Samacheer Kalvi Class 12 Business Maths Solutions Chapter 9 Applied Statistics Exercise 9.1

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 9 Applied Statistics Ex 9.1

 

Question 1. Define Time series.
Answer: A Time Series is a set of data arranged in time order. Croxton and Cowden say it's data listed by when it happened. Wessel and Wallet explain that when numbers are put in the order they happened, it creates a Time Series. Basically, a time series is a list of observations put in time order, either from earliest to latest or vice versa. The main goal is to find changes and try to remove them, so we can guess or predict what will happen in the future. This method helps us understand how things change over time, like daily temperatures or monthly sales.
In simple words: A time series is data collected over time and listed in the order it occurred. It helps us see patterns and predict future events.

🎯 Exam Tip: When defining key terms, always include multiple viewpoints or applications to show a comprehensive understanding.

 

Question 2. What is the need for studying time series?
Answer: Time series analysis is a math tool that helps us find patterns in data collected over a period. It's important to look at old data to see what changes have happened, both in the past and now. This helps us spot if certain things happen regularly or randomly over time. By studying these patterns, businesses and governments can make smarter choices for the future.
In simple words: We study time series to find patterns in past data, understand current changes, and predict what might happen in the future.

🎯 Exam Tip: Focus on the predictive and analytical benefits when discussing the importance of time series studies.

 

Question 3. State the uses of time series.
Answer: Some uses of time series are:
(i) It helps us look at what happened before.
(ii) It helps us guess what might happen next and plan for it.
(iii) It helps us check how well things are doing right now.
(iv) It helps compare different periods, like sales this year versus last year. So, time series is useful for studying data over time in business, money matters, and factories. Understanding past trends is key for making predictions about future events.
In simple words: Time series helps us understand past events, make future predictions, evaluate current performance, and compare different time periods in various fields.

🎯 Exam Tip: List key applications such as forecasting, planning, evaluation, and comparative studies to score full marks.

 

Question 4. Mention the components of the time series.
Answer: Time series data is made of four main parts:
(i) Secular Trend
(ii) Seasonal variations
(iii) Cyclic variations
(iv) Irregular variations
Each component helps statisticians break down complex data into easier-to-understand parts.
In simple words: The four parts of a time series are secular trend, seasonal variations, cyclic variations, and irregular variations.

🎯 Exam Tip: Remember all four main components as they are fundamental to understanding time series data.

 

Question 5. Define secular trend.
Answer: Secular trend is the long-term upward, downward, or steady movement of data over a long time. For example, things like a country's population, production, sales, prices, or income often show an upward trend. A downward trend might be seen in death rates, epidemics, electronic gadget prices, or water levels. This long-term change doesn't mean the movement is always in the exact same direction every single year. A secular trend helps us see the big picture without getting lost in small, short-term changes.
In simple words: Secular trend is the general long-term direction of data, either increasing, decreasing, or staying flat over many years.

🎯 Exam Tip: Emphasize the "long-term" nature and provide clear examples of both upward and downward trends to demonstrate understanding.

 

Question 6. Write a brief note on seasonal variations
Answer: Seasonal variations are changes that happen due to natural causes and repeat regularly at certain times each year. These changes happen within a year and are measured over a certain time interval, like a month or a quarter. Things like weather, holidays, and cultural traditions can cause these variations. They cause numbers to rise and fall in a steady and regular way. For example, umbrella sales go up in the rainy season, cold drinks sell more in summer, firecrackers during Diwali, new clothes during festivals, and sugarcane during Pongal. These predictable changes help businesses plan their stock and sales much better.
In simple words: Seasonal variations are regular, repeating changes within a year, often caused by weather or holidays, like increased umbrella sales in the rainy season.

🎯 Exam Tip: Focus on the "less than one year" period and the recurring, predictable nature of seasonal changes.

 

Question 7. Explain cyclic variations.
Answer: Cyclic variations are changes that go up and down over time but don't always follow the same exact pattern or happen at regular intervals. Usually, one complete cycle lasts between 7 to 9 years, but there's no strict rule for how long it takes. A good example is the business cycle, which goes from a boom to a depression and then recovery. Other factors include changes in government money rules and interest rates. Unlike seasonal changes, cyclic variations are longer-term and less predictable in their exact timing.
In simple words: Cyclic variations are long-term, wave-like changes in data that are not always regular, such as the ups and downs of a business cycle.

🎯 Exam Tip: Distinguish cyclic variations from seasonal ones by highlighting their longer period (more than a year) and less rigid pattern.

