every questionnaire.

2. The fewest questions possible should be asked.

3. The questions should be clear, short, and easy to answer. There shouldn't be any doubt about what the right answer is.

4. The questions shouldn't be so personal or private that the person answering would have to give away private or confidential information.

5. The questions should be set up in a way that makes sense, so that the answers flow from one to the next and the respondent doesn't have to keep going back to the questions that came before.

6. Some of these kinds of questions should be in the questionnaire to make sure that the information given is correct.

4. The means of two large samples of sizes 1000 and 2000 are 67.5 and 68.0 respectively. Test the quality of the means of the two populations each with standard deviation of 2.5. (z table value at α0.05= -1.96).

Ans) Mean of two samples 1000 and 2000 are 67.5 and 68 inches respectively.

To find: The samples that can be regarded as drawn when the standard deviation is 2.5 inches.

Step-by-step explanation:

Therefore, the samples cannot be regarded as drawn from same population.

Final answer: The samples be regarded as drawn from the same population of standard deviation of 2.5 inches.

5. Write short notes on any two of the following: -

(a) Stratified Sampling

Ans) Stratified sampling is a method of obtaining a representative sample from a population that researchers have divided into relatively similar subpopulations (strata). Researchers use stratified sampling to ensure specific subgroups are present in their sample. It also helps them obtain precise estimates of each group’s characteristics. Many surveys use this method to understand differences between subpopulations better. Stratified sampling is also known as stratified random sampling.

The stratified sampling process starts with researchers dividing a diverse population into relatively homogeneous groups called strata, the plural of stratum. Then, they draw a random sample from each group (stratum) and combine them to form their complete representative sample. Learn more about representative samples. When researchers use non-random selection to choose subjects from the strata, it is known as Quota Sampling.

Strata are subpopulations whose members are relatively similar to each other compared to the broader population. Researchers can create strata based on income, gender, and race, among many other possibilities. For example, if your research question requires you to compare outcomes between income levels, you might base the strata on income. All members of the population should be in only one stratum.

Example of Stratified Sampling

Stratified sampling involves multiple steps. First, break down the population into strata. From each stratum, use simple random sampling to draw a sample. This process ensures that you obtain observations for all strata.

For example, imagine we’re assessing standardized testing and our research requires us to compare test scores by income. We can use income levels for our strata. Students from households with similar incomes should be relatively similar compared to the overall state population.

While we want a random sample for unbiased estimates overall, we also want to obtain precise estimates for each income level in our population. Using simple random sampling, income levels with a small number of students and random chance could conspire to provide small sample sizes for some income levels. These smaller sample sizes produce relatively imprecise estimates for them.

To avoid this problem, we’ll use stratified sampling. Our sampling plan might dictate that we select 100 students from each income level using simple random sampling. Of course, this plan presupposes that we know the household income level for each student, which might be problematic.

The benefit of stratified sampling is that you obtain reasonably precise estimates for all subgroups related to your research question. The drawback is that analysing these datasets is more complicated. When you use stratified random sampling, you can’t simply take the overall sample average and use it for the general population because you know that the smaller strata are overrepresented. You need to use a weighted average technique.

(c) Exponential Distribution

Ans) In Probability theory and statistics, the exponential distribution is a continuous probability distribution that often concerns the amount of time until some specific event happens. It is a process in which events happen continuously and independently at a constant average rate. The exponential distribution has the key property of being memoryless. The exponential random variable can be either more small values or fewer larger variables. For example, the amount of money spent by the customer on one trip to the supermarket follows an exponential distribution.

Exponential Distribution Applications

One of the widely used continuous distribution is the exponential distribution. It helps to determine the time elapsed between the events. It is used in a range of applications such as reliability theory, queuing theory, physics and so on.

Some of the fields that are modelled by the exponential distribution are as follows:

1. Exponential distribution helps to find the distance between mutations on a DNA strand.

2. Calculating the time until the radioactive particle decays.

3. Helps on finding the height of different molecules in a gas at the stable temperature and pressure in a uniform gravitational field.

4. Helps to compute the monthly and annual highest values of regular rainfall and river outflow volumes.

The exponential distribution assumes that small values occur more frequently than large values. Consequently, it can model things like wait times, transaction times, and failure times. It can also model other variables, such as the size of orders at convenience stores.

Consider whether the memoryless characteristic of events occurring independently at a constant average rate applies to your subject area.

Memorylessness is not valid for all subject areas, preventing the exponential distribution from modelling some phenomena. For instance, machines tend to wear out over time, causing failure rates to increase as time passes. However, accidents generally occur independently at a fixed rate.  When an accident doesn’t happen for one day, it doesn’t indicate that an accident is more or less likely to happen the next day.

This distribution assumes that the average time between events remains constant. Consequently, you cannot use the exponential distribution when the expected time for events increases or decreases as time passes. Other distributions, such as the Weibull distribution, are appropriate in those cases. Using the exponential distribution in reliability studies requires the process to have a consistent failure rate over time.