State the empirical relationship between Mean, Median, and Mode in a moderately skewed distribution.

Empirical Relationship \text{Mode} = 3\text{Median} - 2\text{Mean} \text{or } \text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median})

What is the Coefficient of Variation (C.V.)? State its formula and practical use.

Coefficient of Variation (C.V.) A relative measure of dispersion expressing standard deviation as a percentage of the mean: \text{C.V.} = \frac{\sigma}{\bar{X}} \times 100 • Use: Used to compare consistency, stability, or risk across two or more distributions. The series with lower C.V. is more consistent/stable.

State the mathematical bounds for Karl Pearson's correlation coefficient (r). What does r = -1 signify?

Bounds of Correlation Coefficient -1 \le r \le +1 When r = -1, there is perfect negative linear correlation between the two variables: as one variable increases, the other decreases in exact fixed proportion.

State the four components of a Time Series.

Four Components of Time Series 1. Secular Trend (T): Long-term smooth general direction over many years. 2. Seasonal Variations (S): Regular short-term patterns recurring within a year (e.g., peak hotel bookings in autumn). 3. Cyclical Fluctuations (C): Medium-term multi-year macroeconomic boom and bust cycles. 4. Irregular / Random Variations (I): Unpredictable erratic shocks (earthquakes, pandemics).

State the Empirical Rule (68-95-99.7) for a Normal Distribution.

Empirical Rule for Normal Distribution • \mu \pm 1\sigma contains approximately 68.26% of all observations. • \mu \pm 2\sigma contains approximately 95.44% of all observations. • \mu \pm 3\sigma contains approximately 99.74% of all observations.

The daily demand for buffet breakfast at a 200-room hotel is normally distributed with a mean (\mu) of 150 guests and a standard deviation (\sigma) of 20 guests. Required: a. What percentage of days will the breakfast demand exceed 180 guests? b. What is the probability that on a randomly selected day, demand is between 130 and 170 guests? c. How many breakfasts should the kitchen prepare so that food shortages occur on no more than 5% of days? (Given: Z = 1.5 \implies \text{Area} = 0.4332; Z = 1.0 \implies \text{Area} = 0.3413; Z = 1.645 \implies \text{Area} = 0.4500).

Normal Distribution Application in Food Service Given \mu = 150, \sigma = 20. a. Probability of demand exceeding 180 guests (X > 180): Z = \frac{X - \mu}{\sigma} = \frac{180 - 150}{20} = \frac{30}{20} = +1.5 P(X > 180) = 0.5000 - P(0 < Z < 1.5) = 0.5000 - 0.4332 = 0.0668 \text{ or } \textbf{6.68\%} b. Probability between 130 and 170 guests: • For X1 = 130: Z1 = \frac{130 - 150}{20} = -1.0 • For X2 = 170: Z2 = \frac{170 - 150}{20} = +1.0 P(130 < X < 170) = P(-1.0 < Z < 0) + P(0 < Z < 1.0) = 0.3413 + 0.3413 = \textbf{0.6826 (68.26\%)} c. Target capacity for shortage probability \le 5\%: Upper tail probability = 0.05 \implies Area from mean to Z is 0.5000 - 0.05 = 0.4500. From standard normal tables, Z = +1.645. X = \mu + Z\sigma = 150 + (1.645 \times 20) = 150 + 32.9 = 182.9 \approx \textbf{183 breakfasts} The kitchen must prepare 183 breakfasts to ensure shortages occur on no more than 5% of mornings.

Explain the 4-Quarter Centered Moving Average method to isolate Seasonal Indices in hotel room occupancy data.

4-Quarter Centered Moving Average Method 1. Compute 4-Quarter Moving Totals: Sum 4 consecutive quarters to smooth out seasonal fluctuations. 2. Center the Totals: Average adjacent pairs of 4-quarter totals to align the figures with specific quarterly mid-points. 3. Compute Centered Moving Average (Trend T): Divide centered totals by 8 to establish the underlying trend value. 4. Compute Seasonal-Irregular Ratios: Under the multiplicative model (Y = T \times S \times C \times I), compute \frac{Y}{T} \times 100. 5. Compute Modified Mean and Adjust: Average the ratios for each respective quarter across multiple years, and scale so the sum of the four indices equals 400.

