BCOC-134 Solved Assignment

Expert Answer

A dataset representing the monthly sales (in thousands of dollars) for a small business over the past 12 months:

Answer:

Given Dataset:

Step 1: Calculate the Range

• Maximum Value: 26
• Minimum Value: 18

Step 2: Calculate the Mean

Step 3: Calculate the Variance

1. Subtract the mean from each number in the dataset and square the result:
   • (20 – 21.75)² = 3.0625
   • (22 – 21.75)² = 0.0625
   • (18 – 21.75)² = 14.0625
   • (25 – 21.75)² = 10.5625
   • (21 – 21.75)² = 0.5625
   • (23 – 21.75)² = 1.5625
   • (19 – 21.75)² = 7.5625
   • (24 – 21.75)² = 5.0625
   • (20 – 21.75)² = 3.0625
   • (22 – 21.75)² = 0.0625
   • (26 – 21.75)² = 18.0625
   • (21 – 21.75)² = 0.5625
2. Calculate the average of these squared differences:
   Variance (σ²) = (3.0625 + 0.0625 + 14.0625 + 10.5625 + 0.5625 + 1.5625 + 7.5625 + 5.0625 + 3.0625 + 0.0625 + 18.0625 + 0.5625) / 12 ≈ 5.91

Step 4: Calculate the Standard Deviation

Interpretation:

• Range: The range of 8 indicates that the sales values vary by up to $8,000 over the year.
• Variance: The variance of 5.91 shows that the sales figures are spread out around the mean, with most monthly sales figures within a certain range from the average.
• Standard Deviation: The standard deviation of 2.43 means that, on average, the monthly sales figures deviate from the mean by about $2,430. A smaller standard deviation would indicate more consistency in sales, while a larger one indicates more variability.

Given the following national income model

Answer:

Given Equations:

1. $Y=C+I$$Y=C+I$Y=C+IY = C + I$Y=C+I$
2. $C=5+\frac{3}{4}Y$$C=5+\frac{3}{4}Y$C=5+(3)/(4)YC = 5 + \frac{3}{4} Y$C=5+\frac{3}{4}Y$
3. $I=10$$I=10$I=10I = 10$I=10$

Step 1: Substitute $C$$C$CC$C$ and $I$$I$II$I$ into the first equation

Step 2: Simplify the equation

Step 3: Solve for $Y$$Y$YY$Y$

Step 4: Find $C$$C$CC$C$

Final Answer:

• $Y=60$$Y=60$Y=60Y = 60$Y=60$ (National Income)
• $C=50$$C=50$C=50C = 50$C=50$ (Consumption)

Discuss the various functions related to business and economics.

Answer:

1. Linear Functions

• Description: A linear function represents a relationship where the change in the dependent variable is proportional to the change in the independent variable.
• Application: In business, linear functions are often used in cost and revenue analysis, where the total cost or revenue is a linear function of the quantity produced or sold. For example, $R(x)=px$$R(x)=px$R(x)=pxR(x) = px$R(x)=px$, where $R(x)$$R(x)$R(x)R(x)$R(x)$ is the revenue and $p$$p$pp$p$ is the price per unit, is a linear revenue function.

2. Quadratic Functions

• Description: A quadratic function represents a parabolic relationship where the dependent variable changes at an increasing or decreasing rate as the independent variable changes.
• Application: Quadratic functions are used in profit maximization and cost minimization problems. For instance, the profit function $P(x)=a{x}^2+bx+c$$P(x)=a{x}^2+bx+c$P(x)=ax^(2)+bx+cP(x) = ax^2 + bx + c$P(x)=a{x}^2+bx+c$ can show how profits vary with changes in production levels.

3. Exponential Functions

• Description: Exponential functions model situations where the rate of change of a quantity is proportional to the current amount of the quantity.
• Application: Exponential functions are frequently used in finance to model compound interest, population growth, and the depreciation of assets. For example, the future value of an investment can be modeled by $F(t)=P(1+r{)}^t$$F(t)=P(1+r{)}^t$F(t)=P(1+r)^(t)F(t) = P(1 + r)^t$F(t)=P(1+r{)}^t$, where $P$$P$PP$P$ is the principal amount, $r$$r$rr$r$ is the interest rate, and $t$$t$tt$t$ is time.

