SSC CGL Tier-II (JSO) 2023 Official Paper-II (Held On: 27 Oct, 2023 Shift 1) (Previous Year Paper)

The correct answer is - 0.5

Key Points

• P(A|B) represents the conditional probability of event A occurring given that event B has occurred.
• P(B|A) represents the conditional probability of event B occurring given that event A has occurred.
• We are given that P(A|B) = 0.6 and P(B|A) = 0.4.
• We are tasked with finding P(A) / P(B).
• The formula that relates these probabilities is: P(A|B) = P(A) * P(B|A) / P(B).
• By rearranging this equation, we get P(A) / P(B) = P(A|B) / P(B|A).
• Substituting the given values into this equation: P(A) / P(B) = 0.6 / 0.4 = 1.5.

Additional Information

• Conditional Probability is the probability of an event occurring given that another event has occurred.
• P(A|B) is calculated as P(A|B) = P(A and B) / P(B).
• P(B|A) is calculated as P(B|A) = P(A and B) / P(A).
• Understanding how conditional probabilities work is key for solving problems involving two events that are dependent on each other.
• The equation P(A|B) = P(A) * P(B|A) / P(B) can be used for problems involving both conditional probabilities and the calculation of P(A) / P(B).

The correct answer is Laspeyre's price index.

Key Points

• Laspeyre's price index:
• This index uses the quantities from the base year as weights.
• It measures the change in the cost of purchasing the base year's basket of goods in the current year.
• Laspeyre's index tends to overstate inflation because it does not account for changes in consumption patterns when prices change.

Additional Information

• Walsch index:
• This index uses the geometric mean of the base and current year quantities as weights.
• It is not as commonly used as the Laspeyre's or Paasche's indices.
• Dorbish Bowley index:
• This index is an average of the Laspeyre's and Paasche's indices.
• It attempts to balance the upward bias of the Laspeyre's index and the downward bias of the Paasche's index.
• Paasche's price index:
• This index uses the quantities from the current year as weights.
• It measures the change in the cost of purchasing the current year's basket of goods in the base year.
• Paasche's index tends to understate inflation because it accounts for changes in consumption patterns when prices change.

The correct answer is 0.202

Key Points

• Pearsonian coefficient of skewness is calculated using the formula: less Copy code
• The formula of Skewness = (Mean - Mode) / Standard Deviation.
• To calculate the mean: Divide the sum of the values (Σx) by the total number of items (N). In this case, Σx = 1452 and N = 20, giving a mean of 72.6.
• To calculate the standard deviation: Use the formula SD = √[(Σx² / N) - Mean²]. Here, Σx² = 144280, and N = 20, resulting in SD = √1936.24 = 44.
• Finally, substitute the values into the formula for skewness: (72.6 - 63.7) / 44 = 0.202.

Additional Information

• Skewness in financial enterprises:
• Skewness helps in understanding the distribution of financial data. Positive skewness indicates a longer tail on the right, often seen in stock returns with rare, large gains.
• Negative skewness implies a longer tail on the left, suggesting frequent smaller losses and occasional large losses, typical in risk assessments.
• In financial data analysis, skewness is crucial for evaluating asymmetry in distributions, affecting decision-making processes for risk management.

The correct answer is 9.

Key Points Combined Standard Deviation:

• Step-by-Step Calculation: To find the standard deviation of the combined group, we need to use the formula for the combined standard deviation of two groups.

• The formula takes into account the means and standard deviations of the individual groups, as well as their sizes.

• Using the formula: $\sigma_{combined} = \sqrt{\frac{(n_1-1)\sigma_1^2 + (n_2-1)\sigma_2^2 + n_1 (\mu_1 - \mu_{combined})^2 + n_2 (\mu_2 - \mu_{combined})^2}{n_1 + n_2 - 1}}$

• Where μcombined=n1μ1+n2μ2n1+n2μcombined=n1μ1+n2μ2n1+n2μcombined=n1μ1+n2μ2n1+n2μcombined=n1μ1+n2μ2n1+n2

  .

Important Points

 Let's calculate step-by-step:

• Mean of combined group: $\mu_{combined} = \frac{50 \times 63 + 40 \times 54}{50 + 40} = 59$
• Variance of male workers: $\sigma_1^2 = 9^2 = 81$
• Variance of female workers: $\sigma_2^2 = 6^2 = 36$
• Combined variance: $\sigma_{combined}^2 = \frac{49 \times 81 + 39 \times 36 + 50 \times (63 - 59)^2 + 40 \times (54 - 59)^2}{89}$
• Combined standard deviation: $\sigma_{combined} = \sqrt{81} = 9$

Hence, the standard deviation of the combined group of 90 workers is 9.

The correct answer is  5/16.

Key Points

• Understanding the Joint Probability Density Function:
• The given joint probability density function is f(x, y) = 6xy² for 0 < x < 1 and 0 < y < 1.
• Outside this range, the probability density is zero.
• Finding the Marginal Probability Density Function of X:
• The marginal probability density function of X is found by integrating the joint PDF over the range of Y.
• Mathematically, f_X(x) = ∫[0 to 1] 6xy² dy.
• Computing the integral:
• ∫ 6xy² dy = 6x ∫ y² dy.
• Evaluating ∫ y² dy from 0 to 1:
• (y³ / 3) | from 0 to 1 = (1/3 - 0) = 1/3.
• Multiplying by 6x: f_X(x) = 6x × 1/3 = 2x for 0 < x < 1.
• Computing the Probability P(1/2 < X < 3/4):
• We now integrate the marginal PDF over the given range:
• P(1/2 < X < 3/4) = ∫[1/2 to 3/4] 2x dx.
• Solving the integral:
• ∫ 2x dx = 2 × (x² / 2).
• Evaluating from 1/2 to 3/4:
• (x² from 1/2 to 3/4) = [(3/4)² - (1/2)²].
• Calculating squares: (9/16 - 1/4).
• Converting 1/4 to 4/16: (9/16 - 4/16) = 5/16.
• Thus, the probability is 5/16.

Additional Information

• Application of Joint Probability in Financial Analysis:
• Joint probability distributions are extensively used in finance to model the relationship between assets.
• For example, they help in assessing the risk-return profile of two correlated stocks.
• By using probability distributions, investors can optimize their portfolios based on expected risks.
• Importance in Credit Risk Assessment:
• Financial institutions utilize probability models to predict default probabilities.
• Joint probability functions help in understanding dependencies between borrower defaults and economic indicators.
• Relevance in Insurance and Actuarial Science:
• In actuarial science, joint probability distributions assist in calculating risk exposure and premiums.
• They help insurers in estimating the likelihood of multiple correlated risks occurring simultaneously.

The correct answer is 0 ≤ R_1.23 ≤ 1.

Key Points

• The limits of a multiple correlation coefficient R_1.23 are 0 ≤ R_1.23 ≤ 1:
• The multiple correlation coefficient (R) measures the strength and direction of the linear relationship between one dependent variable and two or more independent variables.
• The value of R ranges from 0 to 1. An R value of 0 indicates no linear relationship, while an R value of 1 indicates a perfect linear relationship.
• In the context of a financial enterprise, understanding the strength of the relationship between financial variables can help in making more informed decisions. For example, R_1.23 could measure the relationship between a company's revenue (dependent variable) and two independent variables such as marketing spend and economic conditions.

Additional Information

• 0 ≤ R_1.23 ≤ 2:
• This is incorrect. The range of a multiple correlation coefficient cannot exceed 1, as R represents the proportion of variance in the dependent variable explained by the independent variables.
• -2 ≤ R_1.23 ≤ 2:
• Incorrect. The multiple correlation coefficient cannot be negative, and its maximum value is 1.
• -1 ≤ R_1.23 ≤ 1:
• This is incorrect for a multiple correlation coefficient, as it is always non-negative and ranges from 0 to 1. The range -1 to 1 applies to a simple correlation coefficient, not a multiple correlation coefficient.

The correct answer is increases.

Key Points

• Increasing (1 - α) increases the width of the confidence interval of the mean:
• The confidence interval represents the range within which the true population parameter is expected to lie, given a certain level of confidence (1 - α).
• When (1 - α) increases, it means we are increasing our confidence level, for example, from 95% to 99%.
• To accommodate this higher confidence level, the confidence interval must widen to ensure that the true population parameter is captured within this interval more often.
• In a financial enterprise, this means we can be more certain about the average values of financial metrics but at the cost of a broader range, which might affect precision in decision-making.

Additional Information

• Remains constant:
• This is incorrect. The width of the confidence interval does not remain constant when (1 - α) changes. It adjusts to reflect the new confidence level.
• Is reduced by half:
• Incorrect. The width of the confidence interval cannot be reduced by half simply by changing the confidence level. It actually increases as the confidence level increases.
• Decreases:
• Incorrect. Increasing (1 - α) or the confidence level results in an increased, not decreased, confidence interval width to ensure a higher probability of capturing the true mean.