 

Question 8. Explain the stratified random sampling with a suitable example.
Answer: These kinds of changes do not follow any special pattern and do not happen at regular times. They are unexpected changes that happen by chance and cannot be predicted. Normally, these are short-term changes, but sometimes they are so strong that they can start new cycles or other kinds of changes. Examples include floods, wars, earthquakes, tsunamis, strikes, and lockouts. These sudden and unpredictable events make forecasting challenging but are crucial to acknowledge in data analysis.
In simple words: These are sudden, unexpected changes in data that happen randomly, like natural disasters or strikes, and are hard to predict.

🎯 Exam Tip: Focus on the "random" and "unpredictable" nature of these variations, as they represent external shocks to a system.

 

Question 9. Define seasonal index.
Answer: The seasonal index for each season (whether monthly or quarterly) is found using this formula:
Seasonal Index (S.I) = \( \frac { \text{Seasonal Average} }{ \text{Grand Average} } \times 100 \)
If the data is given monthwise:
Seasonal Index = \( \frac { \text{Monthly Average} }{ \text{Grand Average} } \times 100 \)
If quarterly data is given:
Seasonal Index = \( \frac { \text{Quarterly Average} }{ \text{Grand Average} } \times 100 \)
A seasonal index helps to measure how much a specific season or month affects a value compared to the average.
In simple words: A seasonal index shows how much a particular season's data is different from the overall average, calculated using a simple formula.

🎯 Exam Tip: Clearly state the formula and explain what "Seasonal Average" and "Grand Average" represent in context.

 

Question 10. of fitting a straight line.
Answer:
(i) First, a straight line trend can be shown by the equation \( Y = a + bX \). In this equation, \( Y \) is the actual value, \( X \) is time, and \( a \) and \( b \) are fixed numbers.
(ii) Then, we find the values of \( a \) and \( b \) by solving two special equations:
\( \Sigma Y = n a + b \Sigma X \) (2)
\( \Sigma XY = a \Sigma X + b \Sigma X^2 \) (3)
Here, \( n \) is the total number of years in the data.
(iii) If we set the middle point of time as zero, then the sum of \( X \) becomes zero, so \( \Sigma X = 0 \).
(iv) When \( \Sigma X = 0 \), the two normal equations become simpler:
\( \Sigma Y = na + b(0) \implies a = \frac { \Sigma Y }{n} = \bar { Y } \)
\( \Sigma XY = a(0) + b \Sigma X^2 \implies b = \frac { \Sigma XY }{ \Sigma X^2 } \)
The value \( a \) tells us the average of \( Y \), and \( b \) tells us how fast \( Y \) changes (its slope).
(v) Finally, we put the found values of \( a \) and \( b \) into the trend equation (1) to get the line that best fits the data. This method finds the line that minimizes the sum of squared differences between actual and estimated values.
In simple words: To fit a straight line, we use specific formulas to find values 'a' and 'b'. 'a' gives the average of 'Y', and 'b' shows the rate of change. We then plug these into the equation \( Y = a + bX \) to get the best-fit line.

🎯 Exam Tip: Remember the two normal equations and how they simplify when \( \Sigma X = 0 \), as this is a common shortcut in least squares method calculations.

 

Question 11. State the two normal equations used in fitting a straight line.
Answer: To find the fixed numbers 'a' and 'b' for a straight line, we solve these two standard equations:
\( \Sigma Y = n a + b \Sigma X \) (2)
\( \Sigma XY = a \Sigma X + b \Sigma X^2 \) (3)
Here, 'n' stands for the number of years in our data set. These equations are fundamental to the Least Squares method for finding the best-fit line.
In simple words: The two main equations used to find the best-fit straight line are \( \Sigma Y = n a + b \Sigma X \) and \( \Sigma XY = a \Sigma X + b \Sigma X^2 \).

🎯 Exam Tip: Memorize these two normal equations, as they are the core of the least squares method for linear regression.

 

Question 12. State the different methods of measuring trend.
Answer: The trend can be measured using these different methods:
(i) Freehand or Graphical Method
(ii) Method of Semi-Averages
(iii) Method of Moving Averages
Each method has its own advantages, depending on the type of data and desired accuracy.
In simple words: We can find the trend in data using the freehand method, semi-averages method, or moving averages method.

🎯 Exam Tip: Be ready to briefly explain each method if asked, as they represent different approaches to trend analysis.