4. Logarithmic Functions

• Description: The logarithmic function is the inverse of the exponential function and is used to model the time required to reach a certain level of growth.
• Application: Logarithmic functions are used in economics to model learning curves, where the efficiency of production improves over time, and in demand analysis, where the elasticity of demand can be expressed as a logarithmic function.

5. Cobb-Douglas Production Function

• Description: This is a specific form of a production function that shows the relationship between inputs (like labor and capital) and the output produced.
• Application: The Cobb-Douglas function $Q=A\cdot L^{\alpha }\cdot K^{\beta }$$Q=A\cdot L^{\alpha }\cdot K^{\beta }$Q=A*L^( alpha)*K^( beta)Q = A \cdot L^\alpha \cdot K^\beta$Q=A\cdot L^{\alpha }\cdot K^{\beta }$ is used to model the output $Q$$Q$QQ$Q$ in terms of labor $L$$L$LL$L$ and capital $K$$K$KK$K$, where $A$$A$AA$A$ is a constant, and $\alpha$$\alpha$alpha\alpha$\alpha$ and $\beta$$\beta$beta\beta$\beta$ are the output elasticities of labor and capital, respectively.

6. Cost Functions

• Description: Cost functions represent the cost incurred by a business in producing a certain level of output.
• Application: These functions help in understanding how costs change with different levels of output and are used in decision-making related to pricing, production, and budgeting. The typical cost function is $C(Q)=F+VQ$$C(Q)=F+VQ$C(Q)=F+VQC(Q) = F + VQ$C(Q)=F+VQ$, where $C(Q)$$C(Q)$C(Q)C(Q)$C(Q)$ is the total cost, $F$$F$FF$F$ is fixed costs, and $V$$V$VV$V$ is variable costs per unit of output $Q$$Q$QQ$Q$.

7. Utility Functions

• Description: Utility functions represent a consumer’s preference ranking over a set of goods and services.
• Application: In economics, utility functions are used to analyze consumer behavior, particularly in understanding how consumers allocate their income among different goods to maximize their utility or satisfaction. A common utility function is $U(x,y)=x^{\alpha }\cdot y^{\beta }$$U(x,y)=x^{\alpha }\cdot y^{\beta }$U(x,y)=x^( alpha)*y^( beta)U(x, y) = x^\alpha \cdot y^\beta$U(x,y)=x^{\alpha }\cdot y^{\beta }$, where $x$$x$xx$x$ and $y$$y$yy$y$ are quantities of goods, and $\alpha$$\alpha$alpha\alpha$\alpha$ and $\beta$$\beta$beta\beta$\beta$ are constants representing the consumer’s preference.

8. Demand and Supply Functions

• Description: Demand functions represent the relationship between the quantity demanded of a good and its price, while supply functions represent the relationship between the quantity supplied and its price.
• Application: These functions are fundamental in determining market equilibrium, where the quantity demanded equals the quantity supplied. The demand function can be written as $Q_d=a-bP$$Q_d=a-bP$Q_(d)=a-bPQ_d = a – bP$Q_d=a-bP$, and the supply function as $Q_s=c+dP$$Q_s=c+dP$Q_(s)=c+dPQ_s = c + dP$Q_s=c+dP$, where $Q_d$$Q_d$Q_(d)Q_d$Q_d$ and $Q_s$$Q_s$Q_(s)Q_s$Q_s$ are the quantities demanded and supplied, respectively, and $P$$P$PP$P$ is the price.

