Unit 5 Business Statistics and Research Methods PYQ’s 3rd Sept 2024 Shift 2

The correct answer is ‘A-III, B-I, C-II, D-IV’.

Key Points

●       Joint occurrence of Events (A) matches with Compound Event (III).

o   Explanation: A compound event is an event that involves the joint occurrence of two or more events. For example, rolling a die and getting an even number and a number greater than 3.

●       Outcome of an experiment consisting of all possible events (B) matches with Collective Exhaustive Events (I).

o   Explanation: Collective exhaustive events are a set of events that cover all possible outcomes of an experiment. For example, when rolling a die, the events 1, 2, 3, 4, 5, and 6 are collectively exhaustive.

●       One of the events cannot be expected to occur in preference over the other (C) matches with Equally likely Events (II).

o   Explanation: Equally likely events are events that have the same probability of occurring. For example, when flipping a fair coin, getting heads or tails are equally likely events.

●       The occurrence of one event implies that the other cannot occur (D) matches with Mutually Exclusive Events (IV).

o   Explanation: Mutually exclusive events are events that cannot occur at the same time. For example, when flipping a coin, getting heads and tails are mutually exclusive events.

The correct answer is Poisson Distribution.

Key Points

●       Poisson Distribution:

o   When the number of trials (n) is large and the probability of success (p) is small, the Binomial distribution can be approximated by the Poisson distribution. This is particularly useful in financial enterprises for modeling rare events over a fixed period of time, such as the number of defaults on loans or insurance claims.

o   The Poisson distribution simplifies the computation process, especially in cases where calculating factorials for large numbers in the Binomial formula is impractical.

o   For a financial enterprise, using the Poisson approximation helps in better risk management and in creating more accurate predictive models for rare events.

Additional Information

●       Normal Distribution:

o   The Normal distribution is used to approximate the Binomial distribution when both n is large and p is not too close to 0 or 1. It is not suitable when p is very small, as it does not model the rare events accurately.

●       Hypergeometric Distribution:

o   The Hypergeometric distribution is used for scenarios without replacement, where the probability of success changes with each trial. It is not appropriate for approximating a Binomial distribution with a large number of trials.

●       Bernoulli Distribution:

o   The Bernoulli distribution is a special case of the Binomial distribution where there is only one trial (n=1). It cannot be used to approximate a Binomial distribution with a large number of trials.

The correct answer is  A, B, C & D Only.

Key Points

Let’s analyze each measure:

●       Mean Deviation

o   Mean Deviation, also known as Average Deviation, measures the average absolute deviation of each data point from the mean of the data set.

o   Reason for inclusion: It is a measure of dispersion as it quantifies the spread of data points around the mean.

●       Range

o   Range is the difference between the maximum and minimum values in a data set.

o   Reason for inclusion: It is a measure of dispersion as it indicates the extent of variability in the data set.

●       Standard Deviation

o   Standard Deviation measures the average deviation of each data point from the mean, taking into account the squared deviations.

o   Reason for inclusion: It is a widely used measure of dispersion that provides insight into the variability of a data set.

●       Coefficient of Variation

o   Coefficient of Variation is a standardized measure of dispersion, calculated as the ratio of the standard deviation to the mean, often expressed as a percentage.

o   Reason for inclusion: It allows comparison of the degree of variation from one data series to another, even if the means are drastically different.

●       Coefficient of Correlation

o   Coefficient of Correlation measures the strength and direction of the linear relationship between two variables.

o   Reason for exclusion: It is not a measure of dispersion but rather a measure of the relationship between two variables.

Therefore, the measures of dispersion from the given options are A: Mean Deviation, B: Range, C: Standard Deviation, and D: Coefficient of Variation. This makes option 4: “A, B, C & D Only” the correct choice.

The correct answer is Negatively skewed distribution.

Key Points

●       Negatively skewed distribution:

o   A negatively skewed distribution, also known as a left-skewed distribution, has a long tail on the left side. This means that there are a few extremely low values that pull the mean to the left.

o   In the context of a financial enterprise, this type of distribution might represent scenarios where most of the data points (e.g., returns on investment) are higher, but there are a few significant losses that cause the left tail.

o   This can be important for risk management, as it indicates potential for significant downside risk.