Case Study: Revenue Management & Statistical Demand Forecasting at Annapurna Resorts Annapurna Resorts manages three luxury lodges in Pokhara, Chitwan, and Nagarkot. The Chief Revenue Officer (CRO) wants to move from subjective room pricing to data-driven Predictive Revenue Management. Historical occupancy data over the last 16 quarters reveals high seasonal volatility: • Q1 (Jan–Mar): High spring trekking (Index: 125%) • Q2 (Apr–Jun): Pre-monsoon domestic leisure (Index: 85%) • Q3 (Jul–Sep): Low monsoon season (Index: 60%) • Q4 (Oct–Dec): Peak autumn festival season (Index: 130%) The underlying linear trend equation for total quarterly room sales (Rs in millions) is established as: T = 40 + 2.5t (where t = 1 is Q1 2080). As Revenue Management Analyst: a. Forecast trend values (T) for all four quarters of Year 2082 (t = 9, 10, 11, 12). b. Apply Seasonal Indices to compute seasonally adjusted room revenue forecasts for each quarter of Year 2082. c. Formulate dynamic pricing guidelines for Q3 (low season) versus Q4 (peak season). d. Explain how Poisson and Binomial probability distributions can model no-shows and optimize room overbooking limits.

Statistical Demand Forecasting & Revenue Management Case Analysis a. Trend Value Projections for Year 2082 (t = 9, 10, 11, 12) • Q1 2082 (t = 9): T_9 = 40 + 2.5(9) = 40 + 22.5 = \textbf{Rs 62.5 Million} • Q2 2082 (t = 10): T_{10} = 40 + 2.5(10) = 40 + 25.0 = \textbf{Rs 65.0 Million} • Q3 2082 (t = 11): T_{11} = 40 + 2.5(11) = 40 + 27.5 = \textbf{Rs 67.5 Million} • Q4 2082 (t = 12): T_{12} = 40 + 2.5(12) = 40 + 30.0 = \textbf{Rs 70.0 Million} b. Seasonally Adjusted Forecasts (Y = T \times \text{Index}) • Q1 2082: 62.5 \times 1.25 = \textbf{Rs 78.125 Million} • Q2 2082: 65.0 \times 0.85 = \textbf{Rs 55.250 Million} • Q3 2082: 67.5 \times 0.60 = \textbf{Rs 40.500 Million} • Q4 2082: 70.0 \times 1.30 = \textbf{Rs 91.000 Million} • Total Projected Annual Revenue (2082): Rs 264.875 Million c. Dynamic Pricing Strategies • Q3 (Monsoon Low Season - Index 60%): Implement price unbundling; offer stay-3-pay-2 packages, partner with wellness/spa retreats, and reduce minimum stay restrictions. • Q4 (Peak Autumn Season - Index 130%): Enforce strict non-refundable advance deposits, minimum 3-night stay restrictions on festival weekends, and close discount channels to maximize RevPAR. d. Modeling No-Shows via Probability Distributions • Room cancellations and no-shows follow a Binomial Distribution B(n, p), where n is total bookings and p is probability of no-show, or a Poisson Distribution \text{Pois}(\lambda) for daily walk-ins. • By balancing the cost of an empty room (spoilage cost = forgone contribution) against the cost of walking a guest to another hotel (overbooking cost = re-accommodation + goodwill loss), the revenue manager identifies the mathematically optimal overbooking cushion.

Dean's Office Official Model Question Paper – STT 301

Group A

Brief Answer Questions. Attempt ALL questions. (5 × 2 = 10)

[5*2=10]

State the empirical relationship between Mean, Median, and Mode in a moderately skewed distribution.

[2]
View model solution
Empirical Relationship

$\text{Mode} = 3\text{Median} - 2\text{Mean}$
$\text{or } \text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median})$
What is the Coefficient of Variation (C.V.)? State its formula and practical use.

[2]
View model solution
Coefficient of Variation (C.V.)

A relative measure of dispersion expressing standard deviation as a percentage of the mean:
$\text{C.V.} = \frac{\sigma}{\bar{X}} \times 100$
- Use: Used to compare consistency, stability, or risk across two or more distributions. The series with lower C.V. is more consistent/stable.
State the mathematical bounds for Karl Pearson’s correlation coefficient ( $r$ ). What does $r = -1$ signify?

[2]
View model solution
Bounds of Correlation Coefficient

$-1 \le r \le +1$
When $r = -1$ , there is perfect negative linear correlation between the two variables: as one variable increases, the other decreases in exact fixed proportion.
State the four components of a Time Series.

