SSC CGL Tier-II ( JSO ) 2021 Official Paper ( Held On: 10 August 2022 ) (Previous Year Paper)

Key Points

Simple Moving Average: 

• In time series or moving average is a method of calculating the average of a certain number of items which move through the time series.
• The movement of the items happens in the series such that the top data of the previous averaged group drops and adds into the next successive average.
• It is calculated by adding all the data points during a specific period and dividing the sum by the number of time periods.
• To find out the grand average, one must calculate the percentage of the derived data.
• Investors use this method to track the directions in the trend series.

Calculation: 

• For a 3-year simple moving average, you take the sum of production for a 3-year period and divide by 3. To find the third 3-year simple moving average, you will look at the production in years 3, 4, and 5.
• Production in year 3: 55
• Production in year 4: 94
• Production in year 5: 25
• Third 3-year simple moving average = (Production in year 3 + Production in year 4 + Production in year 5) / 3
• Third 3-year simple moving average = (55 + 94 + 25) / 3 = 58
• So, the third 3-year simple moving average for this production data is 58.

The correct answer is 3.833. 

 Key PointsFrequency Distribution: 

• A tabulation or chart of frequencies of the given data that shows the actual number of observations falling in each range.
• When the data in the chart is represented in terms of percentage of occurrence it is known as relative frequency distribution.

Example: 

• Height of ten students of a class are given (in cm): 139, 145, 150, 145, 136, 150, 152, 144, 138, 138
• Frequency distribution of the same data would look like:

Additional Information 

Types of frequency distribution: 

• Grouped frequency distribution: Often to avoid any human error of calculating a large set of data it is organized into more manageable groups.
• Simple or ungrouped frequency distribution: The frequency distribution explained above is known as ungrouped or simple frequency distribution as the data is not categorized into groups.
• Cumulative frequency distribution: Cumulative frequencies are calculated for each group of data that are arranged in a frequency table.
• Relative frequency distribution: A relative frequency distribution shows the proportion of the total number of observations associated with each value or class of values and is related to a probability distribution, which is extensively used in statistics.
• Relative cumulative frequency distribution: The distribution shows relative cumulative frequency that can be found by dividing the frequency of each interval by the total number of observations.
  

Calculation:

The data that has been organized into groups is grouped data. To calculate the mode for the grouped data use the below formula: 

$Mode = L + \frac{(f_1 - f_0)}{2f_1 - f_0 - f_2}*h$

where, 

L = lower boundary of the mode class interval 

f_o = frequency of the class interval before the mode class

f₁ = frequency of the mode class

f₂ = frequency of the class interval after the mode class 

h = size of the class interval (width) 

 ⇒The class interval 3.5 - 4.5 has the highest frequency is of 4 

Substituting the values, we get,

\(Mode = L + \frac{(f_1 - f_0)}{2f_1 - f_0 - f_2}h\ Mode = 3.5 + \frac{(4 - 3)}{24 - 3 - 2}*1\ Mode = 3.833\)

Hence, the mode is equal to 3.833

The correct answer is 21. 

Key Points Mode: 

• The mode is the measure of central tendency that identifies the category or score that occurs the most frequently within the distribution of data.
• In other words, mode represents the data that occurs the highest number of times.

Types of Mode: 

Four types of mode are seen. These are - 

• Unimodal: The data has one mode. For example, the Mode of data set {14, 15, 16, 17, 20, 15, 31, 42, 15, 69, 15, 18, 15, 19} is 15 as there is only one value repeating itself. Hence, it is a unimodal data set.
• Bimodal: The dataset contains exactly two modes. This implies that exactly two data values are having the highest frequencies in the given data. For example, the mode of data set {1, 0, 20, 8,13, 3, 13,14,15, 13, 17,17,19, 13, 7, 19} is 13 and 17 because both 13 and 17 are repeating themselves in the given set. Hence, it is a bimodal data set.
• Trimodal: The data set having exactly three modes. For instance, the Mode of data set {2, 2, 2, 3, 5, 5, 1, 0, 9, 4, 4, 5, 2, 6, 5,4, 7, 5, 8} is 2, 4, and 5 because all the three values are repeating in the given set. Hence, it is a trimodal data set.
• Multimodal: A set of data with four or more than four modes is known as a Multimodal Mode. For example, the mode of data set {100, 80, 80, 95, 95, 80, 99, 102, 66, 85, 90, 110, 100, 90, 90,100 ,95 } is 80, 90, 95, and 100 because all the four values are repeated multiple times in the given set. Hence, it is a multimodal data set.

Additional Information

• Mean: The average of all the values in the given data set.
• Median: The middle most value of the given data set.
• Relationship between mean, median and mode:
• The relationship between mean median and mode can be expressed by using Karl Pearson’s Formula as:
• (Mean - Median) = ⅓ (Mean - Mode)
• 3 (Mean - Median) = (Mean - Mode)
• Mode = Mean - 3(Mean - Median)
• Mode = 3 Median - 2 Mean
• Mean, median and mode are the three measures of central tendency in Statistics.

​Calculation: 

The mode can be calculated using the below formula: 

\(Mean - Mode = 3*(Mean - Mode)\ 12 - Mode = 3*(12 - 15)\ Mode = 312 - 315\ Mode = 21\)

Hence, mode is 21.

The option 2) represents the wrong relationship between the mean sum of squares. The correct relationship is:

Mean sum of squares of total = Mean sum of square treatment + Mean sum of square of error + residual mean square.

Thus, the right answer is option 2) 

Note - 

1. The option 1) is correct because the degrees of freedom of variation

between group is k - 1

       2. The option 3) is correct because the it represents the formula        3. The option 4) is correct because the degrees of freedom within group is N - k

X is degrees of freedom for the total source of variation. To find the degrees pf freedom use the formula, 

N  - 1 = ( number of levels in Factor A ) * ( number of levels of Factor B)  - 1

N =  3*3 - 1

N = 8

Thus, the value of 8

The F_A is calculated using the below formula, 

$F_a = \frac{MSS}{MSW}\ F_a = \frac{147}{3}\ F_a = 49$

The F_B is calculated using the below formula, 

$F_b = \frac{MS_b}{MS_w}\ F_b = \frac{3}{3}\ F_b = 1$

The correct option is  option 3) (8, 49,1)

The correct answer is 1.0.

Key Points

• The quartile deviation, also known as semi-interquartile range (IQR), is a measure of dispersion in a data set and is calculated as the difference between the third quartile (Q3) and the first quartile (Q1).
• It  is a robust statistic, meaning it is not affected by outliers in the data set as much as the range or standard deviation.
• It is a useful summary statistic when comparing the spread of data sets with different sample sizes, as it is not influenced by sample size in the same way that the standard deviation is.

Additional Information

• Given the first quartile of the data set is 7.5, we can calculate the IQR as follows:
• IQR = Q3 - Q1 Since the data set has 5 observations, Q3 can be calculated as the median of the upper half of the data set, which is (9 + 10) / 2 = 9.5. Therefore, IQR = 9.5 - 7.5 = 2.0
• Finally, the quartile deviation can be calculated as IQR / 2 = 2.0 / 2 = 1.0
• So, the value of quartile deviation is 1.0.
• The quartile deviation (also known as semi-interquartile range) is a measure of the dispersion of a data set and is defined as the difference between the first quartile (Q1) and the third quartile (Q3) divided by 2.
• In this case, the first quartile (Q1) is 7.5. To find the third quartile (Q3), we first sort the data set in ascending order: 7, 8, 8, 9, 10.
• The third quartile (Q3) is the median of the upper half of the data set, so Q3 = 9. The quartile deviation is then calculated as (9 - 7.5) / 2 = 1.0.