Additional Information

●       Positively skewed distribution:

o   A positively skewed distribution has a long tail on the right side, indicating a few extremely high values. This is the opposite of a negatively skewed distribution.

o   It is not the correct answer as it refers to data with a long right tail, not a long left tail.

●       Symmetrical distribution:

o   A symmetrical distribution is one where the left and right sides are mirror images of each other. There is no long tail on either side.

o   This type of distribution is not skewed and is not the correct answer.

●       Mesokurtic distribution:

o   A mesokurtic distribution refers to a distribution with kurtosis similar to that of a normal distribution. It describes the “peakedness” of the distribution.

o   This is not related to the skewness (left or right tail) and therefore is not the correct answer.

The correct answer is A, C & E Only.

Key Points

Let’s analyze each statement:

●       A. Chi-square test is a non-parametric test.

o   Non-parametric tests are those that do not assume a specific distribution for the data.

o   The chi-square test does not assume a normal distribution and is used for categorical data.

o   Reason for inclusion: This statement is correct as chi-square is indeed a non-parametric test.

●       B. Chi-square test was developed by Spearman.

o   In reality, the chi-square test was developed by Karl Pearson, not Spearman.

o   Reason for exclusion: This statement is incorrect.

●       C. Chi-square distribution can never be negative.

o   The chi-square distribution is always non-negative because it is derived from the sum of squared standard normal variables.

o   Reason for inclusion: This statement is correct as the chi-square distribution values are always non-negative.

●       D. Chi-square distribution is a discrete distribution.

o   The chi-square distribution is actually a continuous distribution, not discrete.

o   Reason for exclusion: This statement is incorrect.

●       E. Chi-square distribution is a function of its degree of freedom.

o   The shape of the chi-square distribution depends on the degrees of freedom (df).

o   Reason for inclusion: This statement is correct as the chi-square distribution changes with different degrees of freedom.

The correct answer is  A, C, B, D, E.

Key Points

●       Setting the null and alternative hypothesis (A):

o   The first step in hypothesis testing involves formulating the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis typically represents a statement of no effect or no difference, while the alternative hypothesis represents the effect or difference the test aims to detect.

o   This step is crucial as it sets the foundation for the entire hypothesis testing process by clearly stating what is being tested and the expected outcomes.

●       Determining the appropriate statistical test (C):

o   Once the hypotheses are set, the next step is to choose the appropriate statistical test based on the type of data and the research question. Common tests include t-tests, chi-square tests, and ANOVA, among others.

o   The choice of the statistical test is essential as it influences the accuracy and validity of the test results.

●       Setting the level of significance (B):

o   The level of significance, often denoted by alpha (α), is set next. This value, typically 0.05 or 0.01, represents the threshold for rejecting the null hypothesis. It indicates the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true.

o   Setting the level of significance helps in making objective decisions based on the test results.

●       Setting the decision rule (D):

o   After determining the level of significance, the decision rule is established. The decision rule specifies the criteria for rejecting or failing to reject the null hypothesis. For example, if the p-value obtained from the statistical test is less than the level of significance, the null hypothesis is rejected.

o   This step ensures that the decision-making process is consistent and unbiased.

●       Analysing the collected data (E):

o   The final step involves collecting and analyzing the data using the chosen statistical test. The results are then interpreted to determine whether the null hypothesis should be rejected or not.

o   This step is critical as it provides the empirical evidence needed to support or refute the hypotheses.

 The correct answer is 0.90.

Key Points

●       Formula for Probability of Union of Two Events:

o   The probability of either event occurring is given by the formula:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

o   Here:

●       P(A): Probability that a household has a television = 60% = 0.60.

●       P(B): Probability that a household has a refrigerator = 65% = 0.65.

●       P(A ∩ B): Probability that a household has both = 35% = 0.35.

●       Substitute Values:

o   P(A ∪ B) = 0.60 + 0.65 − 0.35.

o   P(A ∪ B) = 0.90.

●       Interpretation:

o   The probability that a randomly selected household has either a television or a refrigerator is 90%.

o   This means 90% of households have at least one of the two appliances.

Additional Information

●       Use in Business Decisions:

o   Understanding overlap in consumer ownership helps firms identify cross-selling opportunities. For example, a company may target households with televisions to promote refrigerators.

o   The data can guide marketing efforts for launching complementary products.

●       Set Theory Application:

o   This principle of union probability is directly derived from set theory, where overlapping probabilities (intersection) must be subtracted to avoid double counting.