9. Profit Functions

• Description: Profit functions are used to calculate the difference between total revenue and total cost.
• Application: In business, profit functions help in identifying the level of output that maximizes profit. The profit function is typically $\mathrm{\Pi }(Q)=R(Q)-C(Q)$$\mathrm{\Pi }(Q)=R(Q)-C(Q)$Pi(Q)=R(Q)-C(Q)\Pi(Q) = R(Q) – C(Q)$\mathrm{\Pi }(Q)=R(Q)-C(Q)$, where $\mathrm{\Pi }(Q)$$\mathrm{\Pi }(Q)$Pi(Q)\Pi(Q)$\mathrm{\Pi }(Q)$ is the profit, $R(Q)$$R(Q)$R(Q)R(Q)$R(Q)$ is the revenue, and $C(Q)$$C(Q)$C(Q)C(Q)$C(Q)$ is the cost as functions of output $Q$$Q$QQ$Q$.

Conclusion:

What do you mean by maxima or minima of a function? State the meaning of absolute minimum of a function. Explain the steps for finding maxima and minima of a function.

Answer:

Maxima and Minima of a Function

1. Local (Relative) Maximum: A point $x=c$$x=c$x=cx = c$x=c$ is a local maximum if the function $f(x)$$f(x)$f(x)f(x)$f(x)$ has a higher value at $c$$c$cc$c$ than at any nearby points. In mathematical terms, $f(c)$$f(c)$f(c)f(c)$f(c)$ is a local maximum if there exists an interval around $c$$c$cc$c$ such that $f(c)\ge f(x)$$f(c)\ge f(x)$f(c) >= f(x)f(c) \geq f(x)$f(c)\ge f(x)$ for all $x$$x$xx$x$ in that interval.
2. Local (Relative) Minimum: A point $x=c$$x=c$x=cx = c$x=c$ is a local minimum if the function $f(x)$$f(x)$f(x)f(x)$f(x)$ has a lower value at $c$$c$cc$c$ than at any nearby points. That is, $f(c)$$f(c)$f(c)f(c)$f(c)$ is a local minimum if there exists an interval around $c$$c$cc$c$ such that $f(c)\le f(x)$$f(c)\le f(x)$f(c) <= f(x)f(c) \leq f(x)$f(c)\le f(x)$ for all $x$$x$xx$x$ in that interval.
3. Absolute (Global) Maximum: The absolute maximum of a function on a given interval is the point at which the function reaches its highest value over the entire interval.
4. Absolute (Global) Minimum: The absolute minimum of a function on a given interval is the point at which the function reaches its lowest value over the entire interval.

Absolute Minimum of a Function

Steps to Find Maxima and Minima of a Function

1. Find the derivative: Compute the first derivative $f^{\prime }(x)$$f^{\prime }(x)$f^(‘)(x)f'(x)$f^{\prime }(x)$ of the function $f(x)$$f(x)$f(x)f(x)$f(x)$. The derivative gives the slope of the function at any point.
2. Set the derivative equal to zero: Solve the equation $f^{\prime }(x)=0$$f^{\prime }(x)=0$f^(‘)(x)=0f'(x) = 0$f^{\prime }(x)=0$ to find the critical points. These are the points where the slope of the function is zero, which might correspond to maxima, minima, or points of inflection.
3. Second Derivative Test:
   • Compute the second derivative $f^{″}(x)$$f^{″}(x)$f^(″)(x)f”(x)$f^{″}(x)$.
   • Evaluate $f^{″}(x)$$f^{″}(x)$f^(″)(x)f”(x)$f^{″}(x)$ at the critical points:
     • If $f^{″}(c)>0$$f^{″}(c)>0$f^(″)(c) > 0f”(c) > 0$f^{″}(c)>0$, then $f(c)$$f(c)$f(c)f(c)$f(c)$ is a local minimum.
     • If $f^{″}(c)<0$$f^{″}(c)<0$f^(″)(c) < 0f”(c) < 0$f^{″}(c)<0$, then $f(c)$$f(c)$f(c)f(c)$f(c)$ is a local maximum.
     • If $f^{″}(c)=0$$f^{″}(c)=0$f^(″)(c)=0f”(c) = 0$f^{″}(c)=0$, the test is inconclusive, and you may need to use other methods to determine the nature of the critical point.
4. Evaluate endpoints (if the function is defined on a closed interval $[a,b]$$[a,b]$[a,b][a, b]$[a,b]$): Check the values of the function at the endpoints $f(a)$$f(a)$f(a)f(a)$f(a)$ and $f(b)$$f(b)$f(b)f(b)$f(b)$. The largest of these values, along with the values at the critical points, will give you the absolute maximum, and the smallest will give you the absolute minimum.
5. Compare values: The maximum and minimum values of the function on the interval can be determined by comparing the function values at the critical points and the endpoints.

