BECC-107 Solved Assignment

For July 2023 and January 2024 Admission Cycles

1. (a) Calculate mean, median and mode from the following data.

Assignment II

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?
   b) Bring out the major properties of binomial distribution. Mention certain important uses of this distribution.
4. a) Fit a straight line $(Y=a+bX)$$(Y=a+bX)$(Y=a+bX)(Y=a+b X)$(Y=a+bX)$ to the following data. Compare the estimated values of the dependent variable with its actual values.

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

Assignment III

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

Expert Answer

BECC-107 Solved Assignment

For July 2023 and January 2024 Admission Cycles

Answer:

Answer:

Answer:

Distinction Between Sample Survey and Census

1. Census:

• Definition: A census is a method of data collection in which every unit or member of a population is surveyed. It aims to cover the entire population, leaving no one out.
• Characteristics:
  • Complete enumeration: Every individual or unit in the population is included.
  • Accuracy: Since it covers the entire population, the data obtained can be highly accurate, assuming no measurement errors occur.
  • Costly and Time-Consuming: Due to its large scale, conducting a census requires significant time, financial resources, and personnel.
  • Infrequent: Censuses are typically conducted at long intervals (e.g., every 10 years) due to their complexity.
• Example: National population census where every household in the country is surveyed to obtain demographic data.

2. Sample Survey:

• Definition: A sample survey collects data from a subset of the population rather than the entire population. The subset, or sample, is chosen to be representative of the population.
• Characteristics:
  • Partial enumeration: Only a sample, not the entire population, is surveyed.
  • Cost-effective: Sample surveys are generally less expensive and quicker to conduct than censuses.
  • Potential for Sampling Error: Since not everyone is surveyed, the results may have some degree of sampling error, though this can be minimized with good sampling techniques.
  • Frequent: Sample surveys can be conducted more frequently due to their lower cost and faster execution.
• Example: A survey of 1,000 households to determine average household income across a country.

Steps in Collecting Data Through a Sample Survey

1. Define the Objectives of the Survey:

• Clearly identify the purpose of the survey and the kind of information you want to collect. For instance, you might aim to assess the income and expenditure patterns of households in a particular region.

2. Define the Population:

• Determine the target population for your survey. This could be the entire population of a city, a specific age group, or households with certain characteristics.

3. Select a Sampling Method:

• Choose an appropriate sampling method to ensure that the sample is representative of the population. Common methods include:
  • Simple random sampling: Every individual in the population has an equal chance of being selected.
  • Stratified sampling: The population is divided into strata (e.g., income levels), and a random sample is taken from each stratum.
  • Cluster sampling: The population is divided into clusters (e.g., neighborhoods), and entire clusters are randomly selected.

4. Determine the Sample Size:

• Decide how many individuals or units to survey. The sample size should be large enough to be representative but small enough to be manageable within available resources.

5. Design the Questionnaire:

• Create a well-structured questionnaire that collects the necessary data. The questions should be clear, concise, and relevant to the objectives of the survey.

6. Conduct a Pilot Survey:

• Test the questionnaire and survey process on a small sample to identify any potential issues or misunderstandings.

7. Collect the Data:

• Administer the survey to the selected sample. This can be done through face-to-face interviews, telephone interviews, mailed questionnaires, or online surveys.

8. Process and Analyze the Data:

• After collecting the data, process it (e.g., coding responses, checking for errors) and analyze it using statistical methods to draw conclusions about the population.

9. Report the Findings:

• Present the findings in a clear and understandable manner, typically in the form of a report or presentation.