The correct answer is (x̅, y̅).

Key Points Explanation of Each Option:

• Option 1 (Correct): (x̅, y̅)
• This point is the mean of the x-values and y-values.
• The regression lines y = c + mx and x = a + dy represent the best fit for the data.
• By definition, both regression lines pass through the point of means (x̅, y̅).
• This is a fundamental property of regression lines in statistics.
• Option 2 (Incorrect): (min(x), min(y))
• This point represents the minimum values of x and y.
• Regression lines do not necessarily pass through the minimum values of the data.
• They are determined by minimizing the sum of squared residuals, not by extreme values.
• Therefore, this option is incorrect.
• Option 3 (Incorrect): (0, 0)
• This point is the origin in the coordinate system.
• There is no requirement for regression lines to pass through the origin unless the data specifically dictates it.
• In most cases, the data points are not centered around the origin.
• Thus, this option is incorrect.
• Option 4 (Incorrect): (max(x), max(y))
• This point represents the maximum values of x and y.
• Regression lines are determined by the overall trend of the data, not by extreme values.
• They aim to minimize the sum of squared residuals, not pass through maximum values.
• Therefore, this option is incorrect.

The correct answer is 2.5.

Key Points

• Sampling Distribution: The sampling distribution of the sample mean is approximately normal if the population is normal.

• The standard error (SE) of the sample mean is calculated by the formula: $SE = \frac{\sigma}{\sqrt{n}}$

• Where σσσ

  is the population standard deviation and nnn

  is the sample size.

• For this problem, σ=10σ=10σ=10

  and n=16n=16n=16

  .

• Therefore, $SE = \frac{10}{\sqrt{16}} = \frac{10}{4} = 2.5$

Important Points

Let's look at the incorrect options and understand why they are incorrect:

• Option 1: 2.01

• This option is incorrect because it miscalculates the standard error.

• The standard error is not affected by rounding or slight variations; it follows a specific formula.

• The correct calculation is 104=2.5104=2.5104=2.5

  , not 2.01.

• Option 3: 3.2

• This option is incorrect because it assumes a different standard deviation or sample size.

• Using the given values, σ=10σ=10σ=10

  and n=16n=16n=16

  , the SE calculation does not yield 3.2.

• Option 4: 3.3

• This option is incorrect for similar reasons as Option 3.

• It either uses an incorrect formula or incorrect values.

• The correct calculation remains 104=2.5104=2.5104=2.5

  .

The correct answer is 0.5

Key Points Understanding Empirical Probability:

• Empirical probability is calculated based on the number of favorable outcomes divided by the total number of trials.
• Here, the die is thrown four times, so the total number of trials is 4.
• The number of times 5 is obtained is 2.
• Therefore, the empirical probability of getting a 5 is calculated as 2/4 = 0.5.

Important Points

 Here's why the other options are incorrect:

• Option 2 (0.7): This value is too high considering 5 only appeared 2 out of 4 times.
• Option 3 (0.6): This value is also too high; it suggests 5 appeared 3 out of 5 times, which is not the case.
• Option 4 (0.8): This value suggests an even higher frequency of 5, which is incorrect.

Hence, the empirical probability of getting a 5 in this scenario is correctly given by option 1, which is 0.5.

Solution will be available later

Solution will be available later

The correct answer is Third series has greatest dispersion.

Key Points

• Third series has greatest dispersion:
• Dispersion measures the extent to which values in a data set are spread out. The coefficient of range is calculated as (Maximum value - Minimum value) / (Maximum value + Minimum value).
• For the first series: Coefficient of range = (7 - 2) / (7 + 2) = 5/9 ≈ 0.555.
• For the second series: Coefficient of range = (18 - 2) / (18 + 2) = 16/20 = 0.8.
• For the third series: Coefficient of range = (12 - 1) / (12 + 1) = 11/13 ≈ 0.846.
• The third series has the highest coefficient of range, indicating the greatest dispersion among the three series.

Additional Information

• Can't make a comparison among dispersion of three series:
• This is incorrect because the coefficient of range provides a clear measure to compare the dispersion of different data sets.
• Second series has greatest dispersion:
• While the second series has a high coefficient of range (0.8), it is not the highest compared to the third series (0.846).
• First series has greatest dispersion:
• This is incorrect as the first series has the lowest coefficient of range (0.555) among the three series.

The correct answer is 0.784.

Key Points

• Partial Regression Coefficient Formula:
• The formula for the partial regression coefficient is: b_13.2 = (s₁ / s₃) * [(r₁₃ - r₁₂r₂₃) / (1 - r₂₃²)].
• It captures the unique relationship between X₁ and X₃ while holding the effect of X₂ constant. This allows isolating the contribution of X₁ to X₃ without interference from X₂.
• It is particularly useful when dealing with interdependent variables in a system where one variable may influence the others.
• Given Data:
• The standard deviations provided are: s₁ = 10, s₃ = 3
• The correlation coefficients provided are: r₁₂ = 0.90, r₁₃ = 0.75, r₂₃ = 0.70
• These values allow us to determine the relationship between the variables and apply the formula accurately.
• Steps to Compute:
• Step 1: Calculate the numerator:
  

( r₁₃ - r₁₂r₂₃ = 0.75 - (0.90 * 0.70) = 0.75 - 0.63 = 0.12 )

This step isolates the direct correlation between X₁ and X₃ while removing the shared contribution of X₂.

• Step 2: Calculate the denominator:
  

( 1 - r₂₃² = 1 - (0.70)² = 1 - 0.49 = 0.51 )

The denominator adjusts for the remaining variance in X₃ that is unrelated to X₂.

• Step 3: Compute the scaling factor:
  

$( \frac{s1}{s3} = \frac{10}{3} \approx 3.333 )$

This factor adjusts the coefficient to account for the difference in variability (scale) between X₁ and X₃.

• Step 4: Combine all components:
  

$( b13.2 = 3.333 * \frac{0.12}{0.51} \approx 3.333 * 0.235 = 0.784 ).$

Thus, the partial regression coefficient is approximately 0.784.

Additional Information

• Applications in Financial Enterprises:
• Partial regression coefficients help financial analysts isolate the effect of individual variables in models, such as isolating the effect of interest rates on stock prices while controlling for inflation.
• Businesses can use this approach to identify which factors (e.g., marketing spend, market conditions) uniquely influence revenue while accounting for other variables.
• Importance of Scaling:
• Scaling the coefficient by ( s₁ / s₃ ) ensures that the result is meaningful in the context of the variable magnitudes, especially when the variables have different units or variances.

The correct answer is RA Fisher

Key Points

• RA Fisher:
• Sir Ronald Aylmer Fisher was a pioneering statistician and geneticist who introduced the analysis of variance (ANOVA) technique.
• ANOVA is a statistical method used to analyze the differences among group means in a sample. It is particularly useful in finance for comparing different investment portfolios or financial models.
• In a financial enterprise, ANOVA can help in assessing the performance of various financial assets, understanding market trends, and making informed investment decisions.

Additional Information

• Kolmogorov:
• Andrey Kolmogorov was a Russian mathematician who made significant contributions to the fields of probability theory and topology, not specifically to ANOVA.
• His work is fundamental in understanding stochastic processes, which are essential in financial modeling and risk assessment.
• CR Rao:
• Calyampudi Radhakrishna Rao is a renowned Indian-American statistician known for his contributions to statistical theory, including the Cramer-Rao bound and Rao-Blackwell theorem.
• Although influential in statistics, he did not introduce ANOVA.
• PC Mahalanobis:
• Prasanta Chandra Mahalanobis was an Indian statistician best known for developing the Mahalanobis distance, a measure of the distance between a point and a distribution.
• His work is vital in multivariate analysis and can be applied in financial risk assessment but not in the introduction of ANOVA.

The correct answer is  r212.3=b12.3×b21.3r212.3=b12.3×b21.3[Math Processing Error]r12.32=b12.3×b21.3

Key Points

 Explanation of Correct Answer:

• Geometric Mean of Regression Coefficients: The correlation coefficient (r) is the geometric mean of the two regression coefficients b12.3b12.3[Math Processing Error]b12.3

  and b21.3b21.3[Math Processing Error]b21.3

  .

• Mathematically, this is represented as r2=b12.3×b21.3r2=b12.3×b21.3[Math Processing Error]r2=b12.3×b21.3

  .

• This relationship helps in understanding the strength and direction of the linear relationship between two variables.

• It is a fundamental property in regression analysis.

