Business Statistics and Research Methods MCQ Quiz - Objective Question with Answer for Business Statistics and Research Methods - Download Free PDF

The correct answer is - Population size

Key Points

• Population size
  • Confidence intervals are typically calculated using the sample statistics rather than the size of the entire population.
  • In most practical applications, the population is considered infinitely large, making population size an irrelevant factor in determining the width of a confidence interval.
  • The finite population correction (FPC) is only applied when the sample is a significant fraction of the total population (typically >5%), which is uncommon in large populations.
• Sample size
  • Larger sample sizes result in narrower confidence intervals due to reduced standard error.
• Confidence desired
  • Higher confidence levels (e.g., 99% vs 95%) require wider intervals to ensure greater certainty of capturing the true parameter.
• Variability in the population
  • Higher variability (or standard deviation) leads to wider confidence intervals to accommodate the increased uncertainty.

Additional Information

• Formula for confidence interval width
  • For means: CI = x̄ ± Z*(σ/√n)
  • Key variables affecting width:
    • Z*: Depends on the confidence level.
    • σ: Population standard deviation (variability).
    • n: Sample size.
• Finite Population Correction (FPC)
  • Used only when the sample is a large proportion of the population.
  • FPC = √((N - n)/(N - 1)), where N = population size, n = sample size.
  • In most exam scenarios, FPC is negligible, so population size does not affect confidence interval width.
• Common misconceptions
  • Assuming that a larger population always increases uncertainty — this is incorrect unless FPC is applied.
  • Believing population size directly affects width — in standard practice, it does not.

The correct answer is - Quota sample

Key Points

• Quota sample
  • A Quota sample is a type of non-probability sampling method where the researcher selects participants based on specific characteristics or quotas.
  • Unlike probability sampling, quota sampling does not rely on random selection, which means not every individual has an equal chance of being chosen.
  • This method can lead to selection bias, making it less suitable for studies requiring statistical representativeness.
  • Quota sampling is often used in market research or surveys where time and cost constraints are significant.

Additional Information

• Probability sampling methods
  • Stratified sampling: Divides the population into subgroups (strata) and randomly selects samples from each subgroup. This ensures representation across different categories.
  • Cluster sampling: Groups the population into clusters, then randomly selects entire clusters for study. Useful for large, geographically dispersed populations.
  • Systematic random sampling: Selects samples using a fixed interval (e.g., every 5th individual). The starting point is chosen randomly, ensuring randomness in selection.
• Key difference between probability and non-probability sampling
  • Probability sampling: Ensures that every individual has an equal chance of being selected, making the sample statistically representative of the population.
  • Non-probability sampling: Relies on subjective methods, such as convenience or judgment, leading to potential bias and lack of representativeness.

The correct answer is - 1 - α (alpha) is called power of the test

Key Points

• Type-I error (α)
  • A Type-I error occurs when a true null hypothesis is incorrectly rejected.
  • The probability of committing this error is denoted by α (alpha).
  • 1 - α represents the confidence level, not the power of the test.
• Type-II error (β)
  • A Type-II error occurs when a false null hypothesis is not rejected.
  • The probability of committing this error is denoted by β (beta).
• Power of the test
  • The power of a test is defined as the probability of correctly rejecting a false null hypothesis.
  • It is given by 1 - β, not 1 - α.

Additional Information

• Summary of Errors
  • Type-I Error (α):
    • Occurs when we reject a true null hypothesis.
    • Example: Saying a new drug is effective when it is not.
  • Type-II Error (β):
    • Occurs when we fail to reject a false null hypothesis.
    • Example: Saying a new drug is ineffective when it is effective.
• Key Probability Terms
  • α (alpha): Probability of a Type-I error.
  • β (beta): Probability of a Type-II error.
  • 1 - β: Power of the test, representing the test's ability to detect a false null hypothesis.
  • 1 - α: Confidence level, representing the probability of correctly retaining a true null hypothesis.
• Importance of Power of the Test
  • A higher power indicates a greater likelihood of detecting a false null hypothesis.
  • Power depends on several factors, including:
    • Sample size (larger samples increase power).
    • Significance level (α).
    • Effect size (the magnitude of the difference being tested).