The correct answer is 123.33. 

Key Points

• Laspeyres price index: 
• a price index used in statistics for measuring the price development of a certain basket of goods and services consumed in the base period.
• Laspeyres Price Index = ${\sum(P_1Q_0) \over \sum(P_0Q_0)}\times100$

​According to the given data, let's assume that base period = 2019 and current period = 2020

Quantity at base period = quantity at current period = Q

According to Laspeyres Price Index 

= ${185Q \over 150Q}\times100$

= 123.33

Hence, the correct answer is 123.33

The correct answer is 4.8

Key Points

• The third factorial moment is a measure of the skewness of the probability distribution of a random variable. It measures how much the distribution deviates from a perfectly symmetric distribution.
• The formula for the third factorial moment involves the expected value of the product of three powers of the random variable, and it can be derived using the properties of the expected value operator and combinatorics.
• In general, if the third factorial moment is positive, then the distribution is said to be skewed to the right, which means that it has a longer tail on the right side of the mean.
• If the third factorial moment is negative, then the distribution is skewed to the left, which means that it has a longer tail on the left side of the mean.
• If the third factorial moment is zero, then the distribution is symmetric.

Additional Information

• To find the third factorial moment of X, we can use the following formula:
• E [X(X-1)(X-2)] = ∑(x³-3x²+2x)p(x)
• where E[X] is the expected value of X, and p(x) is the probability of X taking the value x.
• Using the values given in the problem, we can calculate the third factorial moment as:
• E[X(X-1)(X-2)] = (0³-3(0)²+2(0))(0.5) + (2³-3(2)²+2(2))(0.3) + (4³-3(4)²+2(4))(0.2) = 0 + 1.2 + 3.6 = 4.8
• Therefore, the third factorial moment of X is 4.8.

Important Points

• It's important to note that the third factorial moment is just one measure of skewness, and there are other measures, such as the coefficient of skewness, that can be used to quantify skewness as well.
• Finally, it's worth noting that the third factorial moment is only defined for random variables that have a third moment. Some distributions, such as the Cauchy distribution, do not have a defined third moment, and therefore the third factorial moment is not applicable to them.

The correct answer is 2/5

Key Points

• Empirical probability is based on actual observations or experiments.
• It is calculated by dividing the number of favorable outcomes by the total number of outcomes.
• The more observations or experiments are conducted, the more reliable the empirical probability becomes.
• Probability
• It is a measure of the likelihood or chance of an event occurring.
• It values range from 0 (impossible) to 1 (certain).
• It  can be expressed as a fraction, decimal, or percentage.

Additional Information

• The empirical probability of getting a number greater than 4 when a dice is rolled can be calculated by adding up the number of times the dice lands on a 5 or 6, and dividing by the total number of rolls:
• Empirical probability = (number of 5s + number of 6s) / total number of rolls = (5 + 1) / 15 = 6 / 15 = 2 / 5
• Therefore, the correct answer is 2/5.
• The theoretical probability of rolling a number greater than 4 on a fair dice is 2/6 or 1/3, since there are two favorable outcomes (rolling a 5 or 6) out of six possible outcomes (rolling a number from 1 to 6).
• Empirical probability can be used to estimate theoretical probability, but the two may not always be equal due to chance or sampling error.

The correct answer is 50%.

Key Points

• The interquartile range is a useful measure of variability that is less affected by outliers than the range, which is the difference between the maximum and minimum values in a dataset.
• T he IQR is sometimes used to identify outliers, which are values that fall more than 1.5 times the IQR below the first quartile or above the third quartile.
• The IQR can be used to define the box in a box plot, which is a graphical representation of the distribution of a dataset.
• The whiskers of the box plot extend to the minimum and maximum values within 1.5 times the IQR of the first and third quartiles, respectively.
• The IQR can be calculated using the formula IQR = Q3 - Q1, where Q1 and Q3 are the first and third quartiles, respectively.

Additional Information

• The interquartile range
• It is a robust measure of variability that is useful for datasets with skewed distributions or outliers.
• It  is a standard measure of variability that is taught in many introductory statistics courses and is commonly used in research and data analysis.
• It  is a measure of variability that describes the spread of a dataset by dividing it into quarters.
• Specifically, the IQR is the difference between the upper quartile (Q3) and the lower quartile (Q1).
• Q1 represents the 25th percentile, which is the value below which 25% of the data falls, while Q3 represents the 75th percentile, which is the value below which 75% of the data falls.
• Therefore, the IQR includes the middle 50% of the data, and excludes both the lowest 25% and the highest 25% of the values.
• In summary, the correct answer is that the interquartile range excludes 50% of the values, with the remaining 50% falling within the range.

The quartile deviation formula is given by : 

$Q.D = \frac{Q_3 - Q_1}{2}$

where, Q3 is given as 9.5. In order to calculate the 1st quartile use following steps:

• Arrange the data in ascending order that is, 7,7,8,9, 10
• The upper half of the data is 7,7 where the 25% of data falls
• Thus, the Q1 is:

$Q_1 = \frac{7+7}{2}\ Q_1 = 7$

Substituting the resultant value in quartile deviation formula we get, 

$Q.D = \frac{9.5 - 7}{2}\ Q.D = 1.25$

The correct answer is option 2) 1.25

Two random variables X and Y are independent if the covariance between X and Y is zero, which is expressed mathematically as

Cov(X, Y) = E[(X - E[X])(Y - E[Y])] = 0.

This is equivalent to the expectation of their product being equal to the product of their individual expectations, or E (XY) = E (X) E (Y).

The correct answer is 1-$\sqrt{0.5}$

Key Points

• The cumulative distribution function (CDF) is a function that describes the probability distribution of a random variable.
• In this case, the CDF is defined by the function F(x), which has different values for different ranges of x.
• The upper quartile point of a data set is the value that corresponds to the 75th percentile.
• To calculate the upper quartile point using the CDF, we need to find the value of x for which F(x) is equal to 0.75.
• To understand this, we can look at the formula for F(x) for the range $0 ≤ x < 1$.
• We can see that this part of the function is a quadratic function with a maximum value of 1 at x = 0.
• In this case, we can see that F(x) is equal to 0.75 when x is equal to$1 - √(1/2)$.

Additional Information

• The upper quartile corresponds to the value of x at which F(x) is equal to 0.75, or the point where 75% of the data is below this value.
• Since the maximum value of the quadratic function is at x = 0, we can calculate the upper quartile by finding the root of the quadratic equation$1 - (1-x)^2/2 = 0.75$, which simplifies to$(1-x)^2 = 0.5$.
• Solving for x, we get x = 1 - √(1/2), which is the upper quartile point.
• Overall, the CDF can be a useful tool for analyzing and modeling probability distributions, and the upper quartile point is a useful measure of variability that can provide insight into the spread and shape of a data set.

The information provided is not enough to calculate skewness. Skewness requires information on the distribution's mean and standard deviation, which are not provided.Thus, the correct answer is does not exist.