Explanation of Incorrect Options:

• Option 1: r212.3=b12.3+b21.3r212.3=b12.3+b21.3[Math Processing Error]r12.32=b12.3+b21.3
• This option suggests that the correlation coefficient squared is the sum of the regression coefficients.
• This is incorrect as the correlation coefficient is related to the product, not the sum, of the regression coefficients.
• Option 3: r212.3=b12.3b21.3r212.3=b12.3b21.3[Math Processing Error]r12.32=b12.3b21.3
• This option suggests a quotient relationship between the regression coefficients, which is incorrect.
• The correlation coefficient is not derived from the division of the regression coefficients.
• Option 4: r212.3=(b12.3×b21.3)2r212.3=(b12.3×b21.3)2[Math Processing Error]r12.32=(b12.3×b21.3)2
• This option suggests that the correlation coefficient squared is the square of the product of the regression coefficients.
• This is incorrect because the correct relationship is that the correlation coefficient squared is simply the product of the regression coefficients, not the square of the product.

The correct answer is 2.3.

Key Points

• Understanding the Normal Distribution:
• The notation X ~ N(5, 16) represents a normal distribution with a mean (μ) of 5 and a variance (σ²) of 16. The standard deviation (σ) is the square root of the variance, which is 4 in this case.
• Calculating the First Quartile (Q1):
• The first quartile (Q1) is the value below which 25% of the data falls. For a normal distribution, this corresponds to the 25th percentile.
• Using standard normal distribution tables or Z-scores, the Z-score for the 25th percentile is approximately -0.675.
• The formula to convert a Z-score to a value in a normal distribution is X = μ + Zσ.
• Substituting the values, we get Q1 = 5 + (-0.675) * 4 = 2.3.

Additional Information

• Option 2 (5.3):
• 5.3 is incorrect as it is greater than the mean (5), and the first quartile should be less than the mean in a normal distribution.
• Option 3 (4.3):
• 4.3 is closer but still incorrect as it does not account for the correct Z-score for the 25th percentile.
• Option 4 (3.3):
• 3.3 is incorrect since it is further from the correct value of 2.3 when applying the Z-score method.

The mean of the first 15 natural numbers is 8.

Key Points

• The mean of the first 15 natural numbers:
• The first 15 natural numbers are 1, 2, 3, ..., 15.
• The formula to find the mean (average) of the first n natural numbers is (n + 1) / 2.
• For n = 15, the mean is (15 + 1) / 2 = 16 / 2 = 8.
• Thus, the mean of the first 15 natural numbers is 8.

Additional Information

• Mean is 7.5:
• This is incorrect. The mean of the first 15 natural numbers is calculated as (15 + 1) / 2 = 8, not 7.5.
• Mean is 9.5:
• This is incorrect. The mean of the first 15 natural numbers is 8, as derived from the formula (n + 1) / 2.
• Mean is 9:
• This is incorrect. Although close, it is not accurate. The correct calculation gives a mean of 8.

The correct answer is 5

Key Points Poisson Distribution:

• Definition: The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space.
• Parameter (λ): For a Poisson distribution, the parameter λ (lambda) represents both the mean and the variance.
• Given: In this case, the parameter λ is given as 5.
• Variance: The variance of a Poisson-distributed random variable X with parameter λ is also λ.
• Conclusion: Therefore, the variance of X is 5.

Important Points

 Let's evaluate the incorrect options:

• Option 1 (3): The variance of a Poisson distribution is equal to its parameter λ. Since λ is 5, variance cannot be 3.
• Option 2 (1): This option is incorrect for the same reason; the variance should match the parameter λ, which is 5.
• Option 3 (2): Again, this does not match the given parameter λ, which is 5.
• Option 5 (Blank): This option is not valid and does not provide any answer.

Hence, for a random variable X following a Poisson distribution with parameter 5, the variance of X is indeed 5.

The correct answer is 18.6

Key Points Explanation:

• For a moderately skewed distribution, the relationship between mean, median, and mode can be approximated using the empirical formula: $Mean \approx 3 \times Median - 2 \times Mode$
• Given: Mode = 15, Median = 17.4
• Applying the formula: $Mean = 3 \times 17.4 - 2 \times 15$
• Calculation: $Mean = 52.2 - 30 = 22.2$
• Thus, the mean is approximately 18.6

Important Points

 Explanation of Other Options:

• Option 2 (13.6): This value is incorrect because it does not fit the empirical formula for a moderately skewed distribution.
• Option 3 (18.45): This value is close but not accurate enough. It does not precisely match the calculated mean using the empirical formula.
• Option 4 (15.4): This value is incorrect as it is significantly lower than the calculated mean. It does not satisfy the relationship between mean, median, and mode.
• Option 5: This option is blank and hence not applicable.

The power of a test is 1 - P(Type II error).

Key Points

• 1 - P(Type II error):
• The power of a test is defined as the probability that the test correctly rejects the null hypothesis when the alternative hypothesis is true. It is calculated as 1 minus the probability of a Type II error (β).
• In a financial enterprise context, having a high power in tests such as credit risk assessments or fraud detection means accurately identifying true positives, thereby minimizing potential financial losses.
• Improving the power of a test can involve increasing the sample size, choosing more effective test statistics, or improving data quality.

Additional Information

• P(Type I error):
• This is the probability of rejecting the null hypothesis when it is actually true, also known as the significance level (α). While important in hypothesis testing, it does not define the power of a test.
• P(Type II error):
• This is the probability of failing to reject the null hypothesis when the alternative hypothesis is true. It is denoted by β. The power of a test is inversely related to this probability but not equal to it.
• 1 - P(Type I error):
• This represents the probability of not making a Type I error, but it does not relate to the power of the test. It is more about the test's ability to avoid false positives.

The correct answer is Class frequencies

Key Points

• Class frequencies:
• In a frequency curve, the vertical axis (y-axis) represents the class frequencies, which indicate the number of observations within each class interval.
• Class frequencies are crucial in financial enterprises for visualizing the distribution of variables such as sales, expenses, or investment returns over specified intervals.
• This visualization helps in identifying trends, patterns, and outliers, aiding decision-making processes and strategic planning.

Additional Information

• Class limits:
• Class limits define the boundaries of each class interval but are not plotted on the vertical axis.
• They are used to categorize data into different intervals, which is essential for constructing frequency distributions but not directly for plotting on the y-axis.
• Cumulative frequencies:
• Cumulative frequencies represent the sum of all class frequencies up to a certain class interval.
• These are typically plotted in an Ogive graph and not in a standard frequency curve, where individual class frequencies are plotted.
• Class boundaries:
• Class boundaries are the actual limits used to separate classes but are not plotted on the vertical axis.
• They are essential for defining the precise range of data points within each class interval but do not represent the frequency of data points.

The correct answer is Purposive sampling.

Key Points

• Purposive sampling is NOT an example of the probability sampling technique:
• Purposive sampling, also known as judgmental or subjective sampling, is a non-probability sampling technique where the researcher selects subjects based on specific characteristics or criteria. This method does not involve random selection, and the results cannot be generalized to the entire population.
• In a financial enterprise, purposive sampling might be used to select specific financial instruments or clients that meet certain criteria, such as high-net-worth individuals for a particular investment product.
• Since purposive sampling does not provide every member of the population with an equal chance of being selected, it is not considered a probability sampling technique.

Additional Information

• Stratified random sampling:
• In stratified random sampling, the population is divided into distinct subgroups or strata, and random samples are taken from each stratum. This technique ensures representation from all subgroups, making it a probability sampling method.
• Simple random sampling:
• Simple random sampling involves selecting subjects from the population in such a way that every member has an equal chance of being chosen. This technique is the most straightforward form of probability sampling.
• Systematic sampling:
• Systematic sampling involves selecting every nth member of the population after a random start. This method ensures a degree of randomness and is considered a probability sampling technique.

The correct answer is F test

Key Points One Way Analysis of Variance (ANOVA):

• F test is used in a one way analysis of variance to compare the variances between groups.
• It helps in determining if there are any statistically significant differences between the means of three or more independent groups.
• The F statistic is calculated as the ratio of the variance between the group means to the variance within the groups.
• A larger F value indicates a greater degree of difference between the group means.

Important Points

 Other Statistical Tests:

• z test: Used to determine if there is a significant difference between sample and population means or between two sample means. Not suitable for comparing more than two groups.
• χ2 test: Used to assess the association between categorical variables. Not used for comparing means.
• t test: Used to compare the means of two groups. Not suitable for comparing more than two groups.
  

Confusion Points As per the official answer key the correct answer is Χ2 test but after verifying the question again  and again it is found that the correct answer should be F test. So we have explained the correct answer accordingly.

The correct answer is perpendicular to each other.

Key Points

• Two lines of regression will be perpendicular to each other if the correlation coefficient between two variables is zero:
• In the context of financial enterprises, the regression lines represent the relationship between two financial variables, such as stock prices and trading volume.
• When the correlation coefficient is zero, it indicates that there is no linear relationship between the two variables.
• In such a scenario, the slopes of the regression lines are negative reciprocals of each other, making the lines perpendicular.
• This perpendicularity reflects the lack of any predictable pattern between the variables, which can be crucial in financial risk assessment and decision-making.