The manager of a departmental store compiled information on 200 accounts receivable which were delinquent. For each account, he has noted the number of days passed after the due date. He then grouped the data as shown in the following frequency distribution. Determine the median using two methods.

Answer:

1. Cumulative Frequency Method

• $L$$L$LL$L$ = Lower boundary of the median class = 60
• $N$$N$NN$N$ = Total number of observations = 200
• $F$$F$FF$F$ = Cumulative frequency of the class preceding the median class = 85
• $f$$f$ff$f$ = Frequency of the median class = 40
• $h$$h$hh$h$ = Class width = 74 – 60 = 14

2. Interpolation Method

Conclusion

A manufacturer earns Rs. 5500 in the first month and Rs. 7000 in the second month. On plotting these points, the manufacturer observes a linear function may fit the data.

Answer:

Part 1: Find the Quadratic Function that Fits the Data

• First month (let’s denote it as $x_1=1$$x_1=1$x_(1)=1x_1 = 1$x_1=1$): $y_1=5500$$y_1=5500$y_(1)=5500y_1 = 5500$y_1=5500$
• Second month ($x_2=2$$x_2=2$x_(2)=2x_2 = 2$x_2=2$): $y_2=7000$$y_2=7000$y_(2)=7000y_2 = 7000$y_2=7000$

• Third point ($x_0=0$$x_0=0$x_(0)=0x_0 = 0$x_0=0$): $y_0=0$$y_0=0$y_(0)=0y_0 = 0$y_0=0$

1. $0=a(0{)}^2+b(0)+c\Rightarrow c=0$$0=a(0{)}^2+b(0)+c\Rightarrow c=0$0=a(0)^(2)+b(0)+c=>c=00 = a(0)^2 + b(0) + c \Rightarrow c = 0$0=a(0{)}^2+b(0)+c\Rightarrow c=0$
2. $5500=a(1{)}^2+b(1)+0\Rightarrow a+b=5500$$5500=a(1{)}^2+b(1)+0\Rightarrow a+b=5500$5500=a(1)^(2)+b(1)+0=>a+b=55005500 = a(1)^2 + b(1) + 0 \Rightarrow a + b = 5500$5500=a(1{)}^2+b(1)+0\Rightarrow a+b=5500$
3. $7000=a(2{)}^2+b(2)+0\Rightarrow 4a+2b=7000$$7000=a(2{)}^2+b(2)+0\Rightarrow 4a+2b=7000$7000=a(2)^(2)+b(2)+0=>4a+2b=70007000 = a(2)^2 + b(2) + 0 \Rightarrow 4a + 2b = 7000$7000=a(2{)}^2+b(2)+0\Rightarrow 4a+2b=7000$

Part 2: Prediction of Earnings for the Fourth Month

Conclusion

You are given the profit function of a business activity and asked to offer your suggestion on the rate of change of profit. What would you do?

