IGNOU Solved Assignment 2023–24 & 2024 BECC-107 : STATISTICAL METHODS FOR ECONOMICS

  IGNOU Solved assignment 

BECC-107 : STATISTICAL METHODS FOR ECONOMICS

Q 1) (a) Calculate mean, median and mode from the following data. Class Interval Frequency 3-4 3 4-5 7 5-6 22 6-7 60 7-8 85 8-9 32 9-10 8 (b) Calculate the coefficient of variation from the data given above.

Ans-  To calculate the mean, median, and mode from the given frequency distribution, let's first find the midpoints of each class interval. Then we can calculate the mean, median, and mode. After that, we'll calculate the coefficient of variation.

(d) Coefficient of Variation (CV):

First, we need to find the standard deviation. Then we can calculate the CV.

Q 2. Bring out the distinction between sample survey and census. Describe the steps you would follow in collecting data though a sample survey. Prepare a small questionnaire for collection of income and expenditure levels of households.

Ans-  

Census vs. Sample Survey

The key difference between a census and a sample survey lies in the scope of data collection:

• Census: Involves collecting data from every member of a population. This provides the most accurate picture but is expensive, time-consuming, and sometimes impractical for large populations.
• Sample Survey: Data is collected from a subset of the population, called a sample. This approach is quicker and cheaper, but the results are estimates that may have a margin of error.

Here's a table summarizing the key differences:

Feature	Census	Sample Survey
Population Coverage	Entire Population	Subset (Sample)
Cost	High	Lower
Time	Time-consuming	Faster
Accuracy	Most Accurate	Estimate with Margin of Error

Steps in Collecting Data Through a Sample Survey

1. Define the Population: Clearly identify the target group you want information about (e.g., all households in a city).
2. Determine Sample Size and Sampling Method: Choose a sample size that balances accuracy and cost. There are different sampling methods to ensure your sample represents the population (e.g., random sampling, stratified sampling).
3. Develop the Questionnaire: Design clear, concise questions to gather income and expenditure data. (See example below)
4. Data Collection: Choose a method to reach your sample (e.g., online survey, phone calls, in-person interviews).
5. Data Analysis: Organize and analyze the collected data to draw conclusions about the population.

Sample Questionnaire on Income and Expenditure

Introduction: Thank you for participating in this survey on household income and expenditure. All your responses will be kept confidential.

Household Information:

1. How many people live in your household (including yourself)?
2. What is the age range of the primary income earner in your household? (e.g., 25-34, 35-44)

Income:

3. Please indicate the total gross monthly income (before taxes) of your household from all sources (e.g., salaries, wages, pensions).

Expenditure:

4. Please estimate your household's average monthly expenditure on the following categories (provide answer choices with ranges or open-ended):

   • Housing (rent/mortgage)
   • Food
   • Transportation
   • Utilities (electricity, water)
   • Healthcare
   • Education
   • Other (please specify)

Q. 3) a) The probability that Rajesh will score more than 90 marks in class test is 0.75. What is the probability that Rajesh will secure more than 90 marks in three out of four class tests?

Ans-  The probability of Rajesh scoring more than 90 marks in three out of four class tests is 0.421875.

This scenario can be modeled using the binomial probability distribution, where each trial represents a class test (success = scoring more than 90 marks, failure = scoring 90 or less). We are interested in the probability of getting exactly 3 successes (more than 90 marks in 3 tests) out of 4 trials.

The provided calculation matches this approach:

• Probability of success (scoring more than 90) = p_success = 0.75
• Number of trials (tests) = n_trials = 4
• Number of successes needed (more than 90 marks) = k = 3

Using the binomial coefficient formula and considering these values results in the probability of 0.421875, indicating a 42.19% chance of Rajesh achieving more than 90 marks in three out of the four class tests.

Q. 3. b) Bring out the major properties of binomial distribution. Mention certain important uses of this distribution.

Ans-  

Major Properties of the Binomial Distribution:

The binomial distribution applies to situations with a fixed number of trials (n) where each trial has only two possible outcomes: success (p) or failure (q = 1 - p). Here are its key properties:

1. Fixed Number of Trials (n): The experiment is repeated a predetermined number of times (n).
2. Two Possible Outcomes: Each trial has exactly two outcomes, categorized as success (with probability p) or failure (with probability q = 1 - p).
3. Independent Trials: The outcome of one trial does not influence the outcome of any other trial.
4. Constant Probability: The probability of success (p) and failure (q) remains constant throughout all trials.