The calculation for Kelly's coefficient of skewness can be performed as follows:

$D_9 = 16 (9th decile)\ D_5 = 10 (5th decile)\ D_1 = 3 (1st decile)\ SK_k = \frac{(D_9 - 2D_5 +D_1)}{(D_9 - D1)}\ SK_k = \frac{(16 - 2 * 10 + 3)}{(16 - 3)}\ SK_k = \frac{(-1)}{13}$

Thus, the correct answer is $\frac{-1}{3}$

The de-seasonalized time-series data is a transformation of original time-series data to remove the impact of seasonality. The resulting time-series will only

have the trend (T), cyclical (C), and irregular (I) components. The equation T.C.I x 100 represents the decomposition of the time-series

data into these three components, where the multiplication by 100 is used to express the values as percentages.

Formula used:

P(A|B) = P(A ∩ B)/P(B)

Calculation:

P(A ∩ B ∩ C) = P(A) × P(B|A) ×  P(C|A ∩ B)

⇒ P(A ∩ B ∩ C) = 0.4 × 0.05 × 0.07

⇒ P(A ∩ B ∩ C) = 0.0014

∴ The correct answer is option(1).

The method of semi-average is used to measure linear secular trend. It involves dividing the time period into two equal parts, calculating the

average value of the variable for each part, and then subtracting one average from the other. The difference between the two averages

represents the linear secular trend, as it measures the rate of change in the variable over time.

Skewness of a data set can be calculated using the formula:

$S_k = \frac{\sum(x_i- \bar{x})^3}{\sigma^3}$

where deviations deviations from the mean are calculated as (observation - mean). This formula takes into account the asymmetry of

the data set by considering the deviations raised to the power of 3, and then divided by the cube of the standard deviation to normalize the result.

Multiplying every value in the data set by a constant factor will result in the same skewness, as the mean, standard deviation, and

deviations from the mean will all be multiplied by the same constant factor.

The inter-occurrence time between two events follows an exponential distribution with a mean rate equal

to the inverse of the mean number of occurrences of that event.

For events A and B, the inter-occurrence time Y1 and Y2 follow exponential distributions with mean rates 1/λ1 and 1/λ2 respectively.

The minimum of these two times, min(Y1, Y2), is the time until the next occurrence of either event A or event B. Since both events

are occurring simultaneously and independently, the total rate of occurrences for both events is the sum of the individual rates: λ1 + λ2.

The mean rate for the exponential distribution of the minimum inter-occurrence time is therefore (λ1 + λ2).

The correct answer is $\rm t = \frac{{{r_p}\sqrt {n - v} }}{{\sqrt {1 - r_p^2} }}$    

Key Points

• Partial correlation is a statistical measure that represents the strength of the relationship between two variables, after controlling for the effect of one or more additional variables.
• The test statistic for significance testing of partial correlation, rp, is used to determine if the partial correlation between two variables is significant and non-zero.
• The formula for the test statistic for partial correlation is given as: $\rm t = \frac{{{r_p}\sqrt {n - v} }}{{\sqrt {1 - r_p^2} }}$, where n is the sample size and v is the number of control variables. The numerator of the formula represents the strength of the partial correlation, and the denominator represents the standard error of the estimate.
• The value of t obtained from this formula can then be compared to the critical value of t from a t-distribution table with (n - v - 2) degrees of freedom, to determine the significance of the partial correlation. If the calculated t-value is greater than the critical value, the partial correlation is considered significant and non-zero.

Additional Information

• Partial correlation is used to study the relationship between two variables after controlling for the effects of other variables.
• The test statistic for partial correlation is a ratio of the strength of the partial correlation to the standard error of the estimate.
• The calculated t-value is compared to a critical value from a t-distribution table to determine the significance of the partial correlation.
• A significant and non-zero partial correlation indicates that there is a relationship between the two variables being studied, after controlling for the effects of the control variables.

The correct answer is 5.

Key Points

• Let's denote the observations as X1, X2, X3, X4, and X5. The mean of the observations can be calculated as:
• mean = $(X1 + X2 + X3 + X4 + X5)/5$
• The mean deviation for each observation is calculated as:
• Mean Deviation (X1) = X1 - mean
• Mean Deviation (X2) = X2 - mean
• Mean Deviation (X3) = X3 - mean
• Mean Deviation (X4) = X4 - mean
• Mean Deviation (X5) = X5 - mean
• The sum of the mean deviations of all observations should be zero as every deviation above the mean should have a corresponding deviation below the mean. This can be expressed as:
• Sum of Mean Deviations = Mean Deviation (X1) + Mean Deviation (X2) + Mean Deviation (X3) + Mean Deviation (X4) + Mean Deviation (X5) = 0
• The coefficient of mean deviation is calculated as the ratio of the mean deviation to the mean and is expressed as a percentage. In this case, it is given as 0.4. This can be expressed as:
• Coefficient of Mean Deviation = Mean Deviation/mean * 100

Additional Information

• Given that the mean deviation and coefficient of mean deviation of 5 observations are 1.2 and 0.4, we can use the following formula to calculate the sum of the observations:
• Sum of observations = (Number of observations) * (Mean)
• Since the coefficient of mean deviation is 0.4, it follows that:
• Mean deviation / Mean = 0.4 Mean deviation = 0.4 * Mean
• Since the mean deviation is 1.2, it follows that:
• 0.4 * Mean = 1.2
• Mean = 1.2 / 0.4 = 3
• Therefore, the sum of the observations is:
• Sum of observations = 5 * 3 = 15
• Since the sum of the first four terms is 10, the fifth term must be equal to:
• Fifth term = Sum of observations - Sum of first four terms Fifth term = 15 - 10 = 5
• In statistics, the mean deviation and coefficient of mean deviation are measures of dispersion in a data set.
• The mean deviation is the average of the absolute deviations from the mean, while the coefficient of mean deviation is the ratio of the mean deviation to the mean.
• The fifth term in the set of 5 observations is equal to 5.
• The mean deviation and coefficient of mean deviation provide valuable information about the dispersion in a data set, and can be used to calculate the mean, as well as individual terms in the data set.

The correct answer is Third moment

Key Points

• The first moment is called the MEAN which describes the center of the distribution.
• The second moment is the VARIANCE which describes the spread of the observations around the center.
• The third moment describes other aspects of a distribution such as how the distribution is skewed from its mean or peaked.
• Skewness is a measure of the asymmetry of a probability distribution about its mean.
• The third moment is often used as a measure of skewness, as it indicates the degree of asymmetry of a distribution.
• A positive third moment indicates that the distribution is skewed to the right, while a negative third moment indicates that the distribution is skewed to the left.

Additional Information

• Skewness is a measure of the asymmetry of a distribution
• Third moment is used as a measure of skewness
• Positive third moment indicates a right-skewed distribution
• Negative third moment indicates a left-skewed distribution
• Other measures of skewness include the mean deviation and the coefficient of mean deviation.
• The third moment should be used with caution as it can be sensitive to outliers, and may not always accurately reflect the shape of a distribution. 
• It is important to consider the overall shape of a distribution and use multiple measures of skewness when analyzing data.

The correct answer is -0.1.

 Key Points

• Spearman's rank correlation coefficient is used to measure the association between two variables when the data are ordinal or ranked.
• The formula for calculating the Spearman's rank correlation coefficient is given by:
• ​$\rho = 1 - {6 \times \sum d_i^2 \over {(n^3 - n)}}$, where n is the sample size and the sum of differences is the sum of squared differences between the ranks of the two variables.
• In this case, given that the difference between the rank of the 4 observations are 2.5, 0.5, -1.5, and -1.5, the sum of squared differences would be.
• Plugging in n = 4, we have: $\rho = 1 - {(6 \times 11) \over (4^3 - 4)} = -0.1$
• The Spearman's rank correlation coefficient measures the association between two variables when the data are ordinal or ranked.
• The formula for calculating Spearman's rank correlation coefficient involves finding the sum of squared differences between the ranks of the two variables and using that value in the formula.