[2]
View model solution
Four Components of Time Series
1. Secular Trend ( $T$ ): Long-term smooth general direction over many years.
2. Seasonal Variations ( $S$ ): Regular short-term patterns recurring within a year (e.g., peak hotel bookings in autumn).
3. Cyclical Fluctuations ( $C$ ): Medium-term multi-year macroeconomic boom and bust cycles.
4. Irregular / Random Variations ( $I$ ): Unpredictable erratic shocks (earthquakes, pandemics).
State the Empirical Rule (68-95-99.7) for a Normal Distribution.

[2]
View model solution
Empirical Rule for Normal Distribution
- $\mu \pm 1\sigma$ contains approximately 68.26% of all observations.
- $\mu \pm 2\sigma$ contains approximately 95.44% of all observations.
- $\mu \pm 3\sigma$ contains approximately 99.74% of all observations.

Group B

Descriptive Answer Questions. Attempt any THREE questions. (3 × 10 = 30)

[3*10=30]

The daily restaurant bill amounts (Rs in hundreds) paid by guests at a heritage hotel in Bhaktapur are distributed as follows:

Bill Amount (Rs '00)	10–20	20–30	30–40	40–50	50–60	60–70
Number of Tables ( $f$ )	6	14	20	30	18	12

Required: a. Calculate the Mean daily bill amount b. Calculate the Standard Deviation c. Calculate the Coefficient of Variation (C.V.)

[10]

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Statistical Computation: Mean, Standard Deviation & C.V.

Let $x$ be mid-points, $A = 35$ , $h = 10$ , $d = \frac{x - 35}{10}$ .

Class	$f$	$x$	$d$	$fd$	$fd^2$
10–20	6	15	-2	-12	24
20–30	14	25	-1	-14	14
30–40	20	35	0	0	0
40–50	30	45	+1	30	30
50–60	18	55	+2	36	72
60–70	12	65	+3	36	108
Total	$N = 100$			$\sum fd = 76$	$\sum fd^2 = 248$

a. Mean ( $\bar{X}$ ):

\bar{X} = A + \left(\frac{\sum fd}{N}\right) \times h = 35 + \left(\frac{76}{100}\right) \times 10 = 35 + 7.6 = \textbf{42.6 (Rs 4,260)}

b. Standard Deviation ( $\sigma$ ):

\sigma = h \times \sqrt{\frac{\sum fd^2}{N} - \left(\frac{\sum fd}{N}\right)^2} = 10 \times \sqrt{\frac{248}{100} - (0.76)^2}

\sigma = 10 \times \sqrt{2.48 - 0.5776} = 10 \times \sqrt{1.9024} = 10 \times 1.379 = \textbf{13.79 (Rs 1,379)}

c. Coefficient of Variation (C.V.):

\text{C.V.} = \frac{\sigma}{\bar{X}} \times 100 = \frac{13.79}{42.6} \times 100 = \textbf{32.37\%}

The following data shows the Advertising Expenditure (Rs in lakhs, $X$ ) and Room Sales Revenue (Rs in crores, $Y$ ) for 5 consecutive years at a resort:

Year	1	2	3	4	5
Advertising ( $X$ )	6	8	10	12	14
Sales Revenue ( $Y$ )	10	12	15	18	20

Required: a. Calculate Karl Pearson’s correlation coefficient between advertising and sales b. Fit the linear regression equation of Sales ( $Y$ ) on Advertising ( $X$ ) c. Estimate sales revenue if advertising expenditure is increased to Rs 20 lakhs.

[10]

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Correlation and Linear Regression Analysis

$\bar{X} = \frac{50}{5} = 10$ , $\bar{Y} = \frac{75}{5} = 15$ .

$X$	$Y$	$x = X - 10$	$y = Y - 15$	$x^2$	$y^2$	$xy$
6	10	-4	-5	16	25	20
8	12	-2	-3	4	9	6
10	15	0	0	0	0	0
12	18	+2	+3	4	9	6
14	20	+4	+5	16	25	20
$\sum = 50$	$\sum = 75$	0	0	40	68	52

a. Karl Pearson’s Correlation Coefficient ( $r$ ):

r = \frac{\sum xy}{\sqrt{\sum x^2 \sum y^2}} = \frac{52}{\sqrt{40 \times 68}} = \frac{52}{\sqrt{2720}} = \frac{52}{52.15} = \textbf{+0.997}

(Nearly perfect positive correlation).

b. Regression Equation of $Y$ on $X$ ( $Y = a + bX$ ):

b = \frac{\sum xy}{\sum x^2} = \frac{52}{40} = 1.30

a = \bar{Y} - b\bar{X} = 15 - (1.30 \times 10) = 15 - 13 = 2.0

\text{Regression Line: } \textbf{Y = 2.0 + 1.30X}

c. Estimated Sales Revenue for Advertising $X = 20$ lakhs:

Y = 2.0 + 1.30(20) = 2.0 + 26.0 = \textbf{Rs 28.0 Crores}

The daily demand for buffet breakfast at a 200-room hotel is normally distributed with a mean ( $\mu$ ) of 150 guests and a standard deviation ( $\sigma$ ) of 20 guests.