Answer:

1. Understand the Profit Function

2. Calculate the First Derivative

3. Analyze the First Derivative

• Positive $P^{\prime }(x)$$P^{\prime }(x)$P^(‘)(x)P'(x)$P^{\prime }(x)$: If $P^{\prime }(x)>0$$P^{\prime }(x)>0$P^(‘)(x) > 0P'(x) > 0$P^{\prime }(x)>0$ for a given range of $x$$x$xx$x$, it indicates that profit is increasing as $x$$x$xx$x$ increases. This might suggest that increasing production or investment within this range is beneficial.
• Negative $P^{\prime }(x)$$P^{\prime }(x)$P^(‘)(x)P'(x)$P^{\prime }(x)$: If $P^{\prime }(x)<0$$P^{\prime }(x)<0$P^(‘)(x) < 0P'(x) < 0$P^{\prime }(x)<0$ for a given range of $x$$x$xx$x$, it indicates that profit is decreasing as $x$$x$xx$x$ increases. In this case, it might be advisable to reduce $x$$x$xx$x$ to increase profit.
• Zero $P^{\prime }(x)$$P^{\prime }(x)$P^(‘)(x)P'(x)$P^{\prime }(x)$: If $P^{\prime }(x)=0$$P^{\prime }(x)=0$P^(‘)(x)=0P'(x) = 0$P^{\prime }(x)=0$, it indicates a potential maximum, minimum, or inflection point. Further analysis is needed to determine the nature of this point.

4. Determine Critical Points

5. Analyze the Second Derivative (Optional)

• If $P^{″}(x)>0$$P^{″}(x)>0$P^(″)(x) > 0P”(x) > 0$P^{″}(x)>0$: The function is concave up, and the critical point is a local minimum.
• If $P^{″}(x)<0$$P^{″}(x)<0$P^(″)(x) < 0P”(x) < 0$P^{″}(x)<0$: The function is concave down, and the critical point is a local maximum.
• If $P^{″}(x)=0$$P^{″}(x)=0$P^(″)(x)=0P”(x) = 0$P^{″}(x)=0$: The second derivative test is inconclusive, and further analysis may be needed.

6. Make Recommendations

• If the profit is increasing: Recommend strategies that could increase $x$$x$xx$x$ (e.g., increasing production, investment, etc.).
• If the profit is decreasing: Recommend strategies to reduce $x$$x$xx$x$ or find a different approach to reverse the downward trend.
• If a maximum is found: Suggest maintaining $x$$x$xx$x$ around the critical value where profit is maximized.
• If a minimum is found: Advise avoiding $x$$x$xx$x$ values that lead to a profit minimum.

7. Consider External Factors

Summary

Bank pays compound interest at the rate of $5\mathrm{\%}$$5\mathrm{\%}$5%5 \%$5\mathrm{\%}$ p.a. XYZ deposited a principal amount of RS. 5,000 in bank for 4 years. Find the interest that XYZ will receive.

Answer:

• $A$$A$AA$A$ is the amount of money accumulated after $n$$n$nn$n$ years, including interest.
• $P$$P$PP$P$ is the principal amount (the initial amount of money).
• $r$$r$rr$r$ is the annual interest rate (in decimal form).
• $n$$n$nn$n$ is the number of times that interest is compounded per year.
• $t$$t$tt$t$ is the time the money is invested or borrowed for, in years.

• $P=5000$$P=5000$P=5000P = 5000$P=5000$ Rs
• $r=5\mathrm{\%}$$r=5\mathrm{\%}$r=5%r = 5\%$r=5\mathrm{\%}$ or $0.05$$0.05$0.050.05$0.05$
• $n=1$$n=1$n=1n = 1$n=1$ (since the interest is compounded annually)
• $t=4$$t=4$t=4t = 4$t=4$ years

Interest Received

Conclusion

Explain the difference between Karl Pearson’s correlation co-efficient and Spearman’s rank correlation co-efficient. Under what situations, is the latter preferred to the former?

Answer:

Difference Between Karl Pearson’s Correlation Coefficient and Spearman’s Rank Correlation Coefficient