Additional Information

• A positive value of Spearman's rank correlation coefficient indicates a positive association between the variables, a negative value indicates a negative association and a value of zero indicates no association.
• Spearman's rank correlation coefficient is often used when Pearson's correlation coefficient cannot be applied, for example, when the data are not normally distributed or are not continuous.
• Spearman's rank correlation coefficient ranges from -1 to 1, with -1 indicating a perfect negative association, 1 indicating a perfect positive association, and 0 indicating no association.
• It is important to choose the appropriate correlation coefficient based on the nature of the data and the research question being studied.
• The interpretation of the correlation coefficient and its statistical significance depends on the sample size and the strength of the association between the variables.

The correct answer is $\frac{8}{16575}$

Key Points

• The probability of drawing a king, a queen and a jack in order is the product of the probabilities of drawing each card.
• There are 4 kings, 4 queens, and 4 jacks in a standard deck of 52 cards, so the probability of drawing any one of them is 4/52.
• The first card can be any of the 3 cards in order. The probability of drawing a king first is $\frac{4}{52}$.
• The second card can be either a queen or a jack, but it cannot be the same as the first card.
• There are 4 queens or jacks left, and 51 cards in total, so the probability of drawing the second card is $\frac{4}{51}$.
• The same logic applies for the third card, and the probability of drawing a queen or a jack is $\frac{4}{50}$.
• Therefore, the probability of drawing a king, a queen, and a jack in order is $\frac{4}{52} * \frac{4}{51} *\frac{4}{50} = \frac{8}{16575}$}

Additional Information

• The number of each card in the deck affects the probability of drawing them.
• The order of drawing the cards affects the final probability.
• This problem assumes that the cards are drawn without replacement, which means that once a card is drawn, it is not returned to the deck.
• The problem calculates the probability of drawing the cards in a specific order, but there are other ways to calculate the probability of drawing a specific combination of cards, regardless of order.

The correct answer is zero.

Key Points

• Population skewness measures the asymmetry of a population distribution.
• A value of zero indicates a symmetric distribution.
• The formula for population skewness is given as: Skewness = 3 * (Mean - Median) / Standard Deviation
• Skewness is one of the measures of shape and central tendency.
• Other measures such as kurtosis and standard deviation are also important in understanding the overall distribution of data.

Additional Information

• The population skewness measures the asymmetry of a population distribution, with a value of zero indicating a symmetric distribution. The formula for population skewness is given as:
• $Skewness = 3 * (Mean - Median) / Standard Deviation$
• In a grouped data set, the standard deviation cannot be calculated, hence the skewness is only estimated.
• However, in this case, the observation of equal frequencies in two of the classes suggests that the distribution is symmetric and the skewness is close to zero.
• It's important to note that skewness is one of the measures of shape and central tendency and can help us understand the distribution of data
• however it's not the only one and other measures such as kurtosis and standard deviation are also important in understanding the overall distribution of data.

The correct answer is$\rm F = \frac{Mean\space sum\space of\space square\space within\space group}{Mean\space sum\space of\space square\space between\space group}$

Key Points

• ANOVA is used to test the equality of means among more than two groups
• The F-statistic is the ratio of the mean sum of squares between groups to the mean sum of squares within groups
• A larger F-statistic indicates a significant difference in the means among the groups and supports rejection of the null hypothesis
• The critical value of the F-statistic is determined based on the degrees of freedom and significance level.
• In ANOVA, the test statistic F is used to test the null hypothesis that all population means are equal. 
• F is calculated as the ratio of the mean sum of squares between groups (MSB) to the mean sum of squares within groups (MSW).

​ Additional Information

• The formula for F is given by:
• F = MSB / MSW
• MSB is the average variance among the group means, and MSW is the average variance within the groups.
• If the null hypothesis is true, then the group means should be similar and the variance within the groups should be larger than the variance between the groups. As a result, the F-statistic should be close to 1.
• However, if the null hypothesis is false, and the group means are significantly different, then MSB will be larger than MSW, resulting in a larger F-statistic.
• In such cases, the F-statistic is used to determine the significance of the results, and the null hypothesis is rejected if the F-statistic is larger than the critical value.

The correct answer is -1.8

Key Points

• r^th factorial moment about origin $(\mu'_r)$
• In probability theory of Statistics, the rth factorial moment is a mathematical quantity defined as the expectation of the falling factorial (here, rth of a random variable.
• Let, X assumes the values x₁, x₂, x₃, ... x_n

• third factorial moment of the given series 
  

= -0.1 - 30.5 + 2(-0.1) 

= -0.1 - 1.5 - 0.2

= -1.8

Hence, the correct answer is (-1.8)

The correct answer is cluster sampling.

Key Points

• Non-probability sampling methods are often used when the population is difficult to define or when the researcher has a specific purpose for selecting a certain group of individuals.
• Probability sampling methods are considered to be more representative of the population and are therefore often considered to be more scientifically rigorous than non-probability sampling methods.
• Cluster sampling
• It is a probability sampling method, not a non-probability sampling method.
• In cluster sampling, the population is divided into smaller groups called clusters, and a random sample of these clusters is selected.
• The individuals within the selected clusters are then surveyed to gather data.
• Cluster sampling is a probability sampling method
• Snowball sampling, quota sampling, and purposive sampling are non-probability sampling methods
• Non-probability sampling methods do not involve random selection of individuals, while probability sampling methods do.

 Additional Information

• Snowball sampling is a non-probability sampling method in which individuals are selected based on referrals from other individuals.
• Quota sampling is a type of non-probability sampling method in which individuals are selected based on specific characteristics such as age, gender, or race.
• The sample is created by dividing the population into subgroups based on these characteristics, and then selecting a set number of individuals from each subgroup.
• Purposive sampling is a type of non-probability sampling method in which the researcher selects individuals based on specific criteria that are deemed important for the study.
• This method is often used when the population is small or when the researcher has a specific purpose for selecting a certain group of individuals.
• The choice between probability and non-probability sampling methods will depend on the research question, the resources available, and the specific needs of the study.
• It is important for researchers to be transparent about their sampling methods and to discuss the limitations of their sample when presenting their results.

The correct answer is 10.

 Key Points

• The median of a set of observations is the value that separates the upper half from the lower half of the data. 
• To find the median of a set of observations, the data must be arranged in numerical order.
• In this case, the observations are: 9, 9, 10, 10, 10, 10, 11, 11, 12, 12.
• Since there are an even number of observations, the median is the average of the two middle values, which in this case are 10 and 10.
• The median is therefore 10.
• The median is the value that separates the upper half from the lower half of the data
• To find the median, the data must be arranged in numerical order
• If the number of observations is odd, the median is the middle value
• If the number of observations is even, the median is the average of the two middle values.

 Additional Information

• The median is one of several measures of central tendency, which describe where the data is centered.
• Other measures of central tendency include the mean (average) and the mode (most common value).
• The median is considered to be a more robust measure of central tendency than the mean, as it is not influenced by outliers or extreme values.
• The choice of measure of central tendency depends on the specific needs of the study and the type of data being analyzed
• It is important to understand the strengths and limitations of each measure of central tendency
• to choose the most appropriate measure for the data being analyzed.

The correct answer is: $\sum_{i=1}^k({\overline y _{i.}} - \overline y )^2$ represents the sum of squares due to treatments.