Required: a. What percentage of days will the breakfast demand exceed 180 guests? b. What is the probability that on a randomly selected day, demand is between 130 and 170 guests? c. How many breakfasts should the kitchen prepare so that food shortages occur on no more than 5% of days? (Given: $Z = 1.5 \implies \text{Area} = 0.4332$ ; $Z = 1.0 \implies \text{Area} = 0.3413$ ; $Z = 1.645 \implies \text{Area} = 0.4500$ ).

[10]
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Normal Distribution Application in Food Service

Given $\mu = 150$ , $\sigma = 20$ .

a. Probability of demand exceeding 180 guests ( $X > 180$ ):

$Z = \frac{X - \mu}{\sigma} = \frac{180 - 150}{20} = \frac{30}{20} = +1.5$
$P(X > 180) = 0.5000 - P(0 < Z < 1.5) = 0.5000 - 0.4332 = 0.0668 \text{ or } \textbf{6.68\%}$
b. Probability between 130 and 170 guests:
- For $X_1 = 130$ : $Z_1 = \frac{130 - 150}{20} = -1.0$
- For $X_2 = 170$ : $Z_2 = \frac{170 - 150}{20} = +1.0$ $P(130 < X < 170) = P(-1.0 < Z < 0) + P(0 < Z < 1.0) = 0.3413 + 0.3413 = \textbf{0.6826 (68.26\%)}$ $
c. Target capacity for shortage probability $\le 5\%$ :

Upper tail probability = 0.05 $\implies$ Area from mean to $Z$ is $0.5000 - 0.05 = 0.4500$ . From standard normal tables, $Z = +1.645$ .
$X = \mu + Z\sigma = 150 + (1.645 \times 20) = 150 + 32.9 = 182.9 \approx \textbf{183 breakfasts}$

The kitchen must prepare 183 breakfasts to ensure shortages occur on no more than 5% of mornings.
Explain the 4-Quarter Centered Moving Average method to isolate Seasonal Indices in hotel room occupancy data.

[10]
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4-Quarter Centered Moving Average Method
1. Compute 4-Quarter Moving Totals: Sum 4 consecutive quarters to smooth out seasonal fluctuations.
2. Center the Totals: Average adjacent pairs of 4-quarter totals to align the figures with specific quarterly mid-points.
3. Compute Centered Moving Average (Trend $T$ ): Divide centered totals by 8 to establish the underlying trend value.
4. Compute Seasonal-Irregular Ratios: Under the multiplicative model ( $Y = T \times S \times C \times I$ ), compute $\frac{Y}{T} \times 100$ .
5. Compute Modified Mean and Adjust: Average the ratios for each respective quarter across multiple years, and scale so the sum of the four indices equals 400.

Group C

Comprehensive Answer / Case Analysis Question. (1 × 20 = 20)

[1*20=20]

Case Study: Revenue Management & Statistical Demand Forecasting at Annapurna Resorts

Annapurna Resorts manages three luxury lodges in Pokhara, Chitwan, and Nagarkot. The Chief Revenue Officer (CRO) wants to move from subjective room pricing to data-driven Predictive Revenue Management. Historical occupancy data over the last 16 quarters reveals high seasonal volatility:
- Q1 (Jan–Mar): High spring trekking (Index: 125%)
- Q2 (Apr–Jun): Pre-monsoon domestic leisure (Index: 85%)
- Q3 (Jul–Sep): Low monsoon season (Index: 60%)
- Q4 (Oct–Dec): Peak autumn festival season (Index: 130%)
The underlying linear trend equation for total quarterly room sales (Rs in millions) is established as:
$T = 40 + 2.5t$
(where $t = 1$ is Q1 2080).