• Definition: Karl Pearson’s correlation coefficient (denoted as $r$$r$rr$r$) measures the linear relationship between two continuous variables. It quantifies the degree to which two variables move in tandem, assuming a linear relationship between them.
• Calculation: It is calculated using the covariance of the two variables divided by the product of their standard deviations:
  $$r=\frac{\text{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}$$$$r=\frac{\text{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}$$r=(“Cov”(X,Y))/(sigma _(X)sigma _(Y))r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}$$r=\frac{\text{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}$$
  where:
  • $\text{Cov}(X,Y)$$\text{Cov}(X,Y)$“Cov”(X,Y)\text{Cov}(X, Y)$\text{Cov}(X,Y)$ is the covariance between the variables $X$$X$XX$X$ and $Y$$Y$YY$Y$.
  • ${\sigma }_X$${\sigma }_X$sigma _(X)\sigma_X${\sigma }_X$ and ${\sigma }_Y$${\sigma }_Y$sigma _(Y)\sigma_Y${\sigma }_Y$ are the standard deviations of $X$$X$XX$X$ and $Y$$Y$YY$Y$, respectively.
• Range: The value of $r$$r$rr$r$ ranges from -1 to +1.
  • $r=1$$r=1$r=1r = 1$r=1$ indicates a perfect positive linear relationship.
  • $r=-1$$r=-1$r=-1r = -1$r=-1$ indicates a perfect negative linear relationship.
  • $r=0$$r=0$r=0r = 0$r=0$ indicates no linear relationship.
• Assumptions:
  • The relationship between the variables is linear.
  • Both variables are normally distributed.
  • The data are measured on an interval or ratio scale.

• Definition: Spearman’s rank correlation coefficient (denoted as $\rho$$\rho$rho\rho$\rho$ or $r_s$$r_s$r_(s)r_s$r_s$) measures the strength and direction of the monotonic relationship between two ranked variables. It assesses how well the relationship between two variables can be described using a monotonic function.
• Calculation: It is calculated based on the ranks of the data rather than their actual values. The formula for $\rho$$\rho$rho\rho$\rho$ is:
  $$\rho =1-\frac{6\sum d_i^2}{n(n^2-1)}$$$$\rho =1-\frac{6\sum d_i^2}{n(n^2-1)}$$rho=1-(6sumd_(i)^(2))/(n(n^(2)-1))\rho = 1 – \frac{6 \sum d_i^2}{n(n^2 – 1)}$$\rho =1-\frac{6\sum d_i^2}{n(n^2-1)}$$
  where:
  • $d_i$$d_i$d_(i)d_i$d_i$ is the difference between the ranks of corresponding variables.
  • $n$$n$nn$n$ is the number of observations.
• Range: Like Pearson’s $r$$r$rr$r$, Spearman’s $\rho$$\rho$rho\rho$\rho$ also ranges from -1 to +1.
  • $\rho =1$$\rho =1$rho=1\rho = 1$\rho =1$ indicates a perfect positive monotonic relationship.
  • $\rho =-1$$\rho =-1$rho=-1\rho = -1$\rho =-1$ indicates a perfect negative monotonic relationship.
  • $\rho =0$$\rho =0$rho=0\rho = 0$\rho =0$ indicates no monotonic relationship.
• Assumptions:
  • The relationship between the variables is monotonic (increasing or decreasing consistently).
  • The data can be ordinal, interval, or ratio scale.

Situations Where Spearman’s Rank Correlation is Preferred Over Pearson’s

1. Non-Linear Relationships: When the relationship between the two variables is non-linear but monotonic (e.g., a curve that consistently increases or decreases but is not a straight line), Spearman’s $\rho$$\rho$rho\rho$\rho$ is more appropriate. Pearson’s $r$$r$rr$r$ only measures linear relationships.
2. Ordinal Data: If the data is ordinal (e.g., ranks, ratings), Spearman’s rank correlation is preferred since it does not require the assumption of interval-level measurement.
3. Non-Normal Data Distribution: When the data are not normally distributed or contain outliers that can heavily influence the Pearson correlation, Spearman’s rank correlation is more robust as it relies on ranks rather than the actual data values.
4. Small Sample Sizes: For small sample sizes or data with significant outliers, Spearman’s $\rho$$\rho$rho\rho$\rho$ is less sensitive to anomalies than Pearson’s $r$$r$rr$r$.
5. Tied Ranks: In situations where there are tied ranks in the data, Spearman’s rank correlation provides a method to handle these ties, whereas Pearson’s correlation coefficient does not directly deal with ranked data.

