AIOU 5412 Solved Assignments Spring 2025

AIOU 5412 Statistics for Management Solved Assignment 1 Spring 2025

---

AIOU 5412 Assignment 1

---

Q1 a). What does statistic mean? Explain its various types.

Understanding Statistics

Statistics is a branch of mathematics concerned with collecting, analyzing, interpreting, presenting, and organizing data. It is widely used in various fields, including economics, business, social sciences, medicine, engineering, and data science, to make informed decisions based on data-driven insights.

Statistics helps in understanding data by summarizing its properties, identifying patterns, and drawing conclusions. In the modern era, statistics plays a crucial role in scientific research, policymaking, and decision-making across industries.

Importance of Statistics

Helps in making predictions based on past trends.

Assists in decision-making by providing numerical evidence.

Enables comparisons between different datasets.

Identifies relationships and correlations within data.

Supports hypothesis testing in scientific research.

Types of Statistics

Statistics is broadly divided into two major categories:

Descriptive Statistics

Descriptive statistics focuses on summarizing and organizing data in a meaningful way. Instead of drawing conclusions, it provides a clear picture of the data by presenting its key features.

Key Components of Descriptive Statistics

Measures of Central Tendency

These measures describe the center or typical value of a dataset.

Mean (Average): The sum of all values divided by the number of values.

Median: The middle value when the data is arranged in ascending order.

Mode: The most frequently occurring value in the dataset.

Measures of Dispersion (Variability)

These measures determine the spread of data and how much it deviates from the mean.

Range: The difference between the highest and lowest values.

Variance: Measures how data points vary from the mean.

Standard Deviation: A more precise measure of dispersion, indicating how data is spread around the mean.

Graphical Representations

Histograms: Used to show the frequency distribution of numerical data.

Pie Charts: Ideal for representing proportions and percentages.

Box Plots: Useful for identifying the spread, median, and outliers in data.

Inferential Statistics

Inferential statistics helps in making predictions or drawing conclusions about a population based on a sample. It enables researchers to test hypotheses and make generalizations beyond the observed data.

Key Techniques in Inferential Statistics

Hypothesis Testing

Hypothesis testing is a statistical method used to determine if there is enough evidence to support a claim or hypothesis.

Null Hypothesis (H₀): Assumes no effect or relationship exists.

Alternative Hypothesis (H₁): Suggests an effect or relationship exists.

P-value: Determines the significance level of results.

Confidence Intervals

Confidence intervals provide a range within which the true population parameter is likely to exist. They help in estimating the population mean, proportion, or variance with a certain level of confidence (e.g., 95% confidence interval).

Regression Analysis

Regression analysis helps understand the relationship between variables. The most common types include:

Linear Regression: Models the relationship between two variables using a straight line.

Multiple Regression: Involves multiple independent variables to predict a dependent variable.

Probability Distributions

Probability distributions describe the likelihood of different outcomes in a dataset.

Normal Distribution: A bell-shaped curve where most values cluster around the mean.

Binomial Distribution: Used for experiments with two possible outcomes (success/failure).

Poisson Distribution: Models events occurring within a fixed interval.

Applications of Statistics

Statistics is applied in various fields, including:

Economics: Forecasting trends, analyzing GDP, inflation rates.

Business and Marketing: Consumer behavior analysis, sales forecasting.

Healthcare: Clinical trials, disease outbreak analysis.

Social Sciences: Survey analysis, demographic studies.

Sports: Player performance analysis, game strategy improvements.

---

Q1 b). Explain what is meant by Classification. What are its basic principles of Classification?

Classification: Classification refers to the systematic organization of objects, ideas, or information into specific groups based on shared characteristics. It is a fundamental tool in various disciplines, including science, business, and education, allowing for efficient analysis, decision-making, and communication.

Meaning of Classification: Classification involves arranging entities into categories that share similar attributes, simplifying complex information for better understanding and application. It plays a crucial role in different fields, such as biology, marketing, and artificial intelligence.