As Revenue Management Analyst: a. Forecast trend values ( $T$ ) for all four quarters of Year 2082 ( $t = 9, 10, 11, 12$ ). b. Apply Seasonal Indices to compute seasonally adjusted room revenue forecasts for each quarter of Year 2082. c. Formulate dynamic pricing guidelines for Q3 (low season) versus Q4 (peak season). d. Explain how Poisson and Binomial probability distributions can model no-shows and optimize room overbooking limits.
[20]
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Statistical Demand Forecasting & Revenue Management Case Analysis

a. Trend Value Projections for Year 2082 ( $t = 9, 10, 11, 12$ )
- Q1 2082 ( $t = 9$ ): $T_9 = 40 + 2.5(9) = 40 + 22.5 = \textbf{Rs 62.5 Million}$
- Q2 2082 ( $t = 10$ ): $T_{10} = 40 + 2.5(10) = 40 + 25.0 = \textbf{Rs 65.0 Million}$
- Q3 2082 ( $t = 11$ ): $T_{11} = 40 + 2.5(11) = 40 + 27.5 = \textbf{Rs 67.5 Million}$
- Q4 2082 ( $t = 12$ ): $T_{12} = 40 + 2.5(12) = 40 + 30.0 = \textbf{Rs 70.0 Million}$
b. Seasonally Adjusted Forecasts ( $Y = T \times \text{Index}$ )
- Q1 2082: $62.5 \times 1.25 = \textbf{Rs 78.125 Million}$
- Q2 2082: $65.0 \times 0.85 = \textbf{Rs 55.250 Million}$
- Q3 2082: $67.5 \times 0.60 = \textbf{Rs 40.500 Million}$
- Q4 2082: $70.0 \times 1.30 = \textbf{Rs 91.000 Million}$
- Total Projected Annual Revenue (2082): Rs 264.875 Million
c. Dynamic Pricing Strategies
- Q3 (Monsoon Low Season - Index 60%): Implement price unbundling; offer stay-3-pay-2 packages, partner with wellness/spa retreats, and reduce minimum stay restrictions.
- Q4 (Peak Autumn Season - Index 130%): Enforce strict non-refundable advance deposits, minimum 3-night stay restrictions on festival weekends, and close discount channels to maximize RevPAR.
d. Modeling No-Shows via Probability Distributions
- Room cancellations and no-shows follow a Binomial Distribution $B(n, p)$ , where $n$ is total bookings and $p$ is probability of no-show, or a Poisson Distribution $\text{Pois}(\lambda)$ for daily walk-ins.
- By balancing the cost of an empty room (spoilage cost = forgone contribution) against the cost of walking a guest to another hotel (overbooking cost = re-accommodation + goodwill loss), the revenue manager identifies the mathematically optimal overbooking cushion.

Empirical Relationship

Coefficient of Variation (C.V.)

Bounds of Correlation Coefficient

Four Components of Time Series

Empirical Rule for Normal Distribution

Statistical Computation: Mean, Standard Deviation & C.V.

a. Mean (Xˉ\bar{X}Xˉ):

b. Standard Deviation (σ\sigmaσ):

c. Coefficient of Variation (C.V.):

Correlation and Linear Regression Analysis

a. Karl Pearson’s Correlation Coefficient (rrr):

b. Regression Equation of YYY on XXX (Y=a+bXY = a + bXY=a+bX):

c. Estimated Sales Revenue for Advertising X=20X = 20X=20 lakhs:

Normal Distribution Application in Food Service

a. Probability of demand exceeding 180 guests (X>180X > 180X>180):

b. Probability between 130 and 170 guests:

c. Target capacity for shortage probability ≤5%\le 5\%≤5%:

4-Quarter Centered Moving Average Method

Statistical Demand Forecasting & Revenue Management Case Analysis

a. Trend Value Projections for Year 2082 (t=9,10,11,12t = 9, 10, 11, 12t=9,10,11,12)

b. Seasonally Adjusted Forecasts (Y=T×IndexY = T \times \text{Index}Y=T×Index)

c. Dynamic Pricing Strategies

d. Modeling No-Shows via Probability Distributions

a. Mean ( $\bar{X}$ ):

b. Standard Deviation ( $\sigma$ ):

a. Karl Pearson’s Correlation Coefficient ( $r$ ):

b. Regression Equation of $Y$ on $X$ ( $Y = a + bX$ ):

c. Estimated Sales Revenue for Advertising $X = 20$ lakhs:

a. Probability of demand exceeding 180 guests ( $X > 180$ ):

c. Target capacity for shortage probability $\le 5\%$ :

a. Trend Value Projections for Year 2082 ( $t = 9, 10, 11, 12$ )

b. Seasonally Adjusted Forecasts ( $Y = T \times \text{Index}$ )