Key Points

• In a completely randomized design, y_ij represents the response from the j^th unit receiving the ith treatment.
• The symbol "Σ" represents the sum of all observations.
• The expression$\sum j^{{n_i}}{y{ij}} = {y_i}$ represents the sum of the responses for a specific treatment (i) and is equal to the mean response for that treatment $(\overline y _{i.}).$
• The expression $\sum _{i = 1}^k{({\overline y _{i.}} - \overline y )^2}$represents the sum of squares due to treatments, which is a measure of the variation in the response variable that can be attributed to the different treatments. This expression calculates the difference between the mean response for each treatment (\overline y _{i.}) and the overall mean response (\overline y), squares these differences, and then sums these squared differences across all treatments.

 Additional Information

• The sum of squares due to treatments is used in the analysis of variance (ANOVA) to determine if there are significant differences in the response variable between the different treatments.
• The sum of squares due to treatments is just one component of the total sum of squares, which represents the total variation in the response variable.
• Understanding the concept of sum of squares and its use in ANOVA is important for conducting and interpreting experiments involving treatment comparisons.
• It is important to have a clear understanding of the experimental design and the data collection process in order to accurately interpret the results of an ANOVA.

The correct answer is Geological classification.

Key Points

• The basis of classification of data can be broadly categorized into four categories, namely, geographical, temporal, qualitative, and quantitative classification.
• Geological classification is not one of the commonly used basis of data classification.
• Geographical classification is based on location or region.
• Temporal classification is based on time or date.
• Qualitative classification is based on non-numeric attributes such as color, shape, and size.
• Quantitative classification is based on numeric attributes such as height, weight, and temperature.

 Additional Information

• The basis of classification of data is an important aspect of data analysis and helps in organizing and summarizing data.
• The choice of the basis of classification depends on the type of data and the objective of the analysis.
• For example, if the objective is to analyze the sales of a company based on different regions, then geographical classification would be an appropriate basis.
• The basis of classification should be chosen based on the type of data and the objective of the analysis.
• A clear and appropriate basis of classification can make the data analysis process easier and more effective.

The correct answer is 1/125

Key PointsGiven:

μ1' = 1; μ2' = 6; μ3' = 15

Calculation:

B1 = $\frac{(μ_3)^2}{(μ_2)^3}$ ____ (1)

Now, μ2 = μ2' - (μ1')2 

⇒ 6 - (1)2  

⇒ 5 

μ3 = μ3' - 3 (μ2') (μ1') + 2 (μ1')3

⇒ 15 - 3 (6) (1) + 2 (1)3

⇒ 15 - 18 + 2

⇒ -1

By putting the above two values in (1)

⇒ B1 = $\frac{(-1)^2}{(5)^3}$

⇒ B1 = 1/125

Additional Information

• The Karl Pearson's coefficient of skewness is a measure of the degree of skewness or asymmetry of a probability distribution.
• It is calculated as the third standardized moment about the mean, which is the ratio of the third central moment and the cube of the standard deviation.
• The coefficient of skewness has the following properties:
• If β1 = 0, then the distribution is symmetric and has no skewness.
• If β1 > 0, then the distribution is positively skewed and has a long right tail.
• If β1 < 0, then the distribution is negatively skewed and has a long left tail.
• The coefficient of skewness is an important descriptive statistic that provides information about the shape of a distribution.
• The coefficient of skewness should be used in conjunction with other measures such as the mean, median, and standard deviation to describe the distribution and understand its properties

The correct answer is 1/15

Key Points

• The sum of all the probabilities is always equal to 1.
• For a discrete random variable X with a probability mass function, the probabilities must sum to 1, which is a requirement for a valid probability distribution. 
• Hence, the value of k can be determined by setting the sum of all the probabilities equal to 1 and solving for k.
• In this case, the sum of all the probabilities is given by
• Hence,$2k + 3k + 4k + 3k + 2k + k = 1$
• ⇒ 15k = 1
• ⇒$k = 1/15$
• A probability mass function is a function that assigns a probability to each value of a discrete random variable.
• The probabilities assigned to the values of a discrete random variable must sum to 1.
• The value of k can be determined by solving for the sum of all the probabilities equal to 1.

 Additional Information

• Once the value of k is determined, the probabilities can be found by multiplying each probability by k. In this case, the probabilities are
• P(X = -2) = 2k = 2/15 
  

P(X = -1) = 3k = 3/15 

P(X = 0) = 4k = 4/15 

P(X = 1) = 3k = 3/15 

P(X = 2) = 2k = 2/15 

P(X = 3) = k = 1/15

• A probability mass function must be a valid probability distribution, meaning that all probabilities must be non-negative and the sum of all probabilities must be equal to 1.
• The value of k can be determined by solving for the sum of all the probabilities equal to 1.
• The probabilities assigned to the values of a discrete random variable provide information about the likelihood of each value of X.

The correct answer is z = 3.6

Key Points

• The test statistic is a measure of the difference between the sample mean and the population mean in terms of standard deviations.
• In this case, the test statistic can be calculated as follows:
• $z = (sample mean - population mean) / (standard deviation / \sqrt(sample size))$
• $z = (90 - 82) / (20 / \sqrt(81))$
• z = 3.6

 Additional Information

• The test statistic is a measure of the difference between the sample mean and the population mean in terms of standard deviations.
• The test statistic is calculated by dividing the difference between the sample mean and the population mean by the standard deviation divided by the square root of the sample size.
• The test statistic can be used to determine the significance of the difference between the sample mean and the population mean.
• If the test statistic is large enough, it can be concluded that the sample mean is significantly different from the population mean.

​ Important Points

• The test statistic
• It is only meaningful if the sample is a random sample.
• It assumes that the population is normally distributed.
• It is only one piece of information used to make inferences about the population mean.
• Other factors, such as the sample size and the variability of the population, must also be taken into account.

The correct answer is -0.8.

Key Points

• The regression coefficient is a measure of the strength of the linear relationship between two variables.
• It is calculated as the covariance between the two variables divided by the variance of the independent variable. In this case, the regression coefficient can be calculated as follows:
• Cov(x, y) / Var(x) = -0.8
• The regression coefficient
• It is a measure of the strength of the linear relationship between two variables.
• It is calculated as the covariance between the two variables divided by the variance of the independent variable.
• It can be used to make predictions about the dependent variable based on the value of the independent variable. A negative regression coefficient indicates a negative linear relationship between the two variables, while a positive regression coefficient indicates a positive linear relationship.
• It is only measures linear relationships between the two variables.
• It assumes that the relationship between the two variables is linear.
• It does not indicate causality between the two variables.

Additional Information

The regression coefficient of y on x (b_yx), also known as the slope of the least-squares regression line, can be computed as follows:

First, we need to compute the covariance of x and y (Cov(x,y)), and the variance of x (Var(x)).

The formula for the slope of the least-squares regression line (b_yx) is:

b_yx = Cov(x, y) / Var(x)

Let's calculate:

x: 1, 2, 3, 4

y: 4, 5, 3, 2

First, compute the means:

mean_x = (1 + 2 + 3 + 4) / 4 = 2.5

mean_y = (4 + 5 + 3 + 2) / 4 = 3.5

Then, compute the covariance of x and y:

Cov(x, y) = ${Σ[(x_i - mean_x) × (y_i - mean_y)]} \over (n - 1)$

= -1.33 (approximately)

Next, compute the variance of x:

Var(x) = $Σ[(xi - mean_x) ^ 2] \over (n - 1)$

= 5 / 3

= 1.67 (approximately)

Finally, compute the regression coefficient b_yx:

b_yx = Cov(x, y) / Var(x)

= -1.33 / 1.67

= -0.8

So, the regression coefficient b_yx (y on x) is -0.8 as you stated. This indicates that for each unit increase in x, y is expected to decrease by 0.8 units, on average.