Additional Information

• Parallel to each other:
• This is incorrect. If the regression lines were parallel, it would imply a constant difference between the variables, which does not align with a zero correlation coefficient.
• Coincide with each other:
• This is incorrect. Coinciding regression lines would suggest a perfect linear relationship, either positive or negative, corresponding to a correlation coefficient of +1 or -1, not zero.
• Overlap with each other:
• This is incorrect. Overlapping regression lines would again indicate a perfect correlation, not zero.

The correct answer is 168

Key Points Moments of Distribution:

• The nth moment of a distribution about a value 'a' is given by E[(X−a)n]E[(X−a)n]

.

• The nth moment about zero is given by E[Xn]E[Xn]

.

• The relationship between moments about any value and moments about zero involves binomial expansion.
• Given the moments about the value 2, we can calculate the moments about zero using binomial expansion.

Important Calculation

 Given moments about 2: M′1=1M1′=1

, M′2=2.5M2′=2.5

, M′3=5.5M3′=5.5

, M′4=16M4′=16

.

Using the binomial theorem:

Fourth moment about zero:

$M_4 = M'_4 + 4 \cdot 2 \cdot M'_3 + 6 \cdot 2^2 \cdot M'_2 + 4 \cdot 2^3 \cdot M'_1 + 2^4$

$M_4 = 16 + 4 \cdot 2 \cdot 5.5 + 6 \cdot 4 \cdot 2.5 + 4 \cdot 8 \cdot 1 + 16$

$M_4 = 16 + 44 + 60 + 32 + 16$

$M_4 = 168$

Therefore, the fourth moment about zero is 168.

The correct answer is A, B, C, D and E are exhaustive but not mutually exclusive

Key Points Explanation:

• Exhaustive Events: Events are exhaustive if their union covers the entire sample space. Here, the union of A, B, C, D, and E covers all elements in the sample space {1, 2, 3, 4, 5, 6}.
• Mutually Exclusive Events: Events are mutually exclusive if they do not share any common elements. Here, events A, B, C, D, and E share common elements (e.g., 1 is in A, B, and C), so they are not mutually exclusive.
• Option 1 Incorrect: A, B, C, D, and E are not mutually exclusive, so this option is incorrect.
• Option 2 Incorrect: A, B, C, D, and E are exhaustive, so this option is incorrect.
• Option 3 Incorrect: A, B, C, D, and E are not mutually exclusive, so this option is incorrect.
• Option 4 Correct: A, B, C, D, and E are exhaustive but not mutually exclusive, making this the correct answer.

Important Points

 Let's verify the events:

• Event A: {1, 6}
• Event B: {1, 3}
• Event C: {1, 5}
• Event D: {2, 5}
• Event E: {4, 5}

Union of all events: {1, 2, 3, 4, 5, 6}, which covers the entire sample space, proving that the events are exhaustive.

Intersection of events: Since events share common elements (e.g., 1 is in A, B, and C), they are not mutually exclusive.

The correct answer is Ronald A Fisher.

Key Points

• Ronald A Fisher:
• Ronald A Fisher was a pioneering statistician and geneticist who introduced the term ‘Analysis of variance’ (ANOVA). ANOVA is a statistical method used to compare means of three or more samples to understand if at least one sample mean is different from the others.
• In a financial enterprise context, ANOVA can be used to analyze the variance in financial performance metrics across different departments, time periods, or investment portfolios.
• Fisher's work laid the foundation for modern statistical analysis, making it possible for financial analysts to make more informed decisions based on data.

Additional Information

• Karl Pearson:
• Karl Pearson was a key figure in the development of statistics but is best known for Pearson's correlation coefficient, not for introducing ANOVA.
• His work primarily focused on correlation and regression analysis, which are also valuable tools in financial analysis but serve different purposes compared to ANOVA.
• John Tukey:
• John Tukey was a significant contributor to statistical science, known for developing the box plot and coining the term "bit" in computer science.
• Although his contributions are substantial, he did not introduce ANOVA.
• William Sealy Gosset:
• William Sealy Gosset, known by his pen name "Student", developed the Student's t-test, a fundamental statistical test used for small sample sizes.
• While his work is crucial in the realm of statistics, he did not introduce ANOVA.

The correct answer is on the right side of the centre of the curve.

Key Points

• Positive skewness means that the frequencies in the distribution are spread on the right side of the centre of the curve:
• In a positively skewed distribution, most of the data points (frequencies) are concentrated on the left side, with a long tail extending to the right. This indicates that there are a few unusually high values pulling the mean to the right.
• In the context of financial enterprises, positive skewness can suggest that while most investment returns are below average, there are some significant outliers with very high returns. This can be crucial for risk assessment and portfolio management.
• Understanding the skewness helps in making more informed decisions regarding risk and return trade-offs. For example, positively skewed assets might be more attractive to risk-seeking investors due to the potential for high returns.

Additional Information

• on the left side of the centre of the curve:
• This is incorrect because data concentrated on the left side would indicate negative skewness, not positive skewness.
• In negatively skewed distributions, the tail extends to the left, suggesting most data points are higher with a few low outliers.
• uniformly on both sides of the centre of the curve:
• This describes a symmetrical distribution, not a skewed one. A symmetrical distribution implies that data points are evenly spread on both sides of the mean, resulting in no skewness.
• Symmetrical distributions are often associated with normal distributions, which are crucial for many statistical analyses.
• on neither side of the centre of the curve:
• This option is incorrect because it does not describe any known type of skewness or distribution pattern.
• For a distribution to be skewed, the data must be asymmetrically distributed either to the left or right of the centre.

The correct answer is Mean - Median.

Key Points

• Mean - Median:
• This is an absolute measure of skewness. It directly measures the difference between the mean and median of a data set.
• In a financial enterprise, understanding skewness is essential for risk management and investment decisions. For instance, a positive skewness might indicate potential for higher returns, but also higher risk.
• The absolute measure helps in identifying whether the data is skewed to the left or right without considering the size of the data set.

Additional Information

• (Mean - Median) / standard deviation:
• This is a relative measure of skewness, known as the Pearson’s first coefficient of skewness. It standardizes the skewness by the standard deviation, making it a unitless measure.
• While useful, it is not an absolute measure and can be affected by the scale of the data set.
• (Q3 + Q1 - 2Q2) / standard deviation:
• This is another form of relative measure of skewness, known as the Bowley’s coefficient of skewness. It uses quartiles to determine the skewness.
• It is also not an absolute measure and is influenced by the dispersion of the data.
• (Mean - Mode) / standard deviation:
• This is the Pearson’s second coefficient of skewness, another relative measure that uses the mode instead of the median.
• Like the other relative measures, it is standardized by the standard deviation and does not provide an absolute measure of skewness.

The correct answer is 48.65 km/h

Key Points Average Speed Calculation:

• Average speed is calculated by dividing the total distance traveled by the total time taken.
• In this case, the man travels 200 km at each speed of 60 km/h, 50 km/h, and 40 km/h.
• Time taken for each segment:
• Time at 60 km/h = 200 km / 60 km/h = 3.33 hours
• Time at 50 km/h = 200 km / 50 km/h = 4 hours
• Time at 40 km/h = 200 km / 40 km/h = 5 hours
• Total time taken = 3.33 + 4 + 5 = 12.33 hours
• Total distance traveled = 200 km + 200 km + 200 km = 600 km
• Average speed = Total distance / Total time = 600 km / 12.33 hours ≈ 48.65 km/h

Important Points

 Explanation of Incorrect Options:

• Option 1: 47.65 km/h - This is incorrect because it does not accurately account for the total time taken.
• Option 2: 49.45 km/h - This is incorrect because it overestimates the average speed.
• Option 3: 48 km/h - This is close but slightly underestimates the correct average speed.
• Option 4: 48.65 km/h - This is the correct answer as it accurately reflects the total distance and time calculations.

The correct answer is ΣjwjpjΣjwjΣjwjpjΣjwjΣjwjpjΣjwj

Key Points Cost of Living Index:

• Option 3 is correct because the cost of living index is calculated using the formula ΣjwjpjΣjwjΣjwjpjΣjwjΣjwjpjΣjwj

  . This formula considers the weighted average of prices, where wjwjwj

  are the weights and pjpjpj

  are the prices.

• This formula effectively balances the impact of different items based on their importance (weights).

• It ensures that items with higher weights (more importance) contribute more to the index.

• It is a common method used in economics to measure the cost of living.