The correct answer is - Reliability

Key Points

• Reliability
  • Reliability refers to the consistency or repeatability of a measurement tool.
  • In the context of the goodness of measures, reliability is a necessary condition, as it ensures that the measurement produces stable results over time or across different scenarios.
  • However, reliability alone is not sufficient because a measure can be consistent without being accurate or valid.
  • For example, a weighing scale that consistently shows incorrect weight is reliable but not valid.

Additional Information

• Validity
  • Validity refers to the accuracy of a measurement tool, i.e., whether it measures what it is intended to measure.
  • While validity is crucial, it is dependent on reliability. A measure cannot be valid if it is not reliable.
• Measurement
  • Measurement refers to the process of assigning numerical values or categories to variables or traits.
  • It is a broader concept and encompasses both reliability and validity as essential components of a good measurement.
• Sensitivity
  • Sensitivity measures the ability of a test or tool to detect true positives or changes in the variable being measured.
  • While sensitivity is an important characteristic, it is not directly related to the necessity of reliability in ensuring consistent results.

The correct answer is - varies between ±1

Key Points

• Definition of the coefficient of correlation:
  • The coefficient of correlation, commonly denoted as r, measures the strength and direction of the linear relationship between two variables.
  • It is a key concept in statistics, particularly in regression and correlation analysis.
• Range of the coefficient of correlation:
  • The value of r always lies within the range of −1 to +1.
  • A value of +1 indicates a perfect positive linear relationship.
  • A value of −1 indicates a perfect negative linear relationship.
  • A value of 0 indicates no linear relationship between the variables.
• Why other options are incorrect:
  • Option 2 (has no limit): Incorrect because the coefficient of correlation is bounded by ±1.
  • Option 3 (can be less than 1): Misleading as it suggests the value can exceed ±1, which is not possible.
  • Option 4 (can be more than 1): Incorrect because the coefficient of correlation cannot exceed +1 or −1.

Additional Information

• Interpretation of the coefficient of correlation:
  • A value closer to +1 indicates a strong positive linear relationship (e.g., as one variable increases, the other also increases).
  • A value closer to −1 indicates a strong negative linear relationship (e.g., as one variable increases, the other decreases).
  • A value near 0 indicates a weak or no linear relationship.
• Types of correlation:
  • Positive correlation: Both variables move in the same direction.
  • Negative correlation: Variables move in opposite directions.
  • No correlation: No apparent relationship between the variables.
• Practical applications:
  • Used in finance to measure the relationship between stock prices or returns.
  • Applied in economics to study the relationship between demand and price or income and expenditure.
  • In social sciences, it helps study relationships between variables like education level and income.

The Experimental Research Design:

1. The experimental design is a way to carefully plan experiments in advance so that your results are both objective and valid.
2. The terms “Experimental Design” and “Design of Experiments” are used interchangeably and mean the same thing.
3. However, the medical and social sciences tend to use the term “Experimental Design” while engineering, industrial, and computer sciences favor the term “Design of experiments.”

Design of experiments involves:

1. The systematic collection of data
2. A focus on the design itself, rather than the results
3. Planning changes to independent (input) variables and the effect on dependent variables or response variables
4. Ensuring results are valid, easily interpreted, and definitive.

The following are the Experimental research design:

1. Before and after with control group design
2. Factorial design
3. Latin Square design

One group before-after design is a type of quasi-experimental design in which the dependent variable is treated once before the intervention is introduced, and then after the implementation of an intervention. A quasi-experimental design is similar to another experimental design, but they lack random assignment, hence they are incomplete experimental designs.

Corporate governance is the collection of mechanisms, processes, and relations by which corporations are controlled and directed.