The correct answer is Qualitative classification.

Key Points

• Qualitative classification is a method of grouping data based on attributes or qualities that are inherent in the data.
• The data is classified into categories based on its characteristics, such as color, shape, size, or texture.
• Qualitative classification groups data based on attributes or qualities inherent in the data.
• The data is classified into categories based on its characteristics.

 Additional Information

• Qualitative classification is often used in fields such as biology, sociology, and psychology to categorize individuals or objects based on specific traits.
• Qualitative classification should be used when the data being classified cannot be easily quantified.
• The categories used in qualitative classification should be mutually exclusive and exhaustive, meaning that each item should only belong to one category and all items should belong to at least one category.
• The number of categories used in qualitative classification should be kept to a minimum to ensure the accuracy and usefulness of the classification.

The correct answer is 5.2.

 Key Points

• Deciles divide a data set into ten equal parts.
• The nth decile separates the lowest (10n - 90)% of the data from the highest (90 - 10n)% of the data.
• To calculate the decile, the data must be sorted and the formula $D_n = X[{{n} \over 10} \times (N+1)]$must be used.
• If the result is not a whole number, interpolation must be used to estimate the decile value.
• Deciles are often used in statistics to summarize a data set and to identify the values that separate the data into different percentiles.
• Deciles are just one type of quantile. Other quantiles, such as quartiles and percentiles, can also be calculated.
• The value of the decile can be affected by outliers and extreme values in the data set. It is important to identify and handle outliers before calculating deciles to ensure accurate results.

Additional Information

• The deciles are a set of quantiles that divide a data set into ten equal parts.
• The 6th decile (D6) is the value that separates the lowest 60% of the data from the highest 40% of the data.
• In other words, 60% of the data values are below D6 and 40% of the data values are above D6.
• To calculate the 6th decile for the given data set, we first need to sort the data in ascending order: 2, 3, 4, 5, 6, 8.
• The formula to calculate the decile is as follows:
• $D_n = X[{{n} \over 10} \times (N+1)]$, where X is the sorted data set, n is the decile (in this case, 6), and N is the number of data points (in this case, 6).
• $D_6 = X[{6 \over 10} \times {(6+1)}] = X[4.2]$
• Since 4.2 is not a whole number, we use interpolation to estimate the 6th decile:
• $D_6 = {0.8 \times X[4] + 0.2 \times X[5])} = {(0.8 \times 5 + 0.2 \times 6) } = 5.2$

The correct answer is -0.125.

 Key Points

• Spearman's correlation coefficient is a measure of the monotonic association between two variables.
• It measures how well the rank order of one variable corresponds to the rank order of the other variable.
• A positive correlation means that as the rank of one variable increases, the rank of the other variable also increases.
• A negative correlation means that as the rank of one variable increases, the rank of the other variable decreases.
• To calculate Spearman's correlation coefficient, we first need to calculate the ranks of the data in X and Y.
• The ranks are assigned based on the order of the data, with the smallest value receiving the rank of 1 and the largest value receiving the rank of the number of data points.
• If there are ties, the average rank is assigned.
• The ranks for X are: 1, 2, 2
• The ranks for Y are: 3, 3, 1
• Next, we calculate the difference between the ranks for each data point, square the differences, and sum them.

The formula is as follows:

$ρ = {1 - {{{6 \times \sum d^2}} \over {N \times (N^2 - 1)}}}$

where N is the number of data points and d is the difference between the ranks of X and Y for each data point.

Putting the value of ds and n, we have ρ = -0.125

• The result of -0.125 means that there is a weak negative correlation between the two variables.

Additional Information

• Spearman's correlation coefficient measures the monotonic association between two variables.
• It measures the correspondence between the rank orders of the variables.
• A positive correlation means that as the rank of one variable increases, the rank of the other variable also increases.
• A negative correlation means that as the rank of one variable increases, the rank of the other variable decreases.

 Important Points

• Spearman's correlation coefficient is a non-parametric measure, meaning that it does not make assumptions about the underlying distribution of the data.
• It can be used with both continuous and ordinal data.
• Spearman's correlation coefficient is only appropriate to use when there is a monotonic relationship between the variables.
• If the relationship is not monotonic, other measures, such as Pearson's correlation coefficient, may be more appropriate.
• The strength of the correlation is determined by the magnitude of the coefficient.
• A value close to +1 or -1 indicates a strong correlation, while a value close to 0 indicates a weak correlation.

The correct answer is 8.0.

 Key Points

• The median is a measure of central tendency that gives us the middle value in a dataset.
• The median is useful when the dataset has outliers or extreme values that can distort the mean.
• To find the median, we need to arrange the data in order and choose the middle value or average of the two middle values.
• It's important to note that the median is not always the same as the mean, which is another measure of central tendency that gives us the average value in a dataset.
• Unlike the median, the mean is sensitive to outliers and extreme values, which can have a significant impact on the result.

Additional Information

• The median is a measure of central tendency that gives us the middle value in a dataset when it is arranged in order.
• To find the median of a dataset, we first need to arrange the data in ascending or descending order, and then choose the middle value.
• If the dataset has an odd number of observations, the median is simply the middle value.
• If the dataset has an even number of observations, the median is the average of the two middle values.
• In this case, the dataset has 10 observations, which is an even number.
• After arranging the data in ascending order, the dataset becomes: 7, 7, 8, 8, 8, 8, 9, 9, 10, 10
• The two middle values are 8 and 8, and the average of these two values is 8.0, which is the median of the dataset.

Important Points 

• The median is a useful measure of central tendency for datasets with outliers or extreme values.
• It's essential to understand the difference between the median and the mean and how to calculate them.
• In practice, different measures of central tendency may give different results, and it's important to choose the appropriate one based on the characteristics of the dataset.

The correct answer is for k treatments and N observations, the degree of freedom of the ratio of the mean sum of the square of treatments and the residual mean sum is N - 1 and k - 1.

Key Points

• The number of degrees of freedom is k-1, N-K, where m is the number of samples and N is the total size of all the samples together.
• The sampling distribution of the ratio of the variances of two samples taken from a normal population is known as the F-distribution. It is directly applied in determining whether two samples originate from populations with the same variance.
• The more significant usage is in variance analysis, even if you may infrequently see it employed to assess equality of variances (ANOVA). ANOVA is used to determine if three or more samples are from populations with the same mean, at least in its most basic form as described in this chapter.
• ANOVA determines whether the samples are from populations with equal means or not by determining whether the variance of the observations is more closely related to the variance of each observation from the mean of its sample or to the variance of the means of the samples from the overall mean.

Additional Information

• The divergence of data points from the mean value is measured by the sum of squares.
• While a smaller result shows little deviation from the mean, a bigger sum of squares suggests higher variability.
• Subtract the data points from the mean, square the differences, and then add the resulting amount to determine the sum of squares.
• Total, residual, and regressive sums of squares are the three types.
• The sum of squares can be used by investors to assist in improving their investing decisions.

is the right solution $\mathbf{SE=\frac{\sqrt{46}}{3}}$

key points

• The formula for the standard error is:$\mathbf{SE=\frac{\sigma}{\sqrt{n}}}$, where is $\sigma$ is the standard deviation and n is the total number of observations.
• Standard deviation is calculated using the following table:

mean = $\overline{X}$=$\frac{15+5+12+10+20+4}{6}$=11

standard deviation =$\sqrt{\frac{\sum(X_{i}-\overline{X})^2}{n}}$=$\sqrt{\frac{92}{3 }}$

standard error : $\frac{\sqrt{\frac{92}{3}}}{\sqrt6}=\frac{\sqrt{46}}{3}$

Thus the correct answer is- The standard error of the given data 15, 5, 12, 10, 20 and 4 is $\mathit{\mathbf{\frac{\sqrt{46}}{3}}}$.