Important Uses of the Binomial Distribution:

The binomial distribution finds application in various scenarios where events have a fixed number of trials with two possible outcomes. Here are some important uses:

• Quality Control: Manufacturing processes often involve inspecting products for defects. The binomial distribution helps determine the probability of finding a specific number of defective items in a sample.
• Pass/Fail Rates: It can be used to analyze pass/fail rates in exams, competitions, or production lines, estimating the likelihood of a certain number of successes within a fixed number of attempts.
• Opinion Polls: Binomial distribution is useful in analyzing opinion polls with yes/no or agree/disagree options. It allows us to estimate the margin of error when inferring population opinion from a sample survey.
• Genetic Testing: This distribution plays a role in genetic testing where specific gene variations determine outcomes. It helps calculate the probability of inheriting a particular trait based on parental gene combinations.
• Insurance and Risk Assessment: Insurance companies utilize the binomial distribution to model risks and determine premiums. It helps estimate the probability of a certain number of claims occurring within a specific timeframe.

By understanding these properties and uses, the binomial distribution becomes a valuable tool for analyzing probabilities in various real-world scenarios.

Q4. Define correlation coefficient. What are its properties?

Ans-  The correlation coefficient is a statistical measure that indicates the strength and direction of a linear relationship between two variables. It provides a value between -1 and +1, where:

• +1: Perfect positive correlation (as one variable increases, the other increases proportionally)
• -1: Perfect negative correlation (as one variable increases, the other decreases proportionally)
• 0: No linear correlation (no predictable relationship between the variables)

Here are some key properties of the correlation coefficient:

1. Range: The correlation coefficient (r) always falls between -1 and +1, inclusive.
2. Symmetry: The correlation between X and Y is the same as the correlation between Y and X (i.e., r(X, Y) = r(Y, X)).
3. Scale Invariance: The correlation coefficient is independent of the scale of the data. Multiplying or dividing all values of X or Y by a constant will not change the correlation coefficient.
4. Linear Relationship: The correlation coefficient only measures linear relationships. It doesn't capture non-linear patterns between variables.
5. Strength vs. Causation: A high correlation doesn't necessarily imply causation. Just because two variables are correlated doesn't mean one causes the other. There might be a third influencing factor.

The correlation coefficient is a valuable tool for understanding how variables change together, but it's important to interpret its value within the context of your data and consider other statistical analyses to draw meaningful conclusions.

Q. 5 What is a life table? Explain its uses and limitations

Ans-  A life table, also called a mortality table or actuarial table, is a statistical tool used in demography and actuarial science. It summarizes the mortality experience of a specific population and provides information about:

• Longevity: How long people in a population can be expected to live on average.
• Survivorship: The probability of a person from a certain age group surviving to reach the next age group.
• Life expectancy: The average number of years a person at a specific age is expected to live.

There are two main types of life tables:

• Period life table: Represents mortality rates during a specific time period for a certain population.
• Cohort life table: Represents the overall mortality rates of a specific generation (people born around the same time) throughout their lifetimes.

Uses of Life Tables

Life tables have various applications in different fields:

• Public health: Assess health trends, identify populations with higher mortality risks, and plan interventions.
• Social Security and insurance: Used by insurance companies and social security programs to calculate premiums and benefits based on life expectancy.
• Demographic studies: Analyze population growth, predict future trends, and understand the impact of mortality on population structure.
• Financial planning: Individuals can use life tables to estimate their life expectancy and plan for retirement needs.

Limitations of Life Tables

While life tables are valuable tools, they have limitations to consider:

• Assumptions: Life tables rely on assumptions about future mortality trends, which may not always be accurate.
• Heterogeneity: They represent averages for a population and don't account for individual differences in health status, lifestyle choices, and socio-economic factors.
• Limited Scope: Life tables typically focus on mortality and may not consider other factors affecting life quality.
• Data Dependence: The accuracy of a life table depends on the quality and completeness of data on mortality rates within the population.

Q6. Write short notes on the following: (a) Bayes’ theorem of probability (b) Age specific birth and death rates (c) Measurement of Skewness

Ans-  

Short Notes:

(a) Bayes' Theorem of Probability

Bayes' theorem is a fundamental concept in probability and statistics used to update probabilities based on new evidence. It allows us to calculate the probability of event A occurring, given that event B has already occurred.