 

Question 13. Compute the average seasonal movement for the following series
Answer: We compute the seasonal index using the method of simple averages.

Grand Average \( = \frac { 3.72 + 4.16 + 3.76 + 4.16 }{ 4 } = \frac { 15.80 }{ 4 } = 3.95 \)
S.I for I Quarter \( = \frac { \text{Average of I Quarter} }{ \text{Grand Average} } \times 100 = \frac { 3.72 }{ 3.95 } \times 100 = 94.1772 \)
S.I for II Quarter \( = \frac { \text{Average of II Quarter} }{ \text{Grand Average} } \times 100 = \frac { 4.16 }{ 3.95 } \times 100 = 105.3165 \)
S.I for III Quarter \( = \frac { \text{Average of III Quarter} }{ \text{Grand Average} } \times 100 = \frac { 3.76 }{ 3.95 } \times 100 = 95.1899 \)
S.I for IV Quarter \( = \frac { \text{Average of IV Quarter} }{ \text{Grand Average} } \times 100 = \frac { 4.16 }{ 3.95 } \times 100 = 105.3165 \)
Seasonal indices are important for removing the effect of seasonal changes from time series data, allowing for better trend analysis.
In simple words: First, we find the average production for each quarter over all years. Then, we calculate the grand average of these quarterly averages. Finally, for each quarter, its seasonal index is found by dividing its average by the grand average and multiplying by 100.

🎯 Exam Tip: Ensure precise calculation of the Grand Average, as any error will propagate through all subsequent seasonal index calculations.

 

Question 14. The following figures relates to the profits of a commercial concern for 8 years Find the trend of profits by the method of three yearly moving averages.
Answer: To find the trend of profits using the three-yearly moving average method, we calculate the average profit over every three consecutive years. The total for each three-year period is then divided by three to get the moving average. This helps smooth out short-term ups and downs to show the general trend over time.

Moving averages help to smooth out short-term fluctuations, making it easier to identify underlying long-term trends in data.
In simple words: We calculate the trend by taking the average profit of three years at a time. This helps to see the main direction of profit change without small ups and downs.

🎯 Exam Tip: Remember that for an odd-numbered moving average (like 3-yearly or 5-yearly), the average value is centered against the middle year of the period.

 

Question 15. Find the trend of production by the method of a five-yearly period of moving average for the following data:
Answer: To find the trend of production using the five-yearly moving average method, we calculate the average production over every five consecutive years. This average is then placed at the center of that five-year period. If the result is a decimal, we might need a 'centered' moving average for clearer placement. A longer moving average period, like five years, helps to reveal broader trends by further dampening the impact of short-term variations.

In simple words: To find the trend, we average the production over five years and place this average in the middle of that five-year period. This helps to smooth out minor changes.

🎯 Exam Tip: When using an odd-numbered moving average, the trend value is always aligned with the central year of the period, so make sure to place it correctly in the table.

 

Question 16. The following table gives the number of small- scale units registered with the Directorate of Industries between 1985 and 1991. Show the growth on a trend line by the free hand method.
Answer: The freehand method involves plotting the given data points on a graph and then drawing a smooth, freehand curve that passes roughly through the middle of these points. This line visually represents the general trend of the data, showing the overall growth pattern of small-scale units over the years. The line aims to balance the points both above and below it.

While not mathematically precise, the freehand method offers a quick visual estimate of the trend, especially useful for initial data exploration.
In simple words: We plot the given data points on a graph and then draw a smooth line through them by hand. This line shows the general growth trend of the units.

🎯 Exam Tip: When using the freehand method, ensure your line visually represents the overall direction of the data, smoothing out minor fluctuations without connecting every point.

 

Question 17. The annual production of a commodity is given as follows: Fit a straight line trend by the method of least squares
Answer: To fit a straight line trend using the method of least squares, we first set up a calculation table. We determine the 'X' values (time) by centering the years around a middle point, usually 0. Then, we calculate \( X^2 \) and \( XY \) for each year. After summing these columns, we use specific formulas to find 'a' (the intercept) and 'b' (the slope). Finally, we put these values into the straight line equation \( Y = a + bX \) to get the trend line. The least squares method is widely used because it provides the statistically 'best' fit, minimizing the total error between the actual and estimated values.