Additional Information

• Standard error (SE) is the estimated standard deviation of a statistical sample population.
• The difference between the computed mean of a population and one that is widely accepted to be true is described by the standard error.
• The standard error decreases when more data points are used in mean computations .

The correct option is Shifts in the level of a time series that cannot be explained are referred to as seasonal cycles. 

Key Points

• When a time series is impacted by seasonal elements like the season or day of the week, a seasonal pattern develops. Seasonality always has a set and recognizable periodicity. The seasonality in the monthly sales of anti-diabetic medications seen above is caused in part by a change in the price of the medications at the end of the year.
• The act of estimating and then eliminating systematic and calendar-related influences from a time series is known as seasonal adjustment. Observed data must be seasonally adjusted because seasonal influences might mask both the series' true underlying movement and some non-seasonal traits that analysts may find interesting.

Additional Information

• Although they are commonly mistaken, seasonal and cyclical behaviour is extremely different. If there is no definite frequency for the fluctuations, they are cyclical; if there is a fixed frequency that is seasonal and linked to a certain aspect of the calendar.
• The average length of a cycle is often longer than the average length of a seasonal pattern, and the magnitudes of cycles are typically more variable than those of seasonal patterns.
• Many time series include trends, cycles, and seasonality. We must first identify the time series patterns present in the data before selecting a forecasting approach that can accurately capture the patterns.

Hence in the level of a time series that cannot be explained are referred to as seasonal cycles are not true in the case of time series.

The correct answer is Chebyshev distribution with degree of freedom 1.

Key Points

• If Z follows a standard normal distribution, then Z² follows a chi-square distribution with 1 degree of freedom.
• The chi-square distribution is a family of distributions that is used to model the sum of the squares of independent standard normal random variables.
• The distribution is characterized by the degree of freedom, which is the number of standard normal variables that are squared and summed.
• F or a chi-square distribution with 1 degree of freedom, the distribution is known as the Chebyshev distribution.
• The Chebyshev distribution has a mean of 1 and a variance of 1. It is a right-skewed distribution with a peak near 1 and an exponential tail.

 Additional Information

• The Chebyshev distribution has important applications in the fields of engineering, mathematics, and statistics.
• For example, it is used to model the distribution of the maximum absolute value of independent standard normal random variables.
• It is not the same as the standard normal distribution.
• The standard normal distribution is symmetrical and bell-shaped, while the Chebyshev distribution is right-skewed and has a peak near 1.
• The Chebyshev distribution is a special case of the chi-square distribution.
• The chi-square distribution has more degrees of freedom and is more flexible, but it also requires more information to specify.
• The Chebyshev distribution is a useful distribution for modeling the behavior of extreme values in a data set.

The correct answer is 190.

Key Points

• The Laspeyres price index is a measure of the average change in prices between two periods, where the prices in the base year are used to weigh the prices in the current year.
• To calculate the Laspeyres price index, we first need to calculate the weighted average of the prices in the current year, using the quantities in the base year as weights.
• Then, we divide this weighted average by the weighted average of the prices in the base year and multiply the result by 100 to obtain the index.

Additional Information

• The calculation is as follows:
• A weighted average of prices in the current year:
• $(50 \times 20 + 60 \times 40 + 70 \times 60) \over (20 \times 20+ 30 \times 40 + 40 \times 60)$
  

$= {(1000 + 2400 + 4200) \over (400 + 1200 + 2400)} \times 100= {7600 \over 4000} \times 100= 1.9 \times 100 = 190$

• So, the Laspeyres price index for the current year is 190.

The correct answer is collecting primary data is not quite expensive both, in the terms of time and money.

Key Points

• The process of acquiring data through surveys, interviews, or experiments is known as primary data collection. Household surveys are a common type of primary data.
• It may be more expensive to obtain primary data than secondary data. This is so that volunteers may be found, or a vendor or research firm can be hired. Sometimes secondary data is even free.
• In so far, as it was gathered with the intention of solving a specific study issue, primary data is frequently trustworthy, real, and unbiased. It is noteworthy that primary data collection is uncommon due to the high implementation costs.

Additional Information

• The Obama Administration determined a few years ago that any research conducted with federal public dollars should be made freely available to the general public. Additionally, data management plans should be in place to keep the data safe and secure for virtually all time. These data sets are available for download and manipulation for study and are published as Supplementary Materials in the scientific literature.
• Your choice of data for your analysis or research is influenced by things like:
• Time constraints
• Expertise in the field of interest
• Existing (secondary) data availability
• Ability and experience of your business to gather primary data. ​

Hence, collecting primary data is not quite expensive both, in the terms of time and money is not a correct statement.

The correct answer is -14

Key Points

• The third moment about the mean is $\mathbf{E((X-\mu)^3)}$.
• Here the mean, symbolized by $\mu$, is the first point near the origin of a random variable.
• Expanding$E((X-\mu)^3)$ we get: $E((X-\mu)^3) = E(X^3) - 3E(X)E(X^2) + 2E(X)^3$
• Here $E(X^3) , E(X^2) ,E(X)(=\mu)$ are the third, second, and first moments about the origin respectively.

Substituting the values respectively in the third moment about the mean expression we get:

\(E((X-\mu)^3) = 18 - (328) + (2*2^3 )\E((X-\mu)^3)=-14 \)

Additional Information

• The method of moments is a technique used in statistics to estimate population parameters. To derive higher moments like skewness and kurtosis, the same method is applied. First, functions of the pertinent parameters are used to express the population moments. The sample moments are then put equal to those expressions.
• These dimensionless, normalized central moments depict the distribution without dependence on any linear scale change.

Hence, if the first, second, and third moments about the origin are 2, 8, and 18 respectively, then the third moment about the mean is -14

The correct answer is 6.20

Key Points

The formula of the mode used in a frequency distribution is:

$\mathbf{M_{0}=\text{l }+(\frac{f_{1}-f_{0}}{2f_{1}-f_{0}-f_{2}})*h }$

where ,$M_{0}$=Mode, l=lower limit of the modal class, h=size of the class interval,$f_{1}$=frequency of modal class.$f_{1}$ =frequency of the class preceding, the modal class and $f_{2}$= The frequency of the class succeeding the modal class.

According to the problem, class 5-7 is the modal class, since it has the highest frequency.

Substituting the respective values in the mode equation gives us:

$M_{0}=5+(\frac{4-1}{(2*4)-1-2})*2\ \implies M_{0}=5+1.20\ \implies M_{0}=6.20$

Additional Information

• The value of h here is symmetrical which is $7-5=2$.
• The real upper limit is produced by adding 0.5 to the highest number, if it is expressed as a whole number, or by adding 0.05 to the highest number if it is represented as a decimal. The upper limit is the highest value of the class interval.
• Similarly to the upper limit, the lower limit is the smallest value in the class interval. To determine the actual lower limit, take the smallest number and either deduct 0.5 from it if it's a whole number or 0.05 from it if it's a decimal.