Sample Questionnaire for Collecting Income and Expenditure Levels of Households

1. Number of Household Members: _______
2. Location of Household (City, Town, Village): ____________________
3. Main Source of Household Income (Tick one):
   • [ ] Salaried Employment
   • [ ] Self-Employment
   • [ ] Agriculture
   • [ ] Pension
   • [ ] Other: ____________________

4. What is your household’s total monthly income?
   • [ ] Less than $500
   • [ ] $500 – $999
   • [ ] $1,000 – $1,499
   • [ ] $1,500 – $1,999
   • [ ] $2,000 and above
5. Does your household receive any additional income from the following sources? (Tick all that apply)
   • [ ] Rental income
   • [ ] Government assistance
   • [ ] Interest or dividends
   • [ ] Remittances from family abroad
   • [ ] Other: ____________________

6. What is your household’s total monthly expenditure?
   • [ ] Less than $500
   • [ ] $500 – $999
   • [ ] $1,000 – $1,499
   • [ ] $1,500 – $1,999
   • [ ] $2,000 and above
7. Please estimate your household’s monthly expenditure on the following categories:

8. Does your household save any portion of its income?
   • [ ] Yes
   • [ ] No
9. If yes, what percentage of your household’s income is saved monthly?
   • [ ] Less than 5%
   • [ ] 5% – 10%
   • [ ] 11% – 15%
   • [ ] More than 15%
10. Does your household invest in any of the following? (Tick all that apply)
    • [ ] Real estate
    • [ ] Stocks or bonds
    • [ ] Mutual funds
    • [ ] Pension funds
    • [ ] Other: ____________________

Assignment II

Answer:

Given Data:

• The probability that Rajesh scores more than 90 marks in a single test, $p$$p$pp$p$, is $0.75$$0.75$0.750.75$0.75$.
• The number of class tests, $n$$n$nn$n$, is 4.
• We are asked to find the probability that Rajesh scores more than 90 marks in exactly 3 out of 4 tests, which means $k=3$$k=3$k=3k = 3$k=3$.

Binomial Probability Formula:

• $(\genfrac{}{}{0ex}{}{n}{k})$$(\genfrac{}{}{0ex}{}{n}{k})$((n)/(k))\binom{n}{k}$(\genfrac{}{}{0ex}{}{n}{k})$ is the binomial coefficient, representing the number of ways to choose $k$$k$kk$k$ successes from $n$$n$nn$n$ trials.
• $p$$p$pp$p$ is the probability of success in a single trial.
• $1-p$$1-p$1-p1 – p$1-p$ is the probability of failure in a single trial.
• $X$$X$XX$X$ is the number of successes.

Step-by-Step Calculation:

1. $n=4$$n=4$n=4n = 4$n=4$, $k=3$$k=3$k=3k = 3$k=3$, and $p=0.75$$p=0.75$p=0.75p = 0.75$p=0.75$.
2. The binomial coefficient $(\genfrac{}{}{0ex}{}{4}{3})$$(\genfrac{}{}{0ex}{}{4}{3})$((4)/(3))\binom{4}{3}$(\genfrac{}{}{0ex}{}{4}{3})$ is calculated as:

3. Substituting the values into the binomial probability formula:

4. Calculate the powers of 0.75 and 0.25:

5. Perform the multiplication:

Conclusion:

Answer:

Major Properties of Binomial Distribution

1. Fixed Number of Trials (n):
   • The binomial distribution is based on a fixed number of trials, denoted by $n$$n$nn$n$. Each trial is independent of the others, and the number of trials is predetermined.
2. Two Possible Outcomes:
   • Each trial has exactly two possible outcomes: success (with probability $p$$p$pp$p$) and failure (with probability $1-p$$1-p$1-p1 – p$1-p$). These outcomes are mutually exclusive and collectively exhaustive.
3. Constant Probability of Success:
   • The probability of success, denoted by $p$$p$pp$p$, remains constant for each trial. Similarly, the probability of failure remains $1-p$$1-p$1-p1 – p$1-p$ throughout the trials.
4. Independence of Trials:
   • The trials are independent, meaning the outcome of one trial does not affect the outcome of any other trial.
5. Discrete Distribution:
   • The binomial distribution is discrete, meaning it deals with outcomes that are countable, such as the number of successes in a series of trials.
6. Probability Mass Function (PMF):
   • The probability of observing exactly $k$$k$kk$k$ successes in $n$$n$nn$n$ trials is given by the binomial probability mass function:
   $$P(X=k)=(\genfrac{}{}{0ex}{}{n}{k})\cdot p^k\cdot (1-p{)}^{n-k}$$$$P(X=k)=(\genfrac{}{}{0ex}{}{n}{k})\cdot p^k\cdot (1-p{)}^{n-k}$$P(X=k)=((n)/(k))*p^(k)*(1-p)^(n-k)P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 – p)^{n – k}$$P(X=k)=(\genfrac{}{}{0ex}{}{n}{k})\cdot p^k\cdot (1-p{)}^{n-k}$$
   Where:
   • $(\genfrac{}{}{0ex}{}{n}{k})$$(\genfrac{}{}{0ex}{}{n}{k})$((n)/(k))\binom{n}{k}$(\genfrac{}{}{0ex}{}{n}{k})$ is the binomial coefficient, representing the number of ways to choose $k$$k$kk$k$ successes from $n$$n$nn$n$ trials.
   • $p$$p$pp$p$ is the probability of success on each trial.
   • $1-p$$1-p$1-p1 – p$1-p$ is the probability of failure on each trial.
7. Mean and Variance:
   • The mean $\mu$$\mu$mu\mu$\mu$ of a binomial distribution is given by:
   $$\mu =n\cdot p$$$$\mu =n\cdot p$$mu=n*p\mu = n \cdot p$$\mu =n\cdot p$$
   • The variance ${\sigma }^2$${\sigma }^2$sigma^(2)\sigma^2${\sigma }^2$ of a binomial distribution is given by:
   $${\sigma }^2=n\cdot p\cdot (1-p)$$$${\sigma }^2=n\cdot p\cdot (1-p)$$sigma^(2)=n*p*(1-p)\sigma^2 = n \cdot p \cdot (1 – p)$${\sigma }^2=n\cdot p\cdot (1-p)$$
8. Symmetry and Shape:
   • The shape of the binomial distribution depends on the value of $p$$p$pp$p$:
     • If $p=0.5$$p=0.5$p=0.5p = 0.5$p=0.5$, the distribution is symmetric.
     • If $p<0.5$$p<0.5$p < 0.5p < 0.5$p<0.5$, the distribution is skewed to the right (positively skewed).
     • If $p>0.5$$p>0.5$p > 0.5p > 0.5$p>0.5$, the distribution is skewed to the left (negatively skewed).

Important Uses of Binomial Distribution

1. Quality Control and Manufacturing:
   • The binomial distribution is used to model the number of defective items in a batch or the number of successful outcomes in a series of quality control tests. For example, it can be used to calculate the probability that a certain number of products out of a sample are defective.
2. Medical Research:
   • In clinical trials, the binomial distribution can be used to model the probability of a certain number of patients responding to a treatment (success) versus not responding (failure). This helps in analyzing the effectiveness of treatments or drugs.
3. Survey Research and Polling:
   • Binomial distribution is useful in analyzing survey results, where the outcome of interest might be a simple yes/no answer. For example, it can be used to calculate the probability that a certain percentage of respondents will support a particular candidate or policy.
4. Reliability Testing:
   • In reliability engineering, the binomial distribution can be used to model the probability of a certain number of components in a system failing within a specified period, assuming each component has the same probability of failure.
5. Genetics:
   • In genetics, the binomial distribution is used to predict the number of offspring with a certain trait in a given number of trials (e.g., the probability that a certain number of offspring will inherit a particular gene).
6. Marketing and Sales:
   • The binomial distribution is used to model the number of successful sales calls or responses in a given number of attempts, helping to analyze the effectiveness of sales strategies or marketing campaigns.
7. Finance and Insurance:
   • Binomial distribution can be used in finance to model the number of successful investments or in insurance to model the number of claims filed out of a certain number of policies.

Conclusion

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