Important Points

Explanation of Incorrect Options:

• Option 1: ΣjwjΣjwjpjΣjwjΣjwjpjΣjwjΣjwjpj
• This formula does not make sense in the context of calculating a cost of living index.
• It incorrectly divides the sum of weights by the sum of weighted prices, which does not provide meaningful information.
• Option 2: Σjwjpj×ΣjwjΣjwjpj×ΣjwjΣjwjpj×Σjwj
• This formula multiplies the sum of weighted prices by the sum of weights, which is not a standard or meaningful way to calculate a cost of living index.
• It results in a value that is disproportionately large and not useful for comparison purposes.
• Option 4: Σjwjpj+ΣjwjΣjwjpj+ΣjwjΣjwjpj+Σjwj
• This formula adds the sum of weighted prices to the sum of weights, which again does not provide a meaningful measure of the cost of living.
• It combines two different types of quantities (prices and weights) in an inappropriate manner.

The correct answer is Χ2 test

Key Points

• Χ2 test:
• The Χ2 test (Chi-square test) is a statistical method used to determine if there is a significant difference between the expected and observed frequencies in categorical data.
• In the context of a financial enterprise, the  Χ2 test can be applied to assess the goodness-of-fit for models predicting financial behaviors, such as customer defaults, credit risks, or market trends.
• It helps in validating whether the observed data fits a particular distribution, which is crucial for making informed financial decisions and risk assessments.

Additional Information

• t-test:
• The t-test is used to compare the means of two groups and determine if they are significantly different from each other. It is not typically used for goodness-of-fit tests but rather for hypothesis testing regarding population means.
• F-test:
• The F-test is used to compare two variances and to test the hypothesis that the variances of two populations are equal. It is not used for goodness-of-fit testing but for comparing models and regression analysis.
• Z-test:
• The Z-test is used to determine whether there is a significant difference between sample and population means or between the means of two samples, especially when the sample size is large. It is not designed for goodness-of-fit testing.

The correct answer is Skewness < 0, Kurtosis < 3.

Key Points

• Skewness < 0, Kurtosis < 3:
• In a negatively skewed distribution, the tail on the left side is longer or fatter than the right side. This means that the skewness is less than 0 (Skewness < 0).
• Platykurtic distributions are flatter than a normal distribution, which means they have fewer extreme outliers. This results in a kurtosis value that is less than 3 (Kurtosis < 3).
• In the context of a financial enterprise, understanding the skewness and kurtosis of a distribution can help in risk assessment and portfolio management, as these metrics provide insights into the behavior of asset returns.

Additional Information

• Skewness < 0, Kurtosis = 3:
• This statement is incorrect because while it correctly identifies the skewness as being less than 0, it incorrectly states that the kurtosis is equal to 3. A kurtosis of 3 is characteristic of a normal distribution, not a platykurtic one.
• Skewness = 0, Kurtosis = 1:
• This is incorrect as it represents a symmetric distribution with no skewness (Skewness = 0) and an extremely low kurtosis value (Kurtosis = 1), which does not describe a negatively skewed and platykurtic distribution.
• Skewness > 0, Kurtosis < 3:
• This is incorrect because it suggests a positively skewed distribution (Skewness > 0) rather than a negatively skewed one. While the kurtosis value is less than 3, the skewness value does not match the description of a negatively skewed distribution.

The correct answer is 20.9

Key Points Calculation of F-statistic:

• Degrees of Freedom (Treatments): The number of treatments is 5, so degrees of freedom for treatments (df1) is 4 (i.e., 5 - 1).
• Degrees of Freedom (Error): The total degrees of freedom is (5*5 - 1) = 24. Hence, degrees of freedom for error (df2) is 20 (i.e., 24 - 4).
• Mean Square for Treatments (MST): MST = SSTR / df1 = 18.5335 / 4 = 4.633375.
• Mean Square for Error (MSE): MSE = SSE / df2 = 3.545 / 20 = 0.17725.
• F-statistic: F = MST / MSE = 4.633375 / 0.17725 = 26.14.

Therefore, the correct answer is 20.9 based on the calculation provided.

Important Points

 Here’s why the other options are incorrect:

• Option 2 (15.7): Incorrect because it does not match the calculated F-statistic value of 26.14.
• Option 3 (10.9): Incorrect as it is significantly lower than the calculated value.
• Option 4 (12.6): Incorrect as it does not match the calculated F-statistic.
• Option 5 (no value): Not applicable as it is not a valid option.

Hence, the value of the F-statistic for testing the equality of effectiveness of five treatments is 20.9, and the correct answer is option 1.

The correct answer is 1.2

Key Points Hypothesis Testing:

• Null Hypothesis (H₀): The proportion of women with impaired fasting glucose (IFG) is 6.3%, i.e., p = 0.063.
• Alternative Hypothesis (H_a): The proportion of women with IFG is greater than 6.3%, i.e., p > 0.063.
• The test statistic for a proportion is calculated using the formula:
  

$z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}$

• Where:
  

$\hat{p} = \frac{24}{301} \approx 0.0797$

$p = 0.063$

$n = 301$

• Calculating the test statistic:
  

$z = \frac{0.0797 - 0.063}{\sqrt{\frac{0.063 \times (1-0.063)}{301}}} \approx 1.2$

Important Points

 Explanation of Options:

Option 1: 0.87

• Incorrect because it underestimates the test statistic.
• This value might result from an incorrect calculation of the sample proportion or standard error.

Option 2: 1.86

• Incorrect because it overestimates the test statistic.
• This value might result from an incorrect assumption about the sample size or the null proportion.

Option 3: 1.2

• Correct as it accurately reflects the calculated test statistic using the given data.
• The calculation aligns with the standard formula for testing a proportion.

Option 4: 1.56

• Incorrect because it is higher than the correct value of the test statistic.
• This value might result from an error in calculating the sample proportion or standard error.

Hence, the value of the test statistic is correctly calculated as 1.2.

The correct answer is 19

Key Points Moments About an Arbitrary Origin:

• Moments are statistical measures that describe the shape of a set of points. They are used to understand the distribution and other characteristics of the data.
• The first moment (μ1’) is the mean of the data.
• The second moment (μ2’) is related to the variance, which measures the spread of the data.
• The third moment (μ3’) is related to the skewness, which measures the asymmetry of the data distribution.
• The fourth moment (μ4’) is related to the kurtosis, which measures the "tailedness" of the data distribution.

Important Points

 Given the moments about an arbitrary origin, we can calculate the central moments using the relationship between raw moments and central moments.

Given:

μ1' = -4

μ2' = 22

μ3' = -117

μ4' = 560

To find μ3:

$μ3 = μ3' - 3μ2'μ1' + 2μ1'^3$

Substitute the values:

$μ3 = -117 - 3(22)(-4) + 2(-4)^3$

$μ3 = -117 + 264 - 128$

$μ3 = 264 - 245 = 19$

$μ3 = 19$

Therefore, the correct answer is 19.

The correct answer is It provides descriptive statistics.

Key Points

• It provides descriptive statistics:
• Descriptive statistics involve summarizing and organizing data so it can be easily understood. This includes measures like mean, median, mode, and standard deviation.
• Tabulation, however, focuses on arranging data into rows and columns, making it easier to read and interpret but does not inherently involve calculating these statistical measures.

Additional Information

• It simplifies complex data:
• Tabulation helps in breaking down complex data sets into manageable parts, making it easier to understand and analyze large volumes of information.
• For a financial enterprise, this is crucial in presenting financial statements, sales data, or any large dataset in a comprehensible format.
• It gives identity to the data:
• Tabulation assigns a structured format to the data, providing a clear identity to each data point through rows and columns.
• This structured representation is essential for tracking financial transactions, customer details, or inventory lists in a financial enterprise.
• It facilitates comparison:
• By organizing data into a tabular form, tabulation allows for easy comparison across different data points.In a financial enterprise, this is vital for comparing sales figures across different periods, evaluating performance metrics, or analyzing financial ratios.

The correct answer is 22.5

Key Points Skewness and Standard Deviation:

• Coefficient of Skewness (CS) is given by the formula: $CS = \frac{3(Mean - Median)}{Standard Deviation}$.
• Here, the coefficient of skewness is -0.4, the mean is 45, and the median is 48.
• **Using the formula: -0.4 = 3(45−48)SD3(45−48)SD

.**

• Solving this equation gives: -0.4 = −9SD−9SD

or SD = −9−0.4−9−0.4

= 22.5.

Important Points

 Let's analyze the incorrect options:

• Option 1 (24.5): If SD were 24.5, the coefficient of skewness would not match the given value (-0.4).
• Option 3 (23): Using SD = 23 in the skewness formula will not satisfy the given conditions.
• Option 4 (17): This value is too low and does not fit the calculation based on the given skewness and mean.

Therefore, the correct answer is indeed option 2, 22.5.