Key PointsExplanation:

1. The term ‘Corporate Governance’ refers to the set of mechanisms which is being used in the relationships between shareholders which is further being used to ascertain and monitor the strategic decisions and performance of the organization.
2. Governance also refers to a means for creating order between the firms’ owners and the top-level managers.
3. Therefore, corporate governance symbolizes the company’s values.
4. In the case of modern corporations, the main aim of corporate governance is to assure and fulfill the interests of top-level managers who are being associated with the interests of the shareholders.

Therefore, Corporations are controlled and directed by Corporate governance.

Additional Information 1. Business ethics (also corporate ethics) is a form of applied ethics or professional ethics that examines ethical principles and moral or ethical problems that arise in a business environment. It applies to all aspects of business conduct and is relevant to the conduct of individuals and entire organizations.

2. Corporate code of conduct (CCC), codified set of ethical standards to which a corporation aims to adhere. Commonly generated by corporations themselves, corporate codes of conduct vary extensively in design and objective. Crucially, they are not directly subject to legal enforcement.

3. Internal Corporate Governance Mechanisms: Internal mechanisms are the ways and methods used by the firms which help the management in enhancing the value of shareholders. The constituents of internal mechanisms include ownership structure, the board of directors, audit committees, compensation board, and so on.

When two variables are related in such a way that a change in the value of one variable affects the value of another variable, then variables are said to be correlated or there is a correlation between these two variables.  

According to the direction of change in variables, there are two types of correlation: 1. Positive Correlation 2. Negative Correlation  

1. Positive Correlation: Correlation between two variables is said to be positive if the values of the variables deviate in the same direction i.e. if the values of one variable increase (or decrease) then the values of another variable also increase (or decrease). Some examples of positive correlation are correlation between:

1. Heights and weights of a group of persons;
2. Household income and expenditure;
3. Amount of rainfall and yield of crops; and
4.  Expenditure on advertising and sales revenue.

In the last example, it is observed that as the expenditure on advertising increases, sales revenue also increases. Thus, the change is in the same direction. Hence the correlation is positive. In the remaining three examples, usually the value of the second variable increases (or decreases) as the value of the first variable increases (or decreases).

2. Negative Correlation: Correlation between two variables is said to be negative if the values of variables deviate in opposite direction i.e. if the values of one variable increase (or decrease) then the values of another variable decrease (or increase). Some examples of negative correlations are correlation between:

1. Volume and pressure of perfect gas;
2. Price and demand for goods;
3. Literacy and poverty in a country; and
4. Time spent on watching TV and marks obtained by students in the examination.

In the first example, pressure decreases as the volume increases or pressure increases as the volume decreases. Thus the change is in opposite direction. Therefore, the correlation between volume and pressure is negative. In the remaining three examples also, the values of the second variable change in the opposite direction of the change in the values of the first variable. 

Now, since the sum total of the values of two variables 'X' and 'Y' is equal for all the observations. i.e. x + y = 'z' (equal), for 'z' to remain same (equal) for all observations, if 'x' increases, 'y' should decrease and vice-versa. Thus, option 2 is the correct answer.

In probability theory and statistics, the Poisson distribution, named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

Using the Poisson probability rule, 

P(X = x) = (λ^x.e^-λ) / x!

where, x = 0,1,2,3,...

λ = mean number of occurrences in the interval

e = Euler's constant

Here, λ = 2 customers/minute. Now,

P(3) = (e^-2 x 2³) / 3!

With e-2 = 0.1353, we get

P(3) = (0.1353 x 8) / 6 

∴ P(3) = 0.1804

Thus, option 1 is the correct answer.

The simplest case of a normal distribution is known as the standard normal distribution. This is a special case when μ = 0 and σ = 1, and it is described by this probability density function:

 \(ϕ (x) = \dfrac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}\)

Here, the factor \(1/\sqrt{2\pi}\) ensures that the total area under the curve ϕ(x) is equal to one. The factor 1/2 in the exponent ensures that the distribution has unit variance (i.e., variance being equal to one), and therefore also unit standard deviation. This function is symmetric around x = 0, where it attains its maximum value \(1/\sqrt{2\pi}\) and has inflection points at x = +1 and x = -1.