The correct answer is B, A, D, E, C.

Key Points

●       Target population should be defined according to objective (B):

o   This initial step involves clearly identifying the group of individuals or elements that the researcher aims to study. It sets the foundation for the sampling design by ensuring that the population aligns with the research objectives.

o   Defining the target population helps in understanding the scope of the study and ensures that the sample drawn will be representative of the population under investigation.

●       Sampling frame must be determined (A):

o   Once the target population is defined, the next step is to establish a sampling frame, which is essentially a list or database from which the sample will be drawn.

o   The sampling frame should be as comprehensive and accurate as possible to ensure that every member of the target population has an equal chance of being selected.

●       Selecting a sampling technique (D):

o   After determining the sampling frame, the researcher must choose an appropriate sampling technique. This could be probability sampling (such as simple random sampling, stratified sampling) or non-probability sampling (such as convenience sampling, judgment sampling).

o   The choice of sampling technique will depend on the research objectives, the nature of the population, and the resources available.

●       Number of elements to be included in the study must be carefully determined (E):

o   The researcher needs to decide on the sample size, which is the number of elements or individuals to be included in the study. The sample size should be large enough to provide reliable and valid results but also feasible in terms of resources and time.

o   Determining the sample size involves statistical considerations and is crucial for the accuracy and generalizability of the study findings.

●       Sampling process must be effectively executed (C):

o   The final step is the actual execution of the sampling process, where the sample is drawn from the sampling frame using the selected technique.

o   This step involves ensuring that the sampling process is carried out systematically and rigorously to maintain the integrity and reliability of the study.

The correct answer is Data should not be presented in percentage or ratio format.

Key Points

●       Data should not be presented in percentage or ratio format:

o   When applying the chi-square test, it is essential that data is in the form of raw frequencies or counts. Percentages or ratios can distort the results of the test because the chi-square test evaluates the observed frequencies against the expected frequencies based on the hypothesis.

o   In a financial enterprise context, suppose you’re analyzing customer purchasing behaviors across different product categories. The chi-square test needs the actual number of purchases in each category rather than the percentage of total purchases.

Additional Information

●       The sample must have at least 100 observations:

o   This is incorrect. While having a larger sample size improves the reliability of the chi-square test, there is no strict requirement that you must have at least 100 observations. The chi-square test can be used on smaller samples, although small sample sizes can affect the test’s power and accuracy.

●       The sample should not be taken at random:

o   This statement is incorrect. Random sampling is crucial for most statistical tests, including the chi-square test, to ensure that the sample accurately represents the population and to avoid biases.

●       In any observation, all the individual observations need not be independent:

o   This is also incorrect. One of the key assumptions of the chi-square test is the independence of observations. This means that the occurrence of one observation should not influence the occurrence of another for the chi-square test to be valid.

The correct answer is A, B, D, C, E.

Key Points

●       Formulate a research question (A):

o   This is the initial step in the research process where the researcher identifies and defines the specific problem or question that they aim to address through their study.

o   It serves as the foundation for the entire research project, guiding the direction and scope of the investigation.

●       Literature review (B):

o   After formulating the research question, the next step is to review existing literature related to the topic.

o   This involves gathering and analyzing previous research, theories, and findings to understand the current state of knowledge and to identify gaps that the new research can fill.

●       Develop a research plan (D):

o   Based on the research question and literature review, the researcher then develops a detailed plan outlining the methodology, tools, and procedures to be used for data collection and analysis.

o   This step ensures that the research process is systematic and organized, providing a clear roadmap for conducting the study.

●       Collect data (C):

o   With the research plan in place, the next step is to collect the necessary data.

o   Data collection can involve various methods such as surveys, experiments, observations, or archival research, depending on the nature of the study.

●       Interpret the result (E):

o   The final step involves analyzing the collected data to draw meaningful conclusions and interpretations.

o   This step helps in answering the research question, validating or refuting hypotheses, and providing insights and recommendations based on the findings.

Additional Information

●       Importance of the Research Process:

o   The research process is a systematic approach that ensures the study is conducted in a structured and coherent manner.

o   Each step builds upon the previous one, contributing to the overall rigor and validity of the research.

●       Iterative Nature:

o   The research process is often iterative, meaning that researchers may need to revisit and refine earlier steps based on new insights and findings.

o   This iterative approach helps in continuously improving the quality and relevance of the research.