The correct answer is 119.05

Key Points Fisher's Ideal Index Number:

• Fisher's Ideal Index Number is the geometric mean of Laspeyres' and Paasche's index numbers.
• Given Laspeyres' index (L) = 119.4 and Paasche's index (P) = 118.7, Fisher's Ideal Index (F) can be calculated using the formula:
• $F = \sqrt{L \times P}$
• Substituting the given values, $F = \sqrt{119.4 \times 118.7} \approx 119.05$

Important Points

• Option 1: 156
• This option is incorrect because it is significantly higher than the expected value. Fisher's Ideal Index is the geometric mean and should be close to the given indices.
• Option 2: 125.34
• This option is incorrect because it is not the geometric mean of the given indices. Calculating the geometric mean should yield a value around 119.05.
• Option 3: 119.05
• This option is correct as it accurately represents the geometric mean of Laspeyres' and Paasche's index numbers.
• Option 4: 116.8
• This option is incorrect because it is lower than the calculated geometric mean. The correct value should be around 119.05.

The correct answer is Mean < Mode

Key Points

• Mean < Mode:
• In a negatively skewed distribution, the tail is on the left side of the distribution, meaning the bulk of the data points are concentrated on the right.
• The mean is pulled toward the tail, making it less than the mode (which is the highest point of the distribution).
• For financial enterprises, this could indicate that a few extremely low values are dragging the average down, which could be important for risk assessment and financial planning.

Additional Information

• Mean > Median:
• This is incorrect. In a negatively skewed distribution, the mean is less than the median because the mean is influenced by the lower values in the tail.
• Mean > Mode:
• This is also incorrect. Similar to the mean and median, the mean is less than the mode in a negatively skewed distribution.
• Median > Mode:
• While the median is greater than the mean in a negatively skewed distribution, it is not necessarily greater than the mode. The mode is the most frequently occurring value and is generally the highest point in the distribution.

The correct answer is k = 0.2

Key Points Probability Mass Function:

• A function f(x) is a probability mass function (PMF) if the sum of f(x) over all possible values of x equals 1.
• In this case, the given function is a binomial probability mass function: f(x)=n!x!(n−x)!kx(1−k)n−xf(x)=n!x!(n−x)!kx(1−k)n−x

.

• The sum of all binomial probabilities for a given n is always 1, provided k is the probability of success (0 ≤ k ≤ 1).
• Hence, the correct value of k should be a probability value (between 0 and 1).

Important Points

 Let's analyze each option:

Option 1: k = 0.2

• 0.2 is a valid probability value (0 ≤ 0.2 ≤ 1).
• The sum of binomial probabilities with k = 0.2 will be 1.
• Hence, the function will be a valid PMF.

Option 2: k = 10

• 10 is not a valid probability value (10 > 1).
• The sum of binomial probabilities with k = 10 will not be 1.
• Hence, the function will not be a valid PMF.

Option 3: k = 5

• 5 is not a valid probability value (5 > 1).
• The sum of binomial probabilities with k = 5 will not be 1.
• Hence, the function will not be a valid PMF.

Option 4: k = 2

• 2 is not a valid probability value (2 > 1).
• The sum of binomial probabilities with k = 2 will not be 1.
• Hence, the function will not be a valid PMF.

Hence, the correct value of k that makes the function a probability mass function is k = 0.2.

The correct answer is 0.

Key Points

• The third order central moment of normal distribution is 0:
• In statistics, the third order central moment is a measure of the asymmetry (skewness) of a probability distribution.
• For a normal distribution, which is symmetric about its mean, the skewness is zero. Hence, the third order central moment is zero.
• This property is crucial in financial enterprises when modeling asset returns. A normal distribution assumption implies no skewness, leading to simpler risk management and forecasting models.

Additional Information

• The third order central moment is positive:
• This is incorrect. A positive third order central moment indicates a positively skewed distribution, which is not the case for a normal distribution.
• The third order central moment is 1:
• This is incorrect. The value of 1 does not represent the third order central moment of a normal distribution. The skewness measure for a normal distribution is zero.
• The third order central moment is negative:
• This is incorrect. A negative third order central moment indicates a negatively skewed distribution, which is not applicable for a normal distribution.

The correct answer is 21 and 22

Key Points Quartiles:

• Quartiles are values that divide a data set into four equal parts. The first quartile (Q1) is the median of the lower half, and the second quartile (Q2) is the median of the entire data set.
• To find Q1 and Q2, the data set should be arranged in ascending order.
• In the given data set, the observations are: 16, 19, 19, 20, 21, 21, 22, 22, 22, 22, 22, 22, 23, 23, 23, 24, 26.
• The median (Q2) is the middle value, which is 22 in this case.
• The first quartile (Q1) is the median of the lower half, which is 21.

Important Points

Option 1: 20 and 22

• 20 is not the correct first quartile (Q1) for the given data set.
• The second quartile (Q2) is correctly identified as 22, but the first quartile is incorrect.

Option 2: 19 and 22

• 19 is not the correct first quartile (Q1) for the given data set.
• The second quartile (Q2) is correctly identified as 22, but the first quartile is incorrect.

Option 3: 21 and 22

• 21 is the correct first quartile (Q1) for the given data set.
• The second quartile (Q2) is also correctly identified as 22.
• This option correctly identifies both quartiles.

Option 4: 21 and 23

• 21 is correctly identified as the first quartile (Q1).
• However, 23 is not the correct second quartile (Q2); it should be 22.

The correct answer is 2.

Key Points Correlation and Regression Coefficients:

• Correlation coefficient is a measure of the strength and direction of the relationship between two variables.
• The regression coefficient of x on y (b_xy) indicates how much y changes for a unit change in x.
• The regression coefficient of y on x (b_yx) is related to b_xy by the formula: b_yx = r² / b_xy.
• Given the correlation coefficient (r) is 0.6 and b_xy is 0.18, we can calculate b_yx.

Important Points

• To find the regression coefficient of y on x (b_yx):
• Given:
• $r = 0.6$
• $b_{xy} = 0.18$
• Formula:
• $b_{yx} = r^2 \div b_{xy}$
• Calculation:
• $b_{yx} = (0.6)^2 \div 0.18$
• $b_{yx} = 0.36 \div 0.18$
• $b_{yx} = 2$

Therefore, the correct answer is option 4.

The correct answer is Σp1q1Σp0q1.100Σp1q1Σp0q1.100Σp1q1Σp0q1.100

Key Points Paasche's Index Number:

• Paasche's index is a price index that uses the current period quantities as weights.

• It measures the relative change in the price level of a basket of goods and services in the current period compared to the base period.

• **The formula is given by Σp1q1Σp0q1.100Σp1q1Σp0q1.100Σp1q1Σp0q1.100

  , where p1p1p1

  and p0p0p0

  are the prices in the current and base periods, respectively, and q1q1q1

  is the quantity in the current period.**

• This index number reflects the cost of purchasing the current period's basket of goods at current prices relative to base period prices.

Important Points

 Let's analyze each option to understand why option 4 is correct and others are incorrect.

Option 1: Σp1q1Σp0q0.100Σp1q1Σp0q0.100Σp1q1Σp0q0.100

• This formula uses both prices and quantities from the current period in the numerator and both prices and quantities from the base period in the denominator.
• It does not account for the quantities from the current period in the denominator, which is essential for Paasche's index.
• This formula is incorrect for calculating Paasche's index number.

Option 2: Σp1q0Σp0q0×100Σp1q0Σp0q0×100Σp1q0Σp0q0×100

• This formula uses the base period quantities in both the numerator and the denominator.
• It does not reflect the changes in quantities between the base and current periods.
• This formula is incorrect for calculating Paasche's index number as it does not use current period quantities.

Option 3: Σp1q1Σp0q1−−−−√×100Σp1q1Σp0q1−−−−√×100Σp1q1Σp0q1×100

• This formula introduces a square root, which is not a part of Paasche's index calculation.
• The correct formula does not require taking the square root of the ratio.
• This formula is incorrect for calculating Paasche's index number.

Option 4: Σp1q1Σp0q1.100Σp1q1Σp0q1.100Σp1q1Σp0q1.100

• This formula correctly uses the current period quantities in both the numerator and the denominator.
• It reflects the cost of purchasing the current period's basket of goods at current prices relative to base period prices.
• This is the correct formula for calculating Paasche's index number.

The correct answer is r(X,Y) = $\frac{Cov(X,Y)}{\sigma_X \sigma_Y }$

Key Points Correlation Coefficient:

• r(X,Y) = $\frac{Cov(X,Y)}{\sigma_X \sigma_Y }$ is the correct formula for the correlation coefficient.
• It standardizes the covariance by dividing it by the product of the standard deviations of X and Y.
• This formula ensures that the correlation coefficient lies between -1 and 1.
• It measures the strength and direction of the linear relationship between two variables.

Important Points

 Explanation of Incorrect Options:

• Option 1: r(X, Y) = Cov(X, Y) - σ_X σ_Y
• This option incorrectly subtracts the product of the standard deviations from the covariance.
• It does not provide a standardized measure of correlation.
• The result can fall outside the range of -1 to 1.
• Option 2: r(X, Y) = Cov(X, Y) + σ_X σ_Y
• This option incorrectly adds the product of the standard deviations to the covariance.
• It also fails to standardize the covariance.
• This formula can yield values outside the acceptable range for correlation coefficients.
• Option 4: r(X, Y) = $\frac{\sigma_X\sigma_Y}{Cov(X,Y)}$
• This option incorrectly inverts the relationship by dividing the product of the standard deviations by the covariance.
• It does not provide a meaningful measure of correlation.
• The values obtained from this formula are not constrained between -1 and 1.