Thus, option 1 is the correct answer.

The correct answer is not correlated

Key Points Xi = explanatory variable/ Independent Variable

U i = error term 

Important Points Assumptions of the Classical Linear Regression Model:

• The regression model is linear, correctly specified, and has an additive error term.
• The error term has a zero population mean.
• All explanatory variables are uncorrelated with the error term.
• Observations of the error term are uncorrelated with each other (no serial correlation).
• The error term has a constant variance (no heteroskedasticity).
• No explanatory variable is a perfect linear function of any other explanatory variables (no perfect multicollinearity).
• The error term is normally distributed (not required).

Increasing returns to scale indicates the above production function

In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs (particularly physical capital and labor) and the amount of output that can be produced by those inputs.

The Cobb–Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas during 1927–1947.

In its most standard form for production of a single good with two factors, the function is Y=AK^αL^β

where:

• Y = total production (the real value of all goods produced in a year or 365.25 days)
• L = labor input (person-hours worked in a year or 365.25 days)
• K = capital input (a measure of all machinery, equipment, and buildings; the value of capital input divided by the price of capital)
• A = total factor productivity
• α and β are the output elasticities of capital and labor, respectively. These values are constants determined by available technology.

1. If α + β = 1 you can say that the returns to scale are constant. It means that doubling the amount of both capital and labor would result in double the output.
2. If α + β < 1 returns to scale are decreasing. The proportional change in factors will result in a smaller proportional change in output.
3. If α + β > 1 returns to scale are increasing. Likewise, the proportional change in factors will lead to a higher proportional change in output.

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

Key PointsUsing statistics, hypothesis testing is a formal process for examining our theories about the world. Scientists most frequently employ it to examine particular hypotheses that result from theories.

Important PointsProcess of Hypothesis Testing are as follows:

1. Setup Null and Alternate Hypothesis
2. Select the Significance Level
3. Select Test Statistics
4. Establish the Decision Rule
5. Performance Computations
6. Draw Conclusions

The correct answer is The co-efficient of quartile deviation remain unchanged.

Key PointsQuartile deviation:

• Half of the difference between the upper and lower quartiles is the Quartile Deviation, which may be calculated numerically.
• Quartile Deviation is also known as the Semi Interquartile range.
• QD denotes quartile deviation, with Q3 denoting the upper quartile and Q1 denoting the lower quartile.

Quartile Deviation Formula

QD = (Q₃ – Q₁)/2

Important Points

Coefficient of Quartile Deviation:

Dispersion can be compared for two or more sets of data using the coefficient of quartile deviation (also known as the quartile coefficient of dispersion).

Formula : \({Q_3-Q_1}\over {Q_3 + Q_1}\)

If one set of data has a higher coefficient of quartile deviation than another, the interquartile dispersion of that data set is higher.

In statistics, for a moderately skewed distribution, there exists a relation between mean, median, and mode. This mean median and mode relationship is known as the “empirical relationship”

• Mean is the average of the data set which is calculated by adding all the data values together and dividing it by the total number of data sets.
• Median is the middle value among the observed set of values and is calculated by arranging the values in ascending order or in descending order and then choosing the middle value.
• The mode is the number from a data set that has the highest frequency and is calculated by counting the number of times each data value occurs.

In the case of a moderately skewed distribution, i.e. in general, the difference between mean and mode is equal to three times the difference between the mean and median. Thus, the empirical relationship as Mean – Mode = 3 (Mean – Median). 

If solved further, Mode = 3 Median - 2 Mean and 3 Median = 2 Mean + Mode

Hence 2 Median - 3 Quartile Deviation = 2 Mean is NOT true in a model distribution.