\( a = \frac { \Sigma Y }{ n } = \frac { 1186 }{ 7 } = 169.428 \)
\( b = \frac { \Sigma XY }{ \Sigma X^2 } = \frac { 92 }{ 28 } = 3.285 \)
Therefore the required equation of the straight line is:
\( Y = a + bX \)
\( Y = 169.428 + 3.285 X \)
\( Y = 169.428 + 3.285 (x - 1998) \)
In simple words: We find values 'a' and 'b' using sums from a table. 'a' is the average of Y, and 'b' is the slope. We then put these into \( Y = a + bX \) to get the trend equation.

🎯 Exam Tip: Always verify your \( \Sigma X \) is zero when using the origin-shifting method; otherwise, you must use the full normal equations.

 

Question 18. Determine the equation of a straight line which best fits the following data Compute the trend values for all years from 2000 to 2004
Answer: First, we set up a table to help calculate the trend values using the least squares method. We assign 'X' values to the years, usually with the middle year as 0. Then we find \( \Sigma Y \), \( \Sigma X \), \( \Sigma X^2 \), and \( \Sigma XY \). Using these sums, we calculate 'a' and 'b' from their formulas. Once we have the trend line equation \( Y = a + bX \), we can find the trend value (Yt) for each year by plugging in its corresponding 'X' value. Trend values help businesses forecast future performance and adjust their strategies based on predicted long-term movements.

\( a = \frac { \Sigma Y }{ n } = \frac { 270 }{ 5 } = 54 \)
\( b = \frac { \Sigma XY }{ \Sigma X^2 } = \frac { 54 }{ 10 } = 5.4 \)
Therefore, the required equation of the straight line trend is given by:
\( Y = a + bX \)
\( Y = 54 + 5.4X \)
\( Y = 54 + 5.4 (x - 2002) \)
The trend value can be obtained as follows:
When \( x = 2000 \)
\( Yt = 54 + 5.4 (2000 - 2002) \)
\( Y = 54 + 5.4 (-2) \)
\( = 54 - 10.8 \)
\( = 43.2 \)
When \( x = 2001 \)
\( Yt = 54 + 5.4 (2001 - 2002) \)
\( Y = 54 + 5.4 (-1) \)
\( = 54 - 5.4 \)
\( = 48.6 \)
When \( x = 2002 \)
\( Yt = 54 + 5.4 (2002 - 2002) \)
\( Y = 54 + 5.4 (0) \)
\( = 54 \)
When \( x = 2003 \)
\( Yt = 54 + 5.4 (2003 - 2002) \)
\( Y = 54 + 5.4 (1) \)
\( = 54 + 5.4 \)
\( = 59.4 \)
When \( x = 2004 \)
\( Yt = 54 + 5.4 (2004 - 2002) \)
\( Y = 54 + 5.4(2) \)
\( = 54 + 10.8 \)
\( = 64.8 \)
In simple words: We calculate 'a' and 'b' from the data to get the trend line equation. Then, we use this equation to find the trend value for each year by putting in the specific year's 'X' value.

🎯 Exam Tip: Double-check the X values, especially when the origin is shifted. A mistake here will affect all subsequent calculations of 'a', 'b', and trend values.

 

Question 19. The sales of a commodity in tones varied from January 2010 to December 2010 as follows: in year 2010 Sales (in tones) Fit a trend line by the method of semi-average
Answer: To fit a trend line using the semi-average method, we first divide the data into two equal halves. For data with an even number of periods, like twelve months, this is straightforward (first six, last six). We then calculate the average for the dependent variable (sales, in this case) for each half. These two averages represent the mid-point of each half. A straight line connecting these two mid-points forms the trend line. The semi-average method is simpler than least squares and is useful for quickly identifying a linear trend in data.

In simple words: We split the year's sales data into two equal halves. We then find the average sales for each half and draw a line connecting these two average points to show the trend.

🎯 Exam Tip: Remember to divide the data into two equal parts and calculate the mean for each part, then plot these two means to draw the trend line.

 

Question 19. The sales of a commodity in tones varied from January 2010 to December 2010 as follows:

Fit a trend line by the method of semi-average.
Answer: To find the trend line using the semi-average method for data with an even number of periods, we first divide the data into two equal halves. Then, we calculate the average for each half. For the given 2010 sales data, which has 12 months, we divide it into two groups of 6 months each.
The first half covers January to June, and the second half covers July to December. These two averages represent the semi-averages, which can be used to plot a trend line.

In simple words: We split the yearly sales data into two equal parts: the first six months and the last six months. We then find the average sales for each of these halves. These two average points help us understand the general trend in sales over the year.

🎯 Exam Tip: When using the semi-average method for an even number of data points, divide the data into two perfectly equal halves to calculate the two semi-averages.