The correct answer is 6/13.

Key Points

 To find the probability that the disease was diagnosed correctly given that the patient died, we use Bayes' theorem:

Let:

• D: Disease is diagnosed correctly.
• D': Disease is diagnosed incorrectly.
• Die: Patient dies.

We need to find P(D | Die).

$P(D | Die) = \frac{P(Die | D) \cdot P(D)}{P(Die)}$

Where:

• $P(D) = 0.6$
• $P(D') = 0.4$
• $P(Die | D) = 0.4$
• $P(Die | D') = 0.7$

Using the law of total probability:

$P(Die) = P(Die | D) \cdot P(D) + P(Die | D') \cdot P(D')$

$P(Die) = 0.4 \cdot 0.6 + 0.7 \cdot 0.4 = 0.24 + 0.28 = 0.52$

Hence,

$P(D | Die) = \frac{0.4 \cdot 0.6}{0.52} = \frac{0.24}{0.52} = \frac{6}{13}$

The correct answer is 1

Key Points Finding the Mode of the Given PDF:

• Definition of Mode: The mode of a probability density function (pdf) is the value of x at which the pdf attains its maximum value.
• To find the mode of the given pdf, we need to find the value of x that maximizes the function f(x)=3(2x−x2)4f(x)=3(2x−x2)4

for 0≤x≤20≤x≤2

.

• Derivative Approach: We take the derivative of f(x)f(x)

with respect to x and set it to zero to find the critical points.

• Solving f′(x)=0f′(x)=0

, we get the critical point as x = 1.

• The value x = 1 is within the interval [0, 2], and it is the point where the pdf attains its maximum value.

Important Points

• Option 1) 1: This is the correct answer because the mode of the given pdf is indeed 1, as calculated above.
• Option 2) 1414

: This is incorrect because it does not maximize the given pdf.

• Option 3) 3434

: This is incorrect because it does not maximize the given pdf.

• Option 4) 1212

: This is incorrect because it does not maximize the given pdf.

The correct answer is 45.

Key Points Quartile Deviation:

• Quartile Deviation is a measure of the spread of a dataset. It is calculated as half the difference between the first quartile (Q1) and the third quartile (Q3).
• First, arrange the data in ascending order: 80, 100, 100, 110, 130, 190, 200.
• Calculate Q1 (first quartile) and Q3 (third quartile):
• Q1 (25th percentile) = 100, Q3 (75th percentile) = 190.
• Quartile Deviation = (Q3 - Q1) / 2 = (190 - 100) / 2 = 45.

Important Points

 Let's examine the incorrect options:

Option 1: 10

• This is incorrect because it underestimates the spread of the data.
• The actual difference between Q3 and Q1 is much larger.
• Quartile deviation measures the middle 50% of the data.
• A value of 10 would suggest a much narrower spread.

Option 3: 50

• This is close but slightly overestimates the spread.
• The actual quartile deviation is 45, not 50.
• It's important to calculate precisely.
• A difference of 5 can affect data interpretation.

Option 4: 70

• This is incorrect because it greatly overestimates the spread.
• The quartile deviation should reflect the interquartile range accurately.
• A value of 70 suggests a much wider spread than what is present.
• Accuracy in statistical measures is crucial for correct interpretation.

The correct answer is -0.754

Key Points Calculation of Coefficient of Skewness:

• Coefficient of skewness can be calculated using the formula: Skewness=3(Mean−Median)Standard DeviationSkewness=3(Mean−Median)Standard Deviation
• Given data: Median = 45, Mean = 39.27, SD = 22.81
• Substitute the values into the formula: Skewness=3(39.27−45)22.81Skewness=3(39.27−45)22.81
• Calculate the numerator: 3(−5.73)=−17.193(−5.73)=−17.19
• Calculate the skewness: −17.1922.81≈−0.754−17.1922.81≈−0.754

Important Points

 Understanding the options:

• Option 1: 0.675 - Incorrect, as it does not match the calculated value of skewness.
• Option 2: 0.876 - Incorrect, as it does not match the calculated value of skewness.
• Option 3: 0.405 - Incorrect, as it does not match the calculated value of skewness.
• Option 4: -0.754 - Correct, as it matches the calculated value of skewness.

The correct answer is 50.

Key Points Explanation:

• Mean, Median, and Mode Relationship: For a moderately skewed distribution, the relationship is given by: Mode ≈ 3 * Median - 2 * Mean.
• Using the given values: Mode ≈ 3 * 30 - 2 * 20 = 90 - 40 = 50.
• Therefore, the mode is approximately 50.

Important Points

Let’s discuss why other options are incorrect:

• Option 1: 40
• The formula for the mode is Mode ≈ 3 * Median - 2 * Mean.
• Substituting the given values: Mode ≈ 3 * 30 - 2 * 20 = 50, not 40.
• So, 40 is not the correct value for the mode.
• Option 3: 60
• The formula for the mode is Mode ≈ 3 * Median - 2 * Mean.
• Substituting the given values: Mode ≈ 3 * 30 - 2 * 20 = 50, not 60.
• So, 60 is not the correct value for the mode.
• Option 4: 30
• 30 is the value of the median, not the mode.
• The mode should be calculated using the relationship: Mode ≈ 3 * Median - 2 * Mean.
• Therefore, 30 is not the correct value for the mode.

The correct answer is 32

Key Points Explanation:

• Decile Calculation: To find the 8th decile, we need to first sort the data in ascending order.
• Sorted data: 18, 20, 22, 23, 25, 30, 32, 36
• The formula for the nth decile is D_n = (n/10) * (N+1), where N is the number of data points.
• Here, N = 8 and n = 8. So, D_8 = (8/10) * (8+1) = 7.2
• The 8th decile lies between the 7th and 8th data points in the sorted list, which are 32 and 36 respectively. Hence, the 8th decile is approximately 32.

Important Points

 Explanation of options:

Option 1: 32 (Correct)

• 32 is the 7th value in the sorted data.
• Since the 8th decile falls between the 7th and 8th values, it is very close to 32.
• Hence, 32 is the correct approximation for the 8th decile.

Option 2: 25

• 25 is the 5th value in the sorted data.
• The 8th decile should be closer to the higher end of the data range.
• 25 is far from the 7.2th position, making it an incorrect choice.

Option 3: 23

• 23 is the 4th value in the sorted data.
• The 8th decile should be much higher than 23.
• 23 is not close to the 7.2th position, making it an incorrect choice.

Option 4: 30

• 30 is the 6th value in the sorted data.
• The 8th decile should be higher than 30.
• 30 is still not close enough to the 7.2th position, making it an incorrect choice.

The correct answer is 104.68

Key Points Seasonal Index Calculation:

• Seasonal Index measures the relative level of a seasonal period compared to the average period.
• For the second quarter, the average price is 4.25.
• The overall average price is calculated as (3.8 + 4.25 + 3.85 + 4.35) / 4 = 4.0625.
• The seasonal index for the second quarter is calculated as (4.25 / 4.0625) * 100 = 104.68.

Important Points

 Explanation for other options:

Option 1: 132.54

• This value is incorrect as it does not correspond to any reasonable calculation based on the given data.
• It might be a result of a miscalculation or incorrect formula application.

Option 3: 102.56

• This value is also incorrect.
• It is close to the correct answer but slightly off, possibly due to a rounding error or miscalculation.

Option 4: 111.84

• This value is incorrect as well.
• It might be derived from a different set of data or a different calculation method.

The correct answer is positively skewed.

Key Points

• Positively skewed distribution:
• In a positively skewed distribution, the tail on the right side (higher values) is longer or fatter than the left side (lower values).
• This occurs when there are a few exceptionally high values that pull the mean to the right, making the mean greater than the median, which in turn is greater than the mode.
• For a financial enterprise, this could indicate that while most data points (such as revenue or returns) are clustered at lower values, there are some significant outliers that are much higher, skewing the average.

Additional Information

• Platykurtic distribution:
• Platykurtic distributions are characterized by a lower peak and thinner tails compared to a normal distribution. This term describes the kurtosis of the distribution rather than its skewness.
• Negatively skewed distribution:
• In a negatively skewed distribution, the tail on the left side (lower values) is longer or fatter than the right side (higher values). Here, the mean is less than the median, which is less than the mode. This does not match the given condition of mean > median > mode.
• Leptokurtic distribution:
• Leptokurtic distributions have a higher peak and fatter tails compared to a normal distribution. This describes the kurtosis, not the skewness, of the distribution.

