Samacheer Kalvi Class 12 Business Maths Solutions Chapter 9 Applied Statistics Exercise 9.4

Question 1. A time series is a set of data recorded
(a) Periodically
(b) Weekly
(c) successive points of time
(d) All of the options
Answer: (d) All of the options
In simple words: A time series is like a list of numbers collected over time. These numbers are often taken at regular periods, such as every week or at specific points in time, to see how things change. This helps to observe patterns and make predictions.

🎯 Exam Tip: Remember that a time series specifically deals with data collected at regular intervals, which helps in identifying trends and seasonal changes.

Question 2. A time series consists of
(a) Five components
(b) Four components
(c) Three components
(d) Two components
Answer: (b) Four components
In simple words: A time series has four main parts: trend, seasonal variation, cyclical variation, and irregular variation. These parts together explain how the data changes over time. Understanding each part helps in forecasting.

🎯 Exam Tip: Be sure to know the names and definitions of all four components of a time series: Trend, Seasonal, Cyclical, and Irregular.

Question 3. The components of a time series which is attached to short term fluctuation is
(a) Secular trend
(b) Seasonal variations
(c) Cyclic variation
(d) Irregular variation
Answer: (d) Irregular variation
In simple words: Irregular variations are sudden, unexpected changes in a time series that are not predictable. They are often caused by unusual events like natural disasters or unexpected strikes. These fluctuations are short-term and random.

🎯 Exam Tip: Short-term fluctuations that are unpredictable are typically irregular variations, while predictable, recurring short-term patterns are seasonal.

Question 4. Factors responsible for seasonal variations are
(a) Weather
(b) Festivals
(c) Social customs
(d) All of the options
Answer: (d) All of the options
In simple words: Seasonal changes in data happen because of regular patterns linked to the time of year. Things like weather, holidays, and cultural traditions all make these variations happen. For example, ice cream sales go up in summer due to weather.

🎯 Exam Tip: Seasonal variations occur within a year and are predictable. Think about how daily life changes with seasons, holidays, and common practices.

Question 5. The additive model of the time series with the components T, S, C and I is
(a) \( y = T + S + C \times I \)
(b) \( y = T + S \times C \times I \)
(c) \( y = T + S + C + I \)
(d) \( y = T + S \times C + I \)
Answer: (c) \( y = T + S + C + I \)
In simple words: The additive model assumes that the four parts of a time series (Trend, Seasonal, Cyclical, Irregular) simply add up to make the total value. This means the strength of these parts does not change with the overall level of the data.

🎯 Exam Tip: Remember that in the additive model, components are added together, whereas in the multiplicative model, they are multiplied.

Question 6. Least square method of fitting a trend is
(a) Most exact
(b) Least exact
(c) Full of subjectivity
(d) Mathematically unsolved
Answer: (a) Most exact
In simple words: The least squares method is a mathematical way to find the line that best fits a set of data points. It does this by making the sum of the squared differences between the actual data points and the line as small as possible. This makes it a very precise method.

🎯 Exam Tip: The least squares method is widely used because it provides an objective and mathematically precise way to determine the best-fit line for a trend.

Question 7. The value of 'b' in the trend line \( y = a + bx \) is
(a) Always positive
(b) Always negative
(c) Either positive or negative
(d) Zero
Answer: (c) Either positive or negative
In simple words: In the equation \( y = a + bx \), 'b' stands for the slope of the trend line. This slope can go up (positive), go down (negative), or sometimes stay flat (zero). It shows the direction and strength of the trend in the data.

🎯 Exam Tip: The 'b' value (slope) determines if a trend is increasing (positive), decreasing (negative), or has no linear change (zero).

Question 8. The component of a time series attached to long term variation is known as
(a) Random cause
(b) Secular variations
(c) Irregular variation
(d) Seasonal variations
Answer: (b) Secular variations
In simple words: Secular variations, also known as trends, show the long-term, underlying movement of a time series. This can be an increase, decrease, or staying level over many years. It represents the general direction of the data.

🎯 Exam Tip: Long-term changes and underlying growth or decline over several years are always described by the trend or secular variation component.

Question 9. The seasonal variation means the variations occurring within
(a) A number of years
(b) within a year
(c) within a month
(d) within a week
Answer: (b) within a year
In simple words: Seasonal variations are patterns that repeat over a specific period, usually less than a year. These patterns are often influenced by things like seasons, holidays, or recurring social events. They are predictable and regular.

🎯 Exam Tip: Seasonal variations are characterized by their recurring nature within a single year, making them distinct from cyclical or irregular variations.

Question 10. Another name of consumer's price index number is:
(a) Whole-sale price index number
(b) Cost of living index
(c) Sensitive
(d) Composite
Answer: (b) Cost of living index
In simple words: The Consumer Price Index (CPI) measures changes in the prices of goods and services that households commonly buy. This is also called the cost of living index because it shows how much more or less expensive everyday life has become for people. It helps track inflation.

🎯 Exam Tip: Remember that the Consumer Price Index (CPI) is used to track changes in the cost of a typical basket of goods and services, directly reflecting the cost of living.

Question 11. Cost of living at two different cities can be compared with the help of
(a) Consumer price index
(b) Value index
(c) Volume index
(d) Un-weighted index
Answer: (a) Consumer price index
In simple words: The Consumer Price Index (CPI) helps us compare how much it costs to live in different places. By looking at how prices for common items vary, we can understand which city is more expensive. It measures the average change over time in the prices paid by urban consumers.

🎯 Exam Tip: When comparing living costs between locations, the Consumer Price Index is the most appropriate tool as it reflects the price level of essential goods and services.

Question 12. Laspeyre's index = 110, Paasche's index = 108, then Fisher's Ideal index is equal to:
(a) 110
(b) 108
(c) 100
(d) 109
Answer: (d) 109
In simple words: Fisher's Ideal Index is found by taking the square root of the product of Laspeyre's and Paasche's indices. It combines the strengths of both methods to give a more balanced measure. This helps provide a fairer average of price changes.

🎯 Exam Tip: Always remember the formula for Fisher's Ideal Index: \( \sqrt{\text{Laspeyre's Index} \times \text{Paasche's Index}} \). Round to the nearest whole number if the options are integers.

Question 13. Most commonly used index number is:
(a) Volume index number
(b) Value index number
(c) Price index number
(d) Simple index number
Answer: (c) Price index number
In simple words: Price index numbers are very widely used because they help us understand how prices change over time for goods and services. They are crucial for tracking inflation and how much money is worth. The Consumer Price Index is a well-known example.

🎯 Exam Tip: The most common type of index number is the price index, as it directly impacts economic analysis and daily life through measures like inflation.

Question 14. Consumer price index are obtained by:
(a) Paasche's formula
(b) Fisher's ideal formula
(c) Marshall Edgeworth formula
(d) Family budget method formula
Answer: (d) Family budget method formula
In simple words: Consumer Price Index (CPI) numbers are usually calculated using the family budget method. This method considers the spending habits of a typical household to figure out how much the cost of essential goods and services has changed. It makes sure the index reflects real-world expenses.

🎯 Exam Tip: While various formulas exist for index numbers, the family budget method is specifically used for calculating the Consumer Price Index, reflecting household expenditure patterns.

Question 15. Which of the following Index number satisfy the time reversal test?
(a) Laspeyre's Index number
(b) Paasche's Index number
(c) Fisher Index number
(d) All of the options
Answer: (c) Fisher Index number
In simple words: The time reversal test checks if an index number works correctly when you swap the base period and the current period. Fisher's Ideal Index is special because it passes this test, meaning it gives a consistent result no matter which period you use as the base. This makes it very reliable.

🎯 Exam Tip: Fisher's Ideal Index is considered 'ideal' because it satisfies several important tests, including the time reversal test and factor reversal test.

Question 16. While computing a weighted index, the current period quantities are used in the:
(a) Laspeyre's method
(b) Paasche's method
(c) Marshall Edgeworth method
(d) Fisher's ideal method
Answer: (b) Paasche's method
In simple words: Paasche's method uses the quantities from the current period as weights when calculating an index number. This means it reflects the most recent consumption or production patterns. It helps provide an up-to-date picture of price changes.

🎯 Exam Tip: Distinguish between Laspeyre's and Paasche's methods: Laspeyre's uses base period quantities as weights, while Paasche's uses current period quantities.

Question 17. The quantities that can be numerically measured can be plotted on a
(a) p-chart
(b) c-chart
(c) \( \overline{x} \)-chart
(d) np-chart
Answer: (c) \( \overline{x} \)-chart
In simple words: An \( \overline{x} \)-chart is used to keep track of qualities that you can measure with numbers, like length or weight. It shows the average (mean) of samples taken over time. This helps to see if a process is staying in control.

🎯 Exam Tip: Remember that \( \overline{x} \)-charts (mean charts) and R-charts (range charts) are variable control charts, used for characteristics that can be measured.

Question 18. How many causes of variation will affect the quality of a product?
(a) 4
(b) 3
(c) 2
(d) 1
Answer: (c) 2
In simple words: There are two main reasons why a product's quality might change: common causes and special causes. Common causes are normal, random variations always present in a process. Special causes are unusual, identifiable reasons for variation that are not part of the normal process.

🎯 Exam Tip: Understand the difference between common causes (inherent to the process, random) and assignable/special causes (specific, identifiable, non-random). These are the two types of variation causes.

Question 19. Variations due to natural disorder is known as
(a) random cause
(b) non-random cause
(c) human cause
(d) All of the options
Answer: (a) random cause
In simple words: Variations that happen due to natural, unidentifiable reasons within a process are called random causes or common causes. These are small, unpredictable changes that are always present and cannot be easily fixed without changing the entire process. They are part of the system's inherent variability.

🎯 Exam Tip: Random causes are inherent, unpredictable variations within a stable process, often referred to as common causes, and are not easily eliminated.

Question 20. The assignable causes can occur due to
(a) poor raw materials
(b) unskilled labour
(c) faulty machines
(d) All of the options
Answer: (d) All of the options
In simple words: Assignable causes are specific, identifiable reasons for variation that are not part of the normal, random process. These causes can be found and fixed. Examples include using bad materials, workers who are not trained well, or machines that are broken.

🎯 Exam Tip: Assignable causes are typically external to the normal system operation and can be identified and corrected, leading to process improvement.

Question 21. A typical control chart consists of
(a) CL, LCL
(b) CL, LCL, UCL
(c) UCL, LCL
(d) All of the options
Answer: (b) CL, LCL, UCL
In simple words: A standard control chart has three main lines: a Center Line (CL), an Upper Control Limit (UCL), and a Lower Control Limit (LCL). The CL shows the average of the process. The UCL and LCL show the boundaries within which the process is considered in control. These lines help identify if a process is stable.

🎯 Exam Tip: Always include the Center Line (CL), Upper Control Limit (UCL), and Lower Control Limit (LCL) when describing components of a control chart.

Question 22. \( \overline{x} \)-chart is a
(a) attribute control chart
(b) variable control chart
(c) neither Attribute nor variable control chart
(d) both Attribute and variable control chart
Answer: (b) variable control chart
In simple words: An \( \overline{x} \)-chart is used for characteristics that can be measured on a continuous scale, such as length, weight, or temperature. This makes it a variable control chart. It helps monitor the average value of a process over time.

🎯 Exam Tip: Remember that \( \overline{x} \)-charts are used for variables (measurable data), while p-charts and np-charts are used for attributes (count data, like defects).

Question 23. R is calculated using
(a) \( X_{\text{max}} - X_{\text{min}} \)
(b) \( \overline{X}_{\text{max}} - \overline{X}_{\text{min}} \)
(c) \( \overline{x}_{\text{max}} - \overline{x}_{\text{min}} \)
(d) \( \overline{\overline{x}}_{\text{max}} - \overline{\overline{x}}_{\text{min}} \)
Answer: (a) \( X_{\text{max}} - X_{\text{min}} \)
In simple words: 'R' stands for range, which is a measure of how spread out the data points are within each sample. To find the range, you simply subtract the smallest value (\( X_{\text{min}} \)) from the largest value (\( X_{\text{max}} \)) in a given sample. It helps understand the variability of a process.

🎯 Exam Tip: The range (R) is calculated from the individual observations within each subgroup, representing the difference between the maximum and minimum values in that subgroup.

Question 24. The upper control limit for \( \overline{x} \)-chart is given by
(a) \( \overline{x} + A_2\overline{R} \)
(b) \( \overline{\overline{x}} + A_2R \)
(c) \( \overline{\overline{x}} + A_2\overline{R} \)
(d) \( \overline{x} + A_2\overline{\overline{R}} \)
Answer: (c) \( \overline{\overline{x}} + A_2\overline{R} \)
In simple words: For an \( \overline{x} \)-chart, the Upper Control Limit (UCL) is found by adding the average of the sample means (\( \overline{\overline{x}} \)) to a multiple of the average range (\( \overline{R} \)). The \( A_2 \) factor is a constant that depends on the sample size. This formula helps determine the upper boundary for stable process variation.

🎯 Exam Tip: Always remember the correct formula for UCL of an \( \overline{x} \)-chart: \( \text{UCL}_{\overline{x}} = \overline{\overline{x}} + A_2\overline{R} \), and ensure you use the correct average of sample means and average range.

Question 25. The LCL for R chart is given by
(a) \( D_2\overline{x} \)
(b) \( D_2\overline{\overline{R}} \)
(c) \( D_3\overline{\overline{R}} \)
(d) \( D_3\overline{x} \)
Answer: (d) \( D_3\overline{x} \)
In simple words: For an R-chart, which tracks the range of samples, the Lower Control Limit (LCL) is calculated using the constant \( D_3 \) multiplied by the average range (\( \overline{R} \)). The \( D_3 \) value depends on the sample size. This limit helps identify if the process variability is unusually low.

🎯 Exam Tip: The LCL for an R-chart is \( D_3\overline{R} \). Ensure you use the correct constant \( D_3 \) and the average of the sample ranges for accurate calculations.