For example:

- In biology, organisms are classified based on evolutionary relationships.

- In marketing, consumers are segmented based on purchasing behavior.

- In artificial intelligence, data is categorized to enhance machine learning models.

Basic Principles of Classification: Several fundamental principles guide the classification process:

Identification of Characteristics: Objects or concepts are grouped based on well-defined attributes. In biological classification, traits such as anatomy, genetics, and habitat are considered. In business, factors such as industry type, financial performance, or customer demographics are used.

Mutual Exclusivity: Each category should be distinct, ensuring that objects belong to only one classification group. This prevents redundancy and promotes clarity. For example, scientific taxonomy ensures organisms do not belong to multiple species.

Consistency in Criteria: The classification system must adhere to uniform criteria throughout the grouping process. If inconsistency occurs, it compromises accuracy, leading to misinterpretation.

Hierarchical Structure: Many classification systems use hierarchical organization to arrange broad categories into subcategories. A well-known example is biological classification: Kingdom → Phylum → Class → Order → Family → Genus → Species.

Purpose and Relevance: A classification system must serve a meaningful purpose, such as knowledge organization, data retrieval, or decision-making. Economic classification aids resource allocation, while medical classification supports diagnosis and treatment.

Applications of Classification: Classification is applied across various fields:

- Scientific Classification: Taxonomy categorizes organisms based on evolutionary traits.

- Library Systems: Books are grouped into genres, subjects, and authors for accessibility.

- Data Science and AI: Classification models enable predictive analytics, speech recognition, and image categorization.

- Marketing and Consumer Behavior: Companies classify audiences based on purchasing patterns for targeted advertising.

Conclusion: Classification is essential for organizing and interpreting information efficiently. Its principles—identification of characteristics, mutual exclusivity, consistency, hierarchy, and relevance—ensure logical categorization. Applying classification appropriately enhances decision-making, improves efficiency, and fosters innovation.

---

Q2 a). What Criteria do you apply to judge the merits of an average? Discuss the merits and demerits of different averages.

Understanding the Criteria for Evaluating an Average

Averages are statistical tools used to summarize data sets into a single value that represents the central tendency of the given data. Choosing the right average depends on various factors, which are evaluated through the following criteria:

Representativeness

An average should ideally be a true reflection of the entire dataset. It should be able to summarize the data in a way that is useful for making informed decisions. The arithmetic mean, for example, is widely used because it takes into account all values, but it can be skewed by extreme data points.

Stability

A good average should not fluctuate drastically with small changes in the dataset. Stability ensures that conclusions drawn from the data remain valid even with minor variations. The median is often more stable than the mean when dealing with skewed distributions.

Simplicity & Ease of Calculation

Some averages are easier to calculate than others. The arithmetic mean, median, and mode are relatively simple, while the geometric mean and harmonic mean require more complex calculations.

Applicability

An average should be applicable across different types of data and contexts. The mode is most useful for categorical data, while the arithmetic and geometric means are preferred for numerical data.

Sensitivity to Extreme Values

Averages should not be overly sensitive to outliers, which can distort the representation of the dataset. The mean can be heavily affected by extreme values, whereas the median remains unaffected.

Different Types of Averages: Merits and Demerits

Arithmetic Mean (Average)

The arithmetic mean is the most widely used measure of central tendency. It is calculated by summing up all values and dividing by the total number of data points.

Merits:

- Simple to calculate and understand.

- Uses all values in the dataset, making it a comprehensive measure.

- Useful for statistical analysis in fields like economics, finance, and social sciences.

Demerits:

- Highly sensitive to extreme values (outliers), which can distort results.

- May not be representative of skewed datasets.

Median

The median is the middle value when the data is arranged in ascending order. If the dataset has an even number of values, the median is the average of the two middle numbers.

Merits:

- Not affected by extreme values, making it more representative for skewed distributions.

- Useful in real estate pricing, income distribution studies, and data sets with large variations.

Demerits:

- Does not consider all values in the dataset.

- Not ideal for advanced statistical computations.

Mode

The mode is the most frequently occurring value in a dataset. It is particularly useful for categorical data where numerical averages are meaningless.

Merits:

- Applicable to qualitative data, such as identifying the most popular product sold in a store.

- Easy to understand and interpret.

- Useful in determining market trends and consumer preferences.

Demerits:

- A dataset may have multiple modes or no mode at all.

- Does not consider all data points.

Geometric Mean

The geometric mean is calculated by multiplying all values and then taking the nth root (where n is the number of values). It is primarily used for growth rates and financial returns.

Merits:

- Effective for analyzing rates of change, such as economic growth and investment returns.

- Less affected by extreme values compared to the arithmetic mean.

Demerits:

- Complex to compute.

- Cannot be used when negative values are present in the dataset.

Harmonic Mean

The harmonic mean is calculated by dividing the number of values by the sum of their reciprocals. It is best suited for rates and ratios.

Merits:

- Ideal for averaging speed, efficiency, and rates.

- Less influenced by extreme values than the arithmetic mean.

Demerits:

- Less intuitive to understand.

- Requires positive values in the dataset.

Which Average Should You Use?

- If the data is evenly distributed → Use arithmetic mean.

- If the dataset has outliers → Use median.

- If you need the most frequently occurring value → Use mode.

- For growth rates and investments → Use geometric mean.

- For rates and ratios → Use harmonic mean.

Conclusion

Averages are crucial tools for summarizing data, but not all averages are suitable for every situation. The arithmetic mean is widely used but sensitive to outliers, the median is ideal for skewed data, the mode helps identify popular choices, and the geometric and harmonic means are valuable for financial and efficiency analyses.

Understanding these differences allows us to make informed decisions based on the nature of the data we are analyzing.

---

Q2 b). Explain the Statistical terms i) Sample, ii) Population, iii) Statistics, iv) Parameter.

Sample: A sample refers to a subset of a population chosen for study. In statistics, analyzing an entire population is often impractical due to time, cost, or logistical constraints. Therefore, researchers select a sample that represents the larger population to make inferences. The effectiveness of a sample depends on how well it reflects the characteristics of the entire population.

Samples can be selected through various methods, such as:

• Random Sampling: Every member of the population has an equal chance of being selected.
• Stratified Sampling: The population is divided into subgroups (strata), and samples are taken proportionally.
• Systematic Sampling: Selecting every nth member from a list.
• Convenience Sampling: Using readily available subjects, though this may introduce bias.

The accuracy of statistical conclusions depends on proper sampling techniques. If a sample is biased, the results may not be applicable to the whole population.

---

Population: A population is the entire group of individuals, objects, or items that a study aims to analyze. The population may vary depending on the research question—it can refer to all employees in a company, all voters in a country, or all manufactured items from a factory.

Populations are typically classified into two types:

• Finite Population: A countable number of elements, such as all students in a university.
• Infinite Population: An uncountable number of elements, such as bacteria in a petri dish.

Since collecting data on every single member of a population is often impractical, researchers rely on samples to derive conclusions about the whole group. However, ensuring that the sample accurately reflects the population is crucial for valid results.

---

Statistics: Statistics is the branch of mathematics concerned with collecting, analyzing, interpreting, and presenting numerical data. It plays a critical role in various fields, including economics, healthcare, business, and social sciences.

Statistics is broadly divided into:

• Descriptive Statistics: Summarizing and describing data using measures like mean, median, mode, variance, and standard deviation.
• Inferential Statistics: Making predictions or generalizations about a population based on a sample.

Statistical methods are used to identify patterns, test hypotheses, and provide insights that support decision-making. They are foundational to research and policy-making, helping organizations and individuals understand trends and uncertainties.

---

Parameter: A parameter is a numerical value that describes a characteristic of an entire population. Since populations are often large and difficult to study directly, parameters provide key insights into their overall behavior.

Examples of parameters include:

• Population Mean (μ): The average of all values in a population.
• Population Proportion (P): The proportion of a particular category within a population.
• Population Variance (σ²): The variability of data points within a population.

Unlike statistics, which describe characteristics of a sample, parameters are fixed values that represent an entire population. However, in practice, parameters are often estimated using sample data because obtaining exact population values is usually unfeasible.

---

Q3 a). What is the coefficient of Variation?

The coefficient of variation (CV) is a statistical measure that indicates the relative variability of a dataset compared to its mean. It is particularly useful for comparing the dispersion of data across different datasets, especially when their means are significantly different.

The coefficient of variation is defined as the ratio of the standard deviation (\(\sigma\)) to the mean (\(\mu\)), expressed as a percentage:

\[ CV = \left( \frac{\sigma}{\mu} \right) \times 100\% \]

Where:

• \(\sigma\) is the standard deviation of the dataset.
• \(\mu\) is the mean of the dataset.

Significance and Use Cases of CV

Unlike absolute measures of dispersion such as standard deviation, CV provides a dimensionless quantity that allows comparison across datasets with different units or scales. Some key applications include:

• Finance: Investors use CV to assess the relative risk of different investment portfolios.
• Scientific Research: CV is widely used in biological and medical studies.
• Quality Control: Industries use CV to evaluate the consistency of manufacturing processes.

Interpreting CV Values

• Low CV: Indicates less variability relative to the mean.
• High CV: Suggests greater variability.

Example Calculation

Consider two datasets:

• Dataset A: Mean = 50, Standard Deviation = 5
• \[ CV = \left( \frac{5}{50} \right) \times 100 = 10\% \]
• Dataset B: Mean = 200, Standard Deviation = 40
• \[ CV = \left( \frac{40}{200} \right) \times 100 = 20\% \]

Limitations of CV

• Not appropriate when the mean is close to zero.
• Not suitable for datasets with negative or zero values.
• May not fully capture risk in finance due to extreme values.

Comparison with Other Measures

• Standard Deviation: Absolute measure; does not account for mean.
• Variance: Square of standard deviation.
• Range: Difference between max and min values.

Practical Applications and Decision-Making

• Portfolio Management: Investors compare CV values of different assets.
• Business Strategy: Companies evaluate operational consistency.
• Healthcare: Researchers use CV to compare variability in patient response to treatments.

Final Thoughts

The coefficient of variation is an essential tool in statistical analysis, offering a robust method to assess relative variability. CV helps in making informed comparisons, though it should be used alongside other statistical measures for comprehensive analysis.

---

Q3 b). The following data represent the weights of fish caught. Find coefficient of variation:

Weight (lbs)	0-24	25-49	50-74	75-99	100-124	125-149	150-159
Frequency	5	13	16	20	15	12	8

Find the Coefficient of Variation (CV)

To calculate the coefficient of variation, follow these steps:

Step 1: Find the Mean (\(\bar{x}\))

We determine the midpoint of each weight class and multiply by its frequency:

\[ \text{Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2} \]

Weight Range (lbs)	Midpoint (\(x_i\))	Frequency (\(f_i\))	\( f_i \times x_i \)
0 - 24	12	5	60
25 - 49	37	13	481
50 - 74	62	16	992
75 - 99	87	20	1740
100 - 124	112	15	1680
125 - 149	137	12	1644
150 - 159	154.5	8	1236

Total frequency (\(\sum f_i\)) = 89

Total (\(\sum f_i x_i\)) = 8829

\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{8829}{89} \approx 99.2 \]

Step 2: Find the Standard Deviation (\(\sigma\))

Now, we compute the squared deviations from the mean:

\[ \text{Variance} = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i} \]

Weight Range	Midpoint (\(x_i\))	\( (x_i - \bar{x}) \)	\( (x_i - \bar{x})^2 \)	\( f_i \times (x_i - \bar{x})^2 \)
0 - 24	12	-87.2	7603.84	38019.2
25 - 49	37	-62.2	3860.84	50190.92
50 - 74	62	-37.2	1384.64	22154.24
75 - 99	87	-12.2	148.84	2977.6
100 - 124	112	12.8	163.84	2457.6
125 - 149	137	37.8	1428.84	17146.08
150 - 159	154.5	55.3	3058.09	24464.72

Total (\(\sum f_i (x_i - \bar{x})^2\)) = 157410.36

\[ \sigma = \sqrt{\frac{157410.36}{89}} = \sqrt{1769.9} \approx 42.1 \]

Step 3: Find the Coefficient of Variation (CV)

\[ CV = \left(\frac{\sigma}{\bar{x}}\right) \times 100 = \left(\frac{42.1}{99.2}\right) \times 100 \]

\[ CV \approx 42.46\% \]

Final Answer: The coefficient of variation for the given fish weights is **42.46%**.

This indicates a relative variability of 42.46% compared to the mean.

---

Q4 a). Explain the time and factor reversal tests.

The time reversal test and the factor reversal test are two important consistency checks used in index number theory. They ensure index numbers maintain logical consistency when reversing time periods or swapping price and quantity components.

Time Reversal Test: This test verifies whether an index behaves consistently when the base and current periods are reversed. If an index number is calculated from period 0 to period 1, then calculated again in reverse, the product of both index numbers should ideally equal 1.

The mathematical expression for this test is:

$$ P_{0,1} \times P_{1,0} = 1 $$

For example, if the price index from 2020 to 2025 is \(P_{2020,2025} = 1.2\) and the reverse index is \(P_{2025,2020} = 0.8333\), then:

$$ 1.2 \times 0.8333 = 1 $$

This confirms that the index passes the time reversal test.

Implication: The time reversal test ensures that index numbers remain unbiased regardless of the direction of calculation. Some index formulas, such as the Fisher Price Index, satisfy this test, while others, like Laspeyres and Paasche indices, may not.

Factor Reversal Test: This test checks whether the index maintains logical consistency when price and quantity components are swapped. Multiplying the price index by the quantity index should yield the value index, representing total expenditure or revenue.

The mathematical formulation is:

$$ P \times Q = V $$

Where:

$$ V = \frac{\sum (p_1 q_1)}{\sum (p_0 q_0)} $$

For example, if the price index is 120 and the quantity index is 90, then:

$$ 120 \times 90 = 10800 $$

This should equal the value index.

Implication: The factor reversal test ensures that price and quantity changes are appropriately reflected in index calculations. The Fisher Index satisfies this test, while indices like Laspeyres and Paasche often fail.

Comparison and Importance: Both tests help assess the reliability of index numbers:

• The time reversal test ensures consistency when reversing time periods.
• The factor reversal test ensures proper reflection of price and quantity changes.

These tests are essential for economic analysis, inflation measurement, and price trend assessments.

---

Q4 b). This information describes the unit (000) sales of a bicycle shop for three years:

	Number Sold Year 2018			Price
Model	2013	2015	2017	2013
Sports	45	48	56	89
Touring	64	67	71	104
Cross-Country	28	35	27	138
Spirit	21	16	28	245

Calculate the weighted average of relative quantity indices using the price and quantities from 2013 to compute the value weights with 2013 as the base year.

Calculate the weighted average of relative quantity indices using the price and quantities from 2013 to compute the value weights with 2013 as the base year.

The formula used:

$$W = \frac{\sum (p_0 \cdot q_t / q_0)}{\sum p_0}$$

where:

• \( p_0 \) is the price in 2013 (base year).
• \( q_t \) is the quantity sold in a given year.
• \( q_0 \) is the quantity sold in 2013.
• \( W \) is the weighted average of relative quantity indices.