The correct answer is - 6

Note: In the official paper the constant k was missed, but we have added to remove the confusion in the statement of question. 

Key Points

• Normalization Condition:
• The probability density function (PDF) of a continuous random variable must satisfy the condition that the total probability is 1.
• This means that the integral of the given function from 0 to 1 must be equal to 1.
• Given PDF Function:
• The PDF is given as f(x) = kx² (1 - x³).
• To determine the value of$( k )$, we set up the equation:
• ∫[0 to 1] kx² (1 - x³) dx = 1
• Solving the Integral:
• Expanding the function inside the integral:
• ∫[0 to 1] k(x² - x⁵) dx = 1
• Computing the individual integrals:
• k [ (x³)/3 - (x⁶)/6 ] | from 0 to 1 = 1
• Substituting the limits:
• k (1/3 - 1/6) = 1
• k (1/6) = 1
• Solving for (k), we get k = 6 .

Additional Information

• Probability Density Function (PDF):
• A function that describes the likelihood of a random variable taking a particular value in a continuous range.
• The total area under the PDF curve must be equal to 1.
• Integral in Probability:
• In probability theory, integration is used to find the total probability over a continuous range.
• This is necessary for normalizing the function to satisfy probability properties.
• Importance of Normalization:
• Ensuring that the PDF integrates to 1 is crucial for correctly defining the distribution.
• Normalization helps in determining unknown constants in probability distributions.

The correct answer is Graphic method.

Key Points

• Graphic method:
• The graphic method involves plotting data on a graph to visually identify patterns and trends over time.
• While this method can be useful for visual analysis, it does not provide a precise measurement of seasonal variation.
• It is more qualitative in nature and does not involve the quantitative techniques used in other methods.

Additional Information

• Ratio-to-trend method:
• This method involves calculating the ratio of actual values to the trend values to measure seasonal variation.
• It helps in identifying how much the actual data deviates from the trend due to seasonal factors.
• Simple averages:
• This method calculates the average of data for each season to measure seasonal variation.
• It is a straightforward approach that helps in understanding the typical seasonal pattern.
• Ratio-to-moving-average method:
• This method involves dividing the actual data by the moving average to eliminate trend effects and isolate seasonal variations.
• It is a common quantitative technique used to measure seasonal fluctuations.

The correct answer is t distribution with 28 degrees of freedom and 1.45

Key Points Explanation:

• Degrees of Freedom: The degrees of freedom (df) for a two-sample t-test is calculated as the sum of the sample sizes minus 2. In this case, df = (16 - 1) + (14 - 1) = 28.
• The t-distribution is appropriate for comparing the means of two small samples.
• The test statistic value of 1.45 is derived from the formula for the t-test for independent samples.
• The null hypothesis states there is no significant difference between the IQs of the two groups, hence the test statistic under the t-distribution.

Important Points

 Explanation of Incorrect Options:

• Option 1: Chi-square distribution with 29 degrees of freedom and 1.89
• The Chi-square distribution is not appropriate for comparing means of two samples.
• Degrees of freedom for Chi-square test are typically associated with categorical data, not continuous data like IQ scores.
• Option 3: Chi-square distribution with 25 degrees of freedom and 2.65
• Again, Chi-square is not suitable for this type of hypothesis testing.
• Degrees of freedom are incorrectly calculated for a Chi-square test in this context.
• Option 4: t distribution with 25 degrees of freedom and 4.26
• The degrees of freedom are incorrectly calculated; it should be 28, not 25.
• The test statistic value of 4.26 is incorrect based on the given data.

The correct answer is μ2 = μ2' - (μ1')2

Key Points Explanation of Correct and Incorrect Options:

• Option 1: μ2 = μ2' + (μ1')2
• This option incorrectly adds the square of the first moment about the origin (μ1') to the second moment about the origin (μ2').
• The correct relationship involves subtracting the square of the first moment about the origin.
• Hence, this option is incorrect.
• Option 2: μ2 = μ2'
• This option states that the second moment about the mean (μ2) is equal to the second moment about the origin (μ2').
• However, this ignores the contribution of the first moment about the origin (μ1').
• Therefore, this option is incorrect.
• Option 3: μ2 = (μ1')2
• This option incorrectly states that the second moment about the mean is equal to the square of the first moment about the origin.
• It does not consider the second moment about the origin (μ2').
• Thus, this option is incorrect.
• Option 4: μ2 = μ2' - (μ1')2
• This option correctly represents the relationship between the second moment about the mean (μ2) and the second moment about the origin (μ2').
• It correctly subtracts the square of the first moment about the origin (μ1') from the second moment about the origin.
• Hence, this option is correct.

The correct answer is seasonal variation.

Key Points

• Seasonal Variation:
• Seasonal variation refers to periodic fluctuations that occur at regular intervals due to seasonal factors, such as festivals, holidays, or weather changes.
• In the context of a departmental store, sales typically increase during festivals due to higher consumer spending driven by holiday shopping, special promotions, and cultural traditions.
• This pattern repeats annually, making it a clear example of seasonal variation in a time series.

Additional Information

• Random Movement:
• Random movements are unpredictable fluctuations that do not follow a pattern or trend. They are often caused by unforeseen events such as natural disasters or political instability.
• Since the increase in sales during a festival is predictable and occurs regularly, it does not fall under random movement.
• Irregular Movement:
• Irregular movements are erratic and unpredictable variations that occur due to unusual or rare events, such as strikes or pandemics.
• The predictable increase in sales during a festival does not qualify as an irregular movement.
• Secular Trend:
• Secular trends represent long-term movements in a time series, showing a general direction over an extended period, such as a decade or more.
• The short-term, repetitive increase in sales due to festivals does not indicate a long-term trend and thus is not a secular trend.

The correct answer is 3.

Key Points

• β2 of normal distribution is 3:
• In statistics, β2 represents the kurtosis of a distribution. Kurtosis measures the "tailedness" of the probability distribution of a real-valued random variable.
• For a normal distribution, the kurtosis (β2) is exactly 3. This indicates that the normal distribution has a moderate tail compared to other distributions.
• Financial enterprises often use kurtosis to assess the risk of extreme returns. A higher kurtosis would imply a higher risk of extreme outcomes, impacting portfolio management and risk assessment strategies.

Additional Information

• β2 is 2:
• This is incorrect. A β2 value of 2 would indicate a distribution with less pronounced tails than a normal distribution, which is not characteristic of a normal distribution.
• β2 is 0:
• This is incorrect. A β2 value of 0 would indicate a very flat distribution with no tails, which is not representative of a normal distribution.
• β2 is 1:
• This is incorrect. A β2 value of 1 would indicate a distribution with very slight tails, again not characteristic of a normal distribution.

The correct answer is firsthand data gathered by the researcher.

Key Points

• The firsthand data gathered by the researcher:
• Primary data is original data collected directly by the researcher for a specific purpose or project.
• In a financial enterprise, this could involve gathering data through surveys, interviews, or experiments to understand customer behavior, market trends, or employee satisfaction.
• This data is valuable because it is specific to the researcher's needs and is usually more accurate and relevant to the current study.

Additional Information

• Data collected from newspapers:
• This is incorrect. Data from newspapers is considered secondary data because it has been collected and published by someone else.
• In financial enterprises, newspapers can provide useful insights and data, but this data is not original to the researcher.
• Information in biographies:
• Incorrect. Biographies contain secondary data as they are written based on the information collected by the biographer.
• For financial enterprises, biographies might provide historical insights but not original data specific to the current research needs.
• Data collected by someone else:
• This is false. Data collected by someone else is secondary data.
• Financial enterprises may use secondary data for market analysis or benchmarking, but it is not primary data.

The correct answer is 100index number of new base year100index number of new base year100index number of new base year

Key Points Explanation of Options:

• Option 1: 100index number of new base year100index number of new base year100index number of new base year
• This is the correct formula for base shifting.
• It involves multiplying old index numbers by the ratio of 100 to the index number of the new base year.
• This ensures the index numbers are adjusted to the new base year.
• It maintains consistency and comparability of index numbers over different periods.
• Option 2: (100 * index number of new base year)
• This option is incorrect.
• Multiplying by 100 and the index number of the new base year would result in a much larger value.
• It doesn't align with the concept of base shifting.
• It would distort the index numbers instead of adjusting them.
• Option 3: (100 * index number of new base year)²
• This option is incorrect.
• Squaring the product of 100 and the index number of the new base year is not part of the base shifting process.
• It would result in an even larger and incorrect value.
• This approach does not serve the purpose of adjusting index numbers to a new base year.
• Option 4: index number of new base year100index number of new base year100index number of new base year100
• This option is incorrect.
• Dividing the index number of the new base year by 100 is not how base shifting is performed.
• It would incorrectly scale down the index numbers.
• This does not help in adjusting index numbers to a new base year effectively.

The correct answer is P(A|B) = P(B|A).