Here's the formula:

P(A | B) = (P(B | A) * P(A)) / P(B)

Where:

• P(A | B) - Probability of event A occurring given event B has occurred.
• P(B | A) - Probability of event B occurring given event A has occurred.
• P(A) - Probability of event A occurring (prior probability).
• P(B) - Probability of event B occurring.

This theorem is particularly useful in situations where we have some initial belief about the probability of an event (prior probability) and then receive new information (evidence) that allows us to refine that belief.

(b) Age-Specific Birth and Death Rates

These are demographic rates that measure the frequency of births and deaths within specific age groups of a population. They are expressed as rates per 1,000 people in that age group over a specific period (e.g., per year).

• Age-Specific Birth Rate (ASBR): Indicates the average number of births to women in a particular age group per 1,000 women in that age group during a given period. This helps analyze fertility patterns across different age groups.
• Age-Specific Death Rate (ASDR): Represents the average number of deaths in a specific age group per 1,000 people in that age group during a given period. It helps understand mortality patterns at different life stages.

By analyzing these rates, demographers can gain valuable insights into population dynamics, fertility trends, life expectancy variations across age groups, and plan healthcare interventions for specific age brackets.

(c) Measurement of Skewness

Skewness is a statistical measure that describes the asymmetry of a distribution around its mean. It indicates the "tilt" or "slant" of the data distribution. There are different ways to measure skewness:

• Skewness coefficient: A statistical measure that captures the direction and extent of the asymmetry. A positive value indicates a right skew (tail extends to the right), a negative value indicates a left skew (tail extends to the left), and a value of zero suggests a symmetrical distribution.
• Visual inspection: Histograms and boxplots can be used to visually assess skewness. A right-skewed distribution will have a longer tail on the right side of the peak, while a left-skewed distribution will have a longer tail on the left side.

Understanding skewness is crucial for data analysis as it can affect the selection of appropriate statistical methods and the interpretation of results. For example, some statistical tests assume a normal (symmetrical) distribution, and if the data is skewed, alternative methods might be needed.

Q 7.  Differentiate between the following: (a) Simple random sampling and Stratified random sampling (b) Type I and Type II errors in hypothesis testing (c) Estimator and EstimatE

Ans-  

Differentiating between Key Concepts:

(a) Simple Random Sampling vs. Stratified Random Sampling:

Both methods are used to select a sample from a population, but they differ in their approach:

• Simple Random Sampling: Each member of the population has an equal chance of being selected. This is often achieved through random number generation. It ensures a representative sample if the population is homogenous (similar throughout).

• Stratified Random Sampling: The population is first divided into subgroups (strata) based on a relevant characteristic (e.g., age, gender, occupation). Then, a random sample is drawn from each stratum, typically in proportion to its size in the population. This method ensures representation from all subgroups, even if they are not evenly distributed in the population.

Here's a table summarizing the key differences:

Feature	Simple Random Sampling	Stratified Random Sampling
Selection Method	Random selection from entire population	Random selection from pre-defined subgroups (strata)
Population Homogeneity	Assumed to be homogenous	Can be heterogeneous (subgroups are considered)
Subgroup Representation	Not guaranteed	Ensured for all strata

(b) Type I and Type II Errors in Hypothesis Testing:

Hypothesis testing is a statistical procedure used to assess claims about a population. These errors can occur when making decisions based on the test results:

• Type I Error (Alpha Error): Rejecting a true null hypothesis. This occurs when we mistakenly conclude that there's a significant difference between groups when there truly isn't (false positive).

• Type II Error (Beta Error): Failing to reject a false null hypothesis. This occurs when we miss a real difference between groups due to an insufficient sample size or an insensitive test (false negative).

The goal is to minimize both types of errors. Researchers strive to balance these risks by setting a significance level (alpha) for the test and designing studies with sufficient power to detect true effects.

(c) Estimator vs. Estimate:

These terms are closely related but represent different aspects of statistical inference:

• Estimator: A rule or formula used to calculate a statistic from sample data that serves as an approximation of a population parameter. For example, the sample mean is an estimator of the population mean.

• Estimate: The specific value obtained by applying the estimator to a particular sample. It's the numerical outcome you get when you use the estimator on your data. For example, if you calculate the average height of people in your sample, that average is the estimate of the population mean height.