 

Question 20. Use the method of monthly averages to find the monthly indices for the following data of production of a commodity for the years 2002, 2003 and 2004.
Answer: To find the monthly indices, we first gather the production data for each month over the given years. We then calculate the total and average production for each month across all years. After finding the overall grand average of these monthly averages, we compute the Seasonal Index (S.I.) for each month by dividing its monthly average by the grand average and multiplying by 100. This helps us understand the typical variations in production for each month throughout the year. The table below shows the production data, monthly totals, monthly averages, and the calculated seasonal indices:

The Grand Average is calculated from the sum of the 12 Monthly Averages:
\( Grand \, Average = \frac{Sum \, of \, 12 \, Monthly \, Averages}{12} \)
\( = \frac{17.6+20.3+18+14.3+14+17+20.6+18+17.3+17+16.3+17.6}{12} \)
\( = \frac{208}{12} \)
\( = 17.33 \)

Now we calculate the Seasonal Index (S.I.) for each month:
\( S.I. \, for \, January = \frac{Monthly \, Average \, (for \, Jan)}{Grand \, Average} \times 100 = \frac{17.6}{17.33} \times 100 = 101.56 \)
\( S.I. \, for \, February = \frac{Monthly \, Average \, (for \, Feb)}{Grand \, Average} \times 100 = \frac{20.3}{17.33} \times 100 = 117.14 \)
\( S.I. \, for \, March = \frac{Monthly \, Average \, (for \, Mar)}{Grand \, Average} \times 100 = \frac{18}{17.33} \times 100 = 103.86 \)
\( S.I. \, for \, April = \frac{Monthly \, Average \, (for \, Apr)}{Grand \, Average} \times 100 = \frac{14.3}{17.33} \times 100 = 82.52 \)
\( S.I. \, for \, May = \frac{Monthly \, Average \, (for \, May)}{Grand \, Average} \times 100 = \frac{14}{17.33} \times 100 = 80.78 \)
\( S.I. \, for \, June = \frac{Monthly \, Average \, (for \, Jun)}{Grand \, Average} \times 100 = \frac{17}{17.33} \times 100 = 98.10 \)
\( S.I. \, for \, July = \frac{Monthly \, Average \, (for \, Jul)}{Grand \, Average} \times 100 = \frac{20.6}{17.33} \times 100 = 118.87 \)
\( S.I. \, for \, August = \frac{Monthly \, Average \, (for \, Aug)}{Grand \, Average} \times 100 = \frac{18}{17.33} \times 100 = 103.87 \)
\( S.I. \, for \, September = \frac{Monthly \, Average \, (for \, Sep)}{Grand \, Average} \times 100 = \frac{17.3}{17.33} \times 100 = 100 \)
\( S.I. \, for \, October = \frac{Monthly \, Average \, (for \, Oct)}{Grand \, Average} \times 100 = \frac{17}{17.33} \times 100 = 98.10 \)
\( S.I. \, for \, November = \frac{Monthly \, Average \, (for \, Nov)}{Grand \, Average} \times 100 = \frac{16.3}{17.33} \times 100 = 94.06 \)
\( S.I. \, for \, December = \frac{Monthly \, Average \, (for \, Dec)}{Grand \, Average} \times 100 = \frac{17.6}{17.33} \times 100 = 101.56 \)
In simple words: We find the average production for each month over three years. Then, we find an overall average from all these monthly averages. Finally, we calculate an "index" for each month by comparing its average to the overall average. This index tells us how typical that month's production is compared to the whole year.

🎯 Exam Tip: Ensure that the Grand Average is calculated correctly from the monthly averages, not from the raw data, for accurate seasonal index computation. Rounding too early can affect final index values.

 

Question 21. Use the method of monthly averages to find T the monthly indices for the following data of production of a commodity for the years 2002, 2003 and 2004.
Answer: To determine the seasonal indices for the production data, we first organize the quarterly production values for each year. We then calculate the total and average production for each quarter across all given years. After this, we find the overall grand average of these quarterly averages. Finally, the Seasonal Index (S.I.) for each quarter is computed by dividing its quarterly average by the grand average and multiplying by 100. This process helps us identify typical seasonal patterns in production. The table below provides the quarterly production data, quarterly totals, averages, and seasonal indices:

The Grand Average is calculated from the sum of the 4 Quarterly Averages:
\( Grand \, Average = \frac{Sum \, of \, 4 \, Quarterly \, Averages}{4} \)
\( = \frac{74.4+71.6+72.4+72.8}{4} \)
\( = \frac{291.2}{4} \)
\( = 72.8 \)