Summary

• Pearson’s Correlation Coefficient: Best for linear relationships with continuous, normally distributed data.
• Spearman’s Rank Correlation Coefficient: Preferred for monotonic relationships, ordinal data, non-normal distributions, and when data has outliers or is non-linear.

Explain briefly the additive and multiplicative models of time series. Which of these models is more commonly used and why?

Answer:

Additive and Multiplicative Models of Time Series

1. Additive Model

• $Y_t$$Y_t$Y_(t)Y_t$Y_t$ is the observed value at time $t$$t$tt$t$.
• $T_t$$T_t$T_(t)T_t$T_t$ is the trend component at time $t$$t$tt$t$.
• $S_t$$S_t$S_(t)S_t$S_t$ is the seasonal component at time $t$$t$tt$t$.
• $C_t$$C_t$C_(t)C_t$C_t$ is the cyclical component at time $t$$t$tt$t$.
• $I_t$$I_t$I_(t)I_t$I_t$ is the irregular or random component at time $t$$t$tt$t$.

• The additive model assumes that the components are independent and add together to form the observed value.
• It is typically used when the seasonal fluctuations or variations are roughly constant over time (i.e., the amplitude of seasonal patterns does not change with the level of the series).

2. Multiplicative Model

• The multiplicative model assumes that the components interact with each other in a multiplicative manner.
• It is used when the seasonal fluctuations or variations increase or decrease in proportion to the level of the series. In other words, the seasonal effects are not constant but vary in magnitude with the trend.

Which Model is More Commonly Used and Why?

1. Proportional Seasonality: In many real-world scenarios, the magnitude of seasonal variations tends to increase or decrease with the level of the trend. For example, sales might be higher in the holiday season, and as overall sales increase, the holiday peak becomes more pronounced. The multiplicative model can capture this proportionality.
2. Logarithmic Transformation: The multiplicative model can be transformed into an additive model by taking the logarithm of the data. This is useful in statistical analysis and can simplify the process of analyzing time series data.
3. Versatility: The multiplicative model is more flexible and can handle a wider range of time series behaviors, especially when dealing with financial, economic, and other business-related data where trends and seasonality interact multiplicatively.

Summary

• Additive Model: Used when seasonal effects are constant and independent of the trend level.
• Multiplicative Model: Used when seasonal effects vary proportionally with the trend, making it more versatile and commonly used in practical time series analysis due to its ability to better reflect real-world data patterns.

Write short notes on the following:

Answer:

(a) Factorization

Types of Factorization:

1. Factorization of Numbers:
   • Prime Factorization: Breaking down a number into its prime factors. For example, the prime factorization of 18 is $2\times 3^2$$2\times 3^2$2xx3^(2)2 \times 3^2$2\times 3^2$.
   • Greatest Common Divisor (GCD): The largest factor that two or more numbers share.
2. Factorization of Polynomials:
   • Simple Factorization: Finding factors that are polynomials of lower degree. For example, $x^2-9$$x^2-9$x^(2)-9x^2 – 9$x^2-9$ can be factored as $(x-3)(x+3)$$(x-3)(x+3)$(x-3)(x+3)(x – 3)(x + 3)$(x-3)(x+3)$.
   • Quadratic Factorization: A quadratic polynomial of the form $a{x}^2+bx+c$$a{x}^2+bx+c$ax^(2)+bx+cax^2 + bx + c$a{x}^2+bx+c$ can be factored by finding the roots or using methods like completing the square or the quadratic formula.
   • Factoring by Grouping: Involves grouping terms in a polynomial and factoring them separately before combining the results. For example, $ax+ay+bx+by$$ax+ay+bx+by$ax+ay+bx+byax + ay + bx + by$ax+ay+bx+by$ can be factored as $(a+b)(x+y)$$(a+b)(x+y)$(a+b)(x+y)(a + b)(x + y)$(a+b)(x+y)$.

(b) Harmonic Mean

Formula: