All questions of Chapter 13: Statistical representation and Sampling for CA Foundation Exam

Given Information:

- Total number of women = 40%
- Percentage of coffee drinkers = 45%
- Percentage of male coffee drinkers = 20%

To Find: Percentage of female not-coffee drinkers

Solution:

Let's assume that the total number of people in the area is 100.

- Total number of women = 40% of 100 = 40
- Total number of men = 100 - 40 = 60

We know that the percentage of coffee drinkers is 45%.

- Total number of coffee drinkers = 45% of 100 = 45

We also know that the percentage of male coffee drinkers is 20%.

- Total number of male coffee drinkers = 20% of 100 = 20

Using the above information, we can calculate the total number of female coffee drinkers.

- Total number of female coffee drinkers = Total number of coffee drinkers - Total number of male coffee drinkers
- Total number of female coffee drinkers = 45 - 20 = 25

Now, we can calculate the total number of female not-coffee drinkers.

- Total number of female not-coffee drinkers = Total number of women - Total number of female coffee drinkers
- Total number of female not-coffee drinkers = 40 - 25 = 15

Finally, we need to find the percentage of female not-coffee drinkers.

- Percentage of female not-coffee drinkers = (Total number of female not-coffee drinkers / Total number of people) x 100%
- Percentage of female not-coffee drinkers = (15 / 100) x 100%
- Percentage of female not-coffee drinkers = 15%

Therefore, the correct answer is option (B) 15.

Data presentation is an essential aspect of data analysis. Effective data presentation can help in understanding complex data sets and communicating insights to stakeholders. Different methods of data presentation are available, and each method has its advantages and disadvantages. The most attractive method of data presentation is diagrammatic. The reasons are explained below:

Advantages of Diagrammatic Data Presentation:

1. Easy to understand: Diagrammatic data presentation is easy to understand because it uses visual aids like graphs, charts, and diagrams to represent data. These visual aids help in presenting complex data sets in a simplified and easy-to-understand format.

2. Provides a clear picture: Diagrammatic data presentation provides a clear picture of the data, making it easy to identify trends, patterns, and outliers. This helps in making informed decisions based on the data.

3. Useful for comparisons: Diagrammatic data presentation is useful for making comparisons between different variables or data sets. It allows the viewer to compare data sets side by side, making it easy to identify differences and similarities.

4. Attractive: Diagrammatic data presentation is attractive and visually appealing. It can capture the viewer's attention and keep them engaged, making it more likely that they will retain the information presented.

Disadvantages of Tabular and Textual Data Presentation:

1. Difficult to understand: Tabular and textual data presentation can be difficult to understand, especially for complex data sets. It often requires a lot of effort and time to interpret the data correctly.

2. Lack of clarity: Tabular and textual data presentation can lack clarity, making it challenging to identify trends, patterns, and outliers.

3. Not suitable for comparisons: Tabular and textual data presentation is not suitable for making comparisons between different variables or data sets. It can be challenging to compare data sets side by side, making it difficult to identify differences and similarities.

Conclusion:

In conclusion, diagrammatic data presentation is the most attractive method of data presentation because it is easy to understand, provides a clear picture of the data, useful for making comparisons, and visually appealing. Tabular and textual data presentation, on the other hand, can be challenging to understand, lack clarity, and are not suitable for making comparisons.

• Attribute: An attribute is a qualitative characteristic of an object or a person. Attributes are categorical in nature and cannot be measured on a numerical scale. Examples of attributes include gender, nationality, and hair color. Annual income is a quantitative characteristic, as it can be measured and expressed numerically. Thus, it is not an attribute.


• Discrete Variable: A discrete variable is a type of quantitative variable that takes on a finite set of distinct values. These values are typically integers or whole numbers and have gaps between them. Examples of discrete variables include the number of students in a class, the number of cars in a parking lot, or the number of siblings a person has. Annual income, however, can take on any value within a range and is not restricted to whole numbers. Therefore, it is not a discrete variable.


• Continuous Variable: A continuous variable is a type of quantitative variable that can take on an infinite number of values within a given range. Continuous variables are measured on a continuous scale and can have decimal or fractional values. Examples of continuous variables include height, weight, and temperature. Annual income can take on any value within a range and can be measured to varying degrees of precision (e.g., dollars, cents, or even fractions of a cent). Thus, annual income is a continuous variable.

No of person earning above Rs.1500 are 43(36+7)Total no.of persons 86as we have to find the percentage 43÷86×100=50 Therefore the ans is option a)50.

Let's assume the total population is 100.
Out of which 40% are women.
ie; 40 women & 60 men.
Given that, 45 %of the total population drink coffee.
20% of the population who drink coffee are Male.
so the percentage of the female coffee drinker (45-20) = 25%.
If females = 40% and 25% drinks coffee.
Then females who don't drink Coffee = (40%-25%) = 15.

Explanation:

- Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data.
- The word "statistics" is derived from the Latin word "status," which means "political state" or "condition."
- The word "statista" is not an Italian word, and it has no connection to the origin of the word "statistics."
- The French word for statistics is "statistique," which is similar to the English word.
- Therefore, the false statement is option C, which claims that "Statistics is derived from the French word Statistik."

Arrangement of Diagrams

Pie-diagram, bar-diagram and cubic diagram are different types of diagrams used for data representation. The arrangement of these diagrams can be done based on the dimension of the data being represented. The correct arrangement of the diagrams is as follows:

Dimension 1: 2, 1, 3

- The first dimension is represented by a pie diagram.
- The second dimension is represented by a bar diagram.
- The third dimension is represented by a cubic diagram.

Explanation

Pie-diagram

- A pie diagram is a circular chart that represents data in slices.
- The size of each slice is proportional to the quantity it represents.
- It is used to represent data that can be divided into parts, such as percentages, fractions, or ratios.
- The first dimension, which consists of three values 2, 1, and 3, can be represented using a pie diagram.

Bar-diagram

- A bar diagram is a chart that represents data using rectangular bars.
- The height or length of each bar is proportional to the quantity it represents.
- It is used to represent data that can be measured or counted, such as sales, population, or temperature.
- The second dimension, which consists of three values 2, 1, and 3, can be represented using a bar diagram.

Cubic diagram

- A cubic diagram is a three-dimensional representation of data using cubes.
- The size of each cube is proportional to the quantity it represents.
- It is used to represent data that has three dimensions, such as sales by product, region, and time.
- The third dimension, which consists of three values 2, 1, and 3, can be represented using a cubic diagram.

Conclusion

The correct arrangement of the diagrams for the given values is 2, 1, 3, which means that the pie diagram represents the first dimension, the bar diagram represents the second dimension, and the cubic diagram represents the third dimension.

A bar chart or bar graph is a chart or graph that presents categorical data with rectangular bars with heights or lengths proportional to the values that they represent. The bars can be plotted vertically or horizontally. A vertical bar chart is sometimes called a line graph.

Explanation:
Cumulative frequency distribution is a method of representing data by showing the number of observations that fall below a certain value on a graph. The graph is called an ogive or cumulative frequency curve.

Steps to create an ogive:
1. Arrange the data in ascending order.
2. Calculate the cumulative frequencies by adding up the frequencies of each value and all the values below it.
3. Draw a graph with the x-axis representing the data values and the y-axis representing the cumulative frequencies.
4. Plot the points with the data values on the x-axis and the corresponding cumulative frequencies on the y-axis.
5. Connect the points with a smooth curve to form an ogive.

Advantages of using an ogive:
1. It gives a clear picture of how the data is distributed.
2. It helps to identify the median and quartiles.
3. It helps to compare two or more data sets.

Disadvantages of using an ogive:
1. It can be difficult to interpret for people who are not familiar with statistics.
2. It may not be suitable for large data sets.

Therefore, the correct answer is option 'B' - Ogive.

Error (statistical error) describes the difference between a value obtained from a data collection process and the 'true' value for the population. The greater the error, the less representative the data are of the population. Data can be affected by two types of error: sampling error and non-sampling error.

Explanation:
- Census reports are published by government authorities after collecting data from individuals or households.
- This data is not collected specifically for any research purpose, but rather for administrative purposes such as population counts and resource allocation.
- Therefore, the data collected on religion from census reports is considered to be secondary data, as it is data that has already been collected and published by another organization for a different purpose.
- Secondary data is data that has been collected by someone else for their own purposes, but which can be reused for research or other purposes.

Frequency Polygon

A frequency polygon is a graphical representation of a frequency distribution. It is a line graph that displays the frequency data of a continuous variable.

Steps to Draw a Frequency Polygon

To draw a frequency polygon, follow the steps below:

1. Calculate the class midpoints
2. Calculate the class frequencies
3. Plot the class midpoints on the x-axis
4. Plot the class frequencies on the y-axis
5. Join the successive points with the line segments

True or False

For obtaining frequency polygon we join the successive points whose abscissa represent the corresponding class frequency.

The statement is False.

Explanation

In a frequency polygon, we join the successive points whose abscissa represent the corresponding class midpoint. The class midpoint is the average of the lower and upper class limits.

The class frequency is the number of observations in each class interval.

Therefore, to obtain a frequency polygon, we need to plot the class midpoints on the x-axis and the class frequencies on the y-axis. We then join the successive points with line segments to form the polygon.

20% are female -30% of female are not workers 

It's so simple,
person with income more than 1500=36+7=43
percentage=43/86*100
percentage=1/2*100=50
answer=50%
So, option A is correct

Errors in Statistics

Errors in statistics are the differences between the actual value and the estimated value of a statistical quantity.

Types of Errors

There are two types of errors in statistics:

1. Sampling Error - It is the error that arises due to the use of a sample to estimate a population parameter. It occurs because the sample is only a part of the population, and there may be some variation in the sample that is not representative of the population.

2. Non-Sampling Error - It is the error that arises due to factors other than sampling. These factors include errors in data collection, errors in data processing, errors in measurement, errors in estimation, etc.

Answer

According to the question, the number of errors in statistics is two, as option 'B' states. However, the question does not specify the type of error. Therefore, it is not possible to determine whether the errors are sampling or non-sampling errors.

-Less than ogive : Plot the points with the upper limits of the class as abscissae and the corresponding less than cumulative frequencies as ordinates. The points are joined by free hand smooth curve to give less than cumulative frequency curve or the less than Ogive. It is a rising curve.

Given:
Total employees in the office = 200
Married employees = 150
Total male employees = 160
Married male employees = 120

To find: Number of female unmarried employees

Solution:
Let's first calculate the number of married female employees:
Married female employees = Total married employees - Married male employees
= 150 - 120
= 30

Now, we can find the number of female unmarried employees:
Female unmarried employees = Total female employees - Married female employees
= (Total employees - Total male employees) - Married female employees
= (200 - 160) - 30
= 10

Therefore, the number of female unmarried employees is 10, which is option B.

Statistics is concerned with both qualitative and quantitative information.

Qualitative Information

Qualitative information is descriptive data that cannot be measured numerically. It is concerned with the qualities, characteristics, and properties of a phenomenon. Examples of qualitative information include:

- Colors
- Tastes
- Shapes
- Smells
- Opinions
- Attitudes
- Emotions

In statistics, qualitative information is often represented using nominal or ordinal scales.

Quantitative Information

Quantitative information is numerical data that can be measured and analyzed statistically. It is concerned with the quantities, amounts, and magnitudes of a phenomenon. Examples of quantitative information include:

- Height
- Weight
- Age
- Income
- Temperature
- Time
- Distance

In statistics, quantitative information is often represented using interval or ratio scales.

Why Both are Important?

Both qualitative and quantitative information are important in statistics because they provide different types of information that can help us understand a phenomenon. Qualitative information can provide insights into the attitudes, beliefs, and experiences of people, while quantitative information can provide precise measurements and statistical analyses that can help us make predictions and decisions. In many cases, a combination of both qualitative and quantitative information is needed to fully understand a phenomenon.

Drinking habit of a person is an attribute.

Explanation:

An attribute is a characteristic that describes an object or a person. It is a qualitative variable that cannot be measured numerically. The drinking habit of a person is a characteristic that describes the person's behavior or tendency to consume alcohol. It is not a variable that can be measured quantitatively or numerically. Therefore, the correct answer is option 'A' - an attribute.

A continuous variable is a variable whose value is obtained by measuring. A random variable is a variable whose value is a numerical outcome of a random phenomenon. A discrete random variable X has a countable number of possible values. Example: Let X represent the sum of two dice.

Continuous Variable:

A continuous variable is a numerical variable that can take any value within a certain range. It can be measured on a continuous scale. The range of values can be divided into smaller and smaller units, and the precision of the measurement can be increased indefinitely. Examples of continuous variables include height, weight, time, temperature, and profit.

Distribution of Profits of a Blue-Chip Company:

The distribution of profits of a blue-chip company is a continuous variable. It can take any value within a certain range, which could be positive, negative, or zero. The profits can be measured on a continuous scale, such as dollars, euros, or yen. The range of profits can be divided into smaller and smaller units, such as cents or pence, and the precision of the measurement can be increased indefinitely.

Why Not Discrete Variable:

A discrete variable is a numerical variable that can take only a finite number of values or a countable number of values. It cannot be measured on a continuous scale. Examples of discrete variables include the number of employees, the number of customers, and the number of products sold. The profits of a blue-chip company cannot be a discrete variable because it can take any value within a certain range, and the range is not countable.

Why Not Attributes:

An attribute is a non-numerical variable that describes a characteristic of an object or a person. Examples of attributes include the color of a car, the gender of a person, and the brand of a product. The profits of a blue-chip company cannot be an attribute because it is a numerical variable, and it describes the financial performance of the company, not a characteristic of the company.

Conclusion:

Therefore, the correct answer is option 'B', which is a continuous variable.

The correct answer is option 'B', which is 9.5 - 14.5. Let us understand why.

Class interval is a range of values that are used to group data into smaller parts. It is commonly used in statistics and helps in the process of data analysis. In this question, we have three class intervals - 10-14, 15-19, and 20-24.

To find the first class interval, we need to determine the lower and upper limits of the first interval. We can do this by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit. This is because class intervals are usually written in terms of midpoints, and the midpoints are calculated by taking the average of the lower and upper limits.

Let us apply this formula to the first class interval, which is 10-14.

Lower limit = 10 - 0.5 = 9.5
Upper limit = 14 + 0.5 = 14.5

Therefore, the first class interval is 9.5 - 14.5.

To summarize, when given a set of class intervals, we can find the limits of each interval by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit. This helps in determining the midpoint of each interval, which is a crucial component in data analysis.

Probability Sampling is a sampling technique in which sample from a larger population are chosen using a method based on the theory of probability. Non-probability sampling is a sampling technique in which the researcher selects samples based on the subjective judgment of the researcher rather than random selection.

I'm sorry, I cannot answer without more context or information. Please provide more details or a specific question.

Given:
- 95% LCL = 48.04
- 95% UCL = 51.96
- Sample size (n) = 100

To find: Population variance

Approach:
The formula to calculate the confidence interval for population mean when the sample size (n) is greater than 30 is:

Confidence Interval (CI) = Sample Mean ± Zα/2 * (Standard Error)

Where
- Zα/2 is the Z value for the level of confidence α/2
- Standard Error = Population Standard Deviation / sqrt(n)

As the population variance is unknown, we will use the sample standard deviation as an estimate of it.

Formula to calculate the sample standard deviation (s):
s = sqrt [ Σ(xi - x̄)2 / (n - 1) ]

Where
- xi is the i-th observation
- x̄ is the sample mean

Once we have the sample standard deviation, we can calculate the standard error and then use the confidence interval formula to find the population mean.

Steps:
1. Calculate the sample mean (x̄):
x̄ = (LCL + UCL) / 2
= (48.04 + 51.96) / 2
= 50

2. Calculate the Z value:
For 95% confidence interval, α = 0.05 and α/2 = 0.025
Using a Z table or calculator, we can find the Z value for 0.025 = 1.96

3. Calculate the sample standard deviation (s):
We do not have the individual observations, so we cannot calculate the sample standard deviation.

4. Calculate the standard error:
Standard Error = s / sqrt(n)
= s / sqrt(100)
= s / 10

5. Use the confidence interval formula to find the population mean:
CI = Sample Mean ± Zα/2 * (Standard Error)
= 50 ± 1.96 * (s / 10)

We know that the confidence interval is (48.04, 51.96), so we can write two equations using the above formula and solve for s:

48.04 = 50 - 1.96 * (s / 10)
51.96 = 50 + 1.96 * (s / 10)

Solving the above equations, we get:
s = 1

6. Calculate the population variance:
Population Variance = s2 * (n - 1)
= 1^2 * (100 - 1)
= 99

Therefore, the value of the population variance when the sample size is 100 is 10 (Option B).

Explanation:

In a frequency distribution table, class marks are used to represent the midpoint of each class interval. The class mark is calculated by adding the lower and upper limits of a class interval and dividing by 2.

For the overlapping classes 0-10, 10-20, 20-30, etc., the class mark of the class 0-10 can be calculated as follows:

Class interval: 0-10
Lower limit: 0
Upper limit: 10
Class mark: (0+10)/2 = 5

Therefore, the correct answer is option 'A' which states that the class mark of the class 0-10 is 5.

Attributes and Variables

Attributes and variables are two important concepts in statistics. An attribute is a characteristic or property of an object or an individual. For example, the color of a car, the gender of a person, and the brand of a smartphone are all attributes. A variable, on the other hand, is a quantity or a value that can vary or change. A variable can be measured or observed and can take on different values.

Types of Variables

There are two main types of variables in statistics: discrete variables and continuous variables.

Discrete variables are variables that can only take on certain specific values. These values are usually integers or counts. For example, the number of children in a family, the number of cars in a parking lot, and the number of students in a classroom are all discrete variables.

Continuous variables, on the other hand, are variables that can take on any value within a certain range. These values are usually measured on a scale, such as time, weight, or temperature. For example, the height of a person, the temperature of a room, and the amount of rainfall in a city are all continuous variables.

Marks of a Student

Marks of a student are an example of a discrete variable. This is because marks can only take on certain specific values, such as 50, 60, 70, and so on. Marks cannot take on any value between these specific values, such as 63.5 or 68.9. Therefore, marks are not a continuous variable.

Conclusion

In conclusion, attributes and variables are important concepts in statistics. Marks of a student are an example of a discrete variable, which is a type of variable that can only take on certain specific values.

Tabulation is a method of presenting data in an organised manner by using tables. A table consists of rows and columns, and each column and row have a specific heading. Caption is one such heading that is used in tables.

Definition of Caption:
Caption is the upper part of a table that describes the column and sub-column.

Explanation:
The caption is a brief explanation or title that describes the table's content. It is placed at the top of the table, and it provides information about the data presented in the table. The caption should be written in a concise and clear manner, and it should give an idea of what the table is about.

The caption includes the following information:
- The title of the table: It should be a brief description of the data presented in the table.
- The column headings: The caption should include the column headings, which describe the data presented in each column.
- The sub-column headings: If there are sub-columns in the table, the caption should include the sub-column headings as well.

Importance of Caption:
The caption is an essential part of the table as it provides context to the data presented in the table. It helps the readers understand the purpose of the table and the data presented in it. A well-written caption can make the table more accessible and easier to understand.

Conclusion:
In conclusion, the caption is the upper part of a table that describes the column and sub-column. It is an essential part of the table, and it should be written in a concise and clear manner to provide context to the data presented in the table.

Open-end Class vs Closed-end Class

Open-end class refers to a class where one end is not specified. This means that the class can be expanded or contracted, and more members can be added to the class as they become available. On the other hand, a closed-end class has a specific number of members, and no more members can be added.

Examples of Open-end Class

Some examples of open-end classes include:

- A club where new members can join at any time
- A university class where students can add or drop the course during the first few weeks of the semester
- A support group for people with a particular condition or illness

Benefits of Open-end Class

Some of the benefits of an open-end class include:

- Flexibility: An open-end class is more flexible than a closed-end class, as more members can be added as needed.
- Diversity: An open-end class can be more diverse, as new members can bring different perspectives and experiences.
- Growth: An open-end class can grow over time, as more members join and contribute to the group.

Conclusion

In summary, an open-end class is a class where one end is not specified, and more members can be added as needed. This type of class offers flexibility, diversity, and the potential for growth.

Systematic sampling is a method of sampling where elements are selected at regular intervals from an ordered list. While this method has its advantages, such as being easy to implement and providing a representative sample, there is a potential drawback that needs to be considered.

Periodicity Bias:
One potential drawback of systematic sampling is that it may introduce periodicity bias. Periodicity bias occurs when there is a consistent pattern or cycle in the population that aligns with the sampling interval. For example, if a population is sorted in a way that every 10th element exhibits a similar characteristic, systematic sampling would consistently select those elements, leading to a biased sample that does not accurately represent the entire population.
By relying solely on a systematic approach, researchers may inadvertently overlook certain segments of the population and fail to capture its full diversity. This can result in skewed results and inaccurate conclusions drawn from the sample.
In order to minimize the risk of periodicity bias, researchers can introduce randomization into the sampling process. By combining systematic sampling with random selection techniques, such as random starting points or randomizing the sampling interval, researchers can reduce the likelihood of falling into a pattern that could introduce bias.
In conclusion, while systematic sampling is a straightforward and efficient sampling method, researchers should be aware of the potential drawback of periodicity bias and take steps to mitigate this risk in order to ensure the validity and reliability of their research findings.

Sampling is a technique used in statistics to gather data from a population. It involves selecting a subset of individuals or items from the population to represent the entire population. This allows for more efficient data collection and analysis, as it is often impractical or impossible to collect data from every individual or item in a population.

There are different methods of sampling, and one of them is cluster sampling. Cluster sampling is a specific type of sampling technique where the population is divided into groups or clusters, and then a subset of clusters is selected for the study. Within each selected cluster, all individuals or items are included in the sample. This is different from other sampling methods where individual elements are directly selected for inclusion in the sample.

Cluster sampling is similar to area sampling, which is the correct answer to the given question. Area sampling is a method where the population is divided into different geographical areas or regions, and then a subset of these areas is selected for the study. Within each selected area, all individuals or items are included in the sample. This is similar to cluster sampling, where the population is divided into clusters and all individuals or items within the selected clusters are included in the sample.

Explanation:
- Sampling is a technique used to gather data from a population.
- Cluster sampling is a specific type of sampling technique.
- Cluster sampling involves dividing the population into clusters and selecting a subset of clusters for the study.
- All individuals or items within the selected clusters are included in the sample.
- Area sampling is a method where the population is divided into geographical areas or regions.
- Within each selected area, all individuals or items are included in the sample.
- Cluster sampling and area sampling are similar because they both involve dividing the population into groups or areas and selecting a subset of these groups or areas for the study.
- The correct answer to the given question is option C, as area sampling is similar to cluster sampling.