Now we calculate the Seasonal Index (S.I.) for each quarter:
\( S.I. \, for \, I \, Quarter = \frac{Average \, of \, I \, Quarter}{Grand \, Average} \times 100 = \frac{74.4}{72.8} \times 100 = 102.20 \)
\( S.I. \, for \, II \, Quarter = \frac{Average \, of \, II \, Quarter}{Grand \, Average} \times 100 = \frac{71.6}{72.8} \times 100 = 98.35 \)
\( S.I. \, for \, III \, Quarter = \frac{Average \, of \, III \, Quarter}{Grand \, Average} \times 100 = \frac{72.4}{72.8} \times 100 = 99.45 \)
\( S.I. \, for \, IV \, Quarter = \frac{Average \, of \, IV \, Quarter}{Grand \, Average} \times 100 = \frac{72.8}{72.8} \times 100 = 100 \)
In simple words: We find the average production for each quarter over the years. Then, we calculate one big average from all these quarterly averages. For each quarter, we find an "index" by comparing its average to the big average. This shows how busy each quarter usually is.

🎯 Exam Tip: When dealing with quarterly data, always ensure your averages are grouped by quarter across all years, not within a single year.

 

Question 22. The following table shows the number of salesmen working for a certain concern:

Use the method of least squares to fit a straight line and estimate the number of salesmen in 1997.
Answer: To fit a straight line trend using the method of least squares, we first set up a calculation table. We define a new time variable X, usually by setting the middle year to 0. For an odd number of years, this is straightforward. We then calculate \( \sum Y \), \( \sum X \), \( \sum X^2 \), and \( \sum XY \). Using these sums, we find the constants 'a' and 'b' for the trend equation \( Y = a + bX \). Finally, we use this equation to estimate future values, like the number of salesmen in 1997. This method minimizes the sum of squared differences between actual and estimated values.

From the table, we have:
\( N = 5 \)
\( \sum Y = 244 \)
\( \sum X = 0 \)
\( \sum X^2 = 10 \)
\( \sum XY = 20 \)

Now, we calculate 'a' and 'b':
\( a = \frac{\sum Y}{n} = \frac{244}{5} = 48.8 \)
\( b = \frac{\sum XY}{\sum X^2} = \frac{20}{10} = 2 \)

The required equation of the straight line trend is:
\( Y = a + bX \)
\( Y = 48.8 + 2X \)
We can also write this as: \( Y = 48.8 + 2(x - 1994) \)

Let's calculate the trend values for each year:
For x = 1992:
\( Y_t = 48.8 + 2(1992 - 1994) \)
\( Y_t = 48.8 + 2(-2) \)
\( Y_t = 48.8 - 4 \)
\( Y_t = 44.8 \)

For x = 1993:
\( Y_t = 48.8 + 2(1993 - 1994) \)
\( Y_t = 48.8 + 2(-1) \)
\( Y_t = 48.8 - 2 \)
\( Y_t = 46.8 \)

For x = 1994:
\( Y_t = 48.8 + 2(1994 - 1994) \)
\( Y_t = 48.8 + 2(0) \)
\( Y_t = 48.8 \)

For x = 1995:
\( Y_t = 48.8 + 2(1995 - 1994) \)
\( Y_t = 48.8 + 2(1) \)
\( Y_t = 48.8 + 2 \)
\( Y_t = 50.8 \)

For x = 1996:
\( Y_t = 48.8 + 2(1996 - 1994) \)
\( Y_t = 48.8 + 2(2) \)
\( Y_t = 48.8 + 4 \)
\( Y_t = 52.8 \)

To estimate the number of salesmen in 1997:
For x = 1997:
\( Y_t = 48.8 + 2(1997 - 1994) \)
\( Y_t = 48.8 + 2(3) \)
\( Y_t = 48.8 + 6 \)
\( Y_t = 54.8 \)
Therefore, the estimated number of salesmen in 1997 is 54.8.
In simple words: We find a straight line that best fits the given sales data. This line helps us predict future sales. We calculate two special numbers, 'a' and 'b', to create this line's equation. Then, we use the equation to guess how many salesmen there would be in 1997.

🎯 Exam Tip: When using the least squares method, carefully choose the origin (the '0' point for X) to simplify calculations, especially when there's an odd number of data points, as this makes \( \sum X = 0 \).