Hence for the given frequency distribution the value of mode is 6.20

The correct answer is in a polygon, an additional line is not drawn at each to bring the graph back to a zero frequency.

Key Points

• A frequency polygon is a line graph that plots class frequency against the class midpoint. You can get it by connecting the midpoints of the rectangles' tops in the histogram.
• One must include the first and last class with zero frequencies in order to confirm that our graph is genuinely a polygon (i.e., the graph is closed-shape). Draw attention to the frequency column and the column with the midpoints for each class.

Additional Information

• A histogram is a bar graph with rectangle-shaped bars displaying the data, but a frequency polygon is a line graph with a curved line showing the data.
• A polygon has flat surfaces throughout. A polygon must have a minimum of three sides. Each side of a line segment must intersect another line segment only at its terminal. By counting the sides of a polygon, we may quickly ascertain its geometry.

Hence, for frequency distribution presentation, the wrong option is in a polygon, an additional line is not drawn at each to bring the graph back to a zero frequency.

The correct answer is 1.7414

Key Points

• Kurtosis = Fourth Moment / (Second Moment)²

Kurtosis = 58.6921/(5.8056)² 

Kurtosis = 1.7414

Important Points

• An attribute of a dataset can be described using the statistical measure kurtosis. On a graph, regularly distributed data typically resembles an upside-down bell when plotted. The bell curve is a name for this. The tails on either side of the curve are often formed by the plotted data that are most dispersed from the data mean. Kurtosis reveals how much information is included in the tails.
• Leptokurtic (below normal), platykurtic (above normal), and mesokurtic (normal) are the three categories of kurtosis (more than normal).
• Only when combined with standard deviation is kurtosis beneficial. Even if an investment has a high kurtosis (poor), the overall standard deviation is likely to be modest (good). On the other hand, a low kurtosis (excellent) investment could have a large total standard deviation (bad).

Hence, if the population kurtosis of the observations 16,12,6,2,4,10 is 1.7414, then the population kurtosis of the 8,6,3,1,2,5 is 1.7414

The correct answer is16.

Key Points 

• The formula used to derive this result is Mean - Mode = 3 (Mean -Median).
  

The solution to this problem is:

$10-Mode=3(10-12) \implies (10+6)=Mode \implies {Mode=16}$

• The Mean in a frequency distribution is the average of all the observations. The average of all the observations is done by adding them up and dividing them by the total number of observations in a frequency distribution.
• The Median is the middlemost value of a distribution when all the values of the observation are arranged from ascending to descending order.
• Mode is the value that has the highest frequency.

Additional Information 

• The relation between Mean, Median and Mode determines the nature of the frequency distribution curve.
• If Mean=Median=Mode then the distribution curve is symmetrical.
• If Mean>Median>Mode then the distribution curve is positively skewed.
• The Mode>Median>Mean then the distribution curve is negatively skewed.

Hence, the mean and median of the distribution are 10 and 12 respectively, then the mode equals to the mean and median of the distribution are 10 and 12 respectively, then the mode is 16.

The correct answer is positively skewed.

Key Points

• The first step is to find out the value of the median. The media can be found out from the relation, Mean-Mode=3(Mean-Median). The values of mean and mode are 32 and 21. Therefore the value of the median is,$32-21=3(32-Median) \implies 11=96-3Median \implies Median=28.33$.
• In this case The value of Mean>Mode>Median . This, according to the nature of the curve of frequency distribution, refers that the distribution is positively skewed.

Thus the correct answer is- The given values of mean and mode, the given distribution is positively skewed. 

Additional Information

• If the Mean<Median<Mode, the distribution would have been negatively skewed.
• If Mean=Median=Mode, the distribution would have been symmetrically distributed.
• The values of mean and mode can also be computed using the given relation.

Hence, If mean and mode of the distribution is 32 and 21, then the distribution is positively skewed.

The correct answer is $\mathbf{\frac{6}{36}}$

Key Points

• The probability of a particular event E occurring is given by $P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes or sample space.}}$
• The sample space of this problem, where two dice are thrown together is 36
• The sum of values on the dice resulting in a total of 7 is given by the following table:

• The number of favourable outcomes, in this case, is 6.
• Let F be the event of getting a 7 when two dice are thrown together.
• Therefore P(F)=$\frac{6}{36}$

Important Points

• Sample space is the set of all the possible outcomes in the given experiment. In this case, there are 36 outcomes possible when 2 dice are thrown together.
• Event denotes the probability of getting a single outcome of an experiment. In this problem, the event is to get the value of 7.
• The outcome is the result of the experiment.

Hence, the probability of getting a total of 7 on two dice thrown together is $\frac{6}{36}$

The correct answer is a cyclical component.

Key Points

• A cyclical industry is one that responds to the business cycle, with revenues typically being higher during periods of economic expansion and prosperity and lower during periods of recession and contraction.
• Businesses in cyclical industries can deal with this type of instability by implementing staff cutbacks and layoffs to make up for lost income during lean economic times, and by giving bonuses and hiring en masse during booming economic times.
• In reality, a recession is a particular kind of vicious cycle with cascading decreases in output, employment, income, and sales that feedback into a further decline in output, quickly spreading from industry to industry and region to region.

Additional Information

• A recession, according to the NBER, lasts anywhere from a few months to more than a year and occurs between the peaks and troughs of a business cycle when there is a significant decline in economic activity across the board.
• Cyclical industries produce or market goods that we can live without or put off purchasing during difficult economic times. Luxury products, leisure vacations, and new construction are a few examples.

Hence,  recession in the industry is associated with the cyclical component.

The correct answer is $\mathbf{\sqrt{\frac{66}{5}}}$

Key Points

• The formula used to derive this formula is \(\mathbf{s}=\mathbf{\sqrt{\frac{n_{1}*s^2_{1}+n_{2}*s^2_{2}+n_{1}d^2_{1}+n_{2}d^2_{2}}{n_{1}+n_{2}}}}\), where \(d_{1}=\overline{x}{1}-\overline{x}\) and \(d{2}=\overline{x}{2}-\overline{x}\) also, \(\overline{x}=\frac{n{1}\overline{x}{1}+n{2}\overline{x}{2}}{n{1}+n_{2}}\) where $n_{1}$is the total observation of the first data set, $n_{2}$ is the total observation of the second data set. Also,$s_{1},x_{1} \text{ and } s_{2},x_{2}$ are the standard deviation and mean for data set 1 and data set 2 respectively.
• Here \(n_{1}=6,n_{2}=9 \text{ and } s_{1}=3 , s_{2}=4 \text{ and } \overline{x}{1}=4 , \overline{x}{2}=4\). The following calculations are done:

​\(\overline{x}=\frac{64+94}{6+9} \ \overline{x}=4 \ {d}{1}=4-4=0 \d{2}=4-4=0\)

• Substituting the values in the expression s, we get:

​ \(s=\sqrt{\frac{69+916+60+90}{15}} \s=\sqrt{\frac{198}{15}}\s=\sqrt{\frac{66}{5}}\)

Additional Information​

• Combined Standard Deviation is the result of calculating the standard deviation of two or more series.
• The number of series that can be included in the Combined Standard Deviation calculation is N.
• If we know the mean and quantity of items in each group of the data, we can calculate the combined arithmetic mean.

Hence, the standard deviation of data sets of sizes 6 and 9 have a standard deviation 3 and 4, respectively, and arithmetic means 4 and 4, respectively. The standard deviation of the combined data set of size 15 is $\mathit{\mathbf{ \sqrt{\frac{66}{5}}}}$