Step 1: Compute Relative Quantity Index for Each Model

• Sports: \( \frac{56}{45} = 1.244 \)
• Touring: \( \frac{71}{64} = 1.109 \)
• Cross-Country: \( \frac{27}{28} = 0.964 \)
• Spirit: \( \frac{28}{21} = 1.333 \)

Step 2: Compute Value Weights (\( p_0 \cdot q_0 \))

• Sports: \( 89 \times 45 = 4005 \)
• Touring: \( 104 \times 64 = 6656 \)
• Cross-Country: \( 138 \times 28 = 3864 \)
• Spirit: \( 245 \times 21 = 5145 \)

Total value weight:

$$4005 + 6656 + 3864 + 5145 = 19670$$

Step 3: Compute Weighted Sum

$$\sum (p_0 \cdot q_t/q_0)$$

$$ (4005 \times 1.244) + (6656 \times 1.109) + (3864 \times 0.964) + (5145 \times 1.333) $$

$$ 4977.4 + 7383.7 + 3729.3 + 6859.4 = 22949.8 $$

Step 4: Calculate Weighted Average Index

$$W = \frac{22949.8}{19670} = 1.167$$

Thus, the weighted average of relative quantity indices for 2017 (using 2013 as the base year) is 1.167 or 116.7%.

This indicates that the overall quantity sold has increased by 16.7% compared to 2013.

---

Q5 a). Define Laws of probabilities with examples.

Definition of Probability: Probability measures how likely an event is to occur. It is expressed as a number between 0 and 1, where:

Formula:

\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

Example: If you toss a fair coin, the probability of getting heads is:

\[ P(\text{Heads}) = \frac{1}{2} = 0.5 \]

Basic Laws of Probability:

Law of Complementation: The probability of an event NOT happening is given by:

\[ P(A') = 1 - P(A) \]

Example: If the probability of rain tomorrow is 0.3, then the probability that it won’t rain is:

\[ P(\text{No Rain}) = 1 - 0.3 = 0.7 \]

Addition Rule of Probability: The probability of either \( A \) or \( B \) happening (if they are not mutually exclusive) is:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Example: If the probability of picking a red ball is 0.4 and a blue ball is 0.5, and if one ball could be both red and blue with a probability of 0.1, then:

\[ P(\text{Red or Blue}) = 0.4 + 0.5 - 0.1 = 0.8 \]

Multiplication Rule of Probability: The probability that both independent events \( A \) and \( B \) occur is:

\[ P(A \cap B) = P(A) \times P(B) \]

Example: The probability of rolling a 4 on a die and flipping heads on a coin is:

\[ P(4 \text{ and heads}) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \]

Conditional Probability and Bayes' Theorem:

Conditional Probability: The probability of event \( A \) occurring given that \( B \) has already occurred:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Example: If the probability of selecting a female employee is 0.6, and the probability of a female being a manager is 0.2, then the probability that a randomly chosen person is a female manager is:

\[ P(\text{Manager | Female}) = \frac{P(\text{Female and Manager})}{P(\text{Female})} = \frac{0.2}{0.6} = 0.333 \]

Bayes' Theorem: Helps reverse probabilities:

\[ P(A|B) = \frac{P(B|A) P(A)}{P(B)} \]

Example: If a medical test is 95% accurate in detecting a disease and the disease occurs in 1% of the population, Bayes' theorem can calculate the probability that a person who tested positive actually has the disease.

Applications of Probability:

• Gaming and Gambling: Probability determines odds in betting.
• Medicine: Diagnostic tests rely on probabilities.
• Weather Forecasting: Meteorologists predict rain probabilities.
• Insurance and Finance: Companies assess risk probabilities.

Conclusion: The laws of probability help us quantify uncertainty and make informed decisions. Whether predicting weather, assessing medical risks, or designing computer algorithms, probability provides a systematic way to handle randomness.

---

Q5 b). Econocon is planning its company picnic. The only thing that will cancel the picnic is a thunderstorm. The Weather Services has predicted dry conditions with a probability of 0.2, moist conditions with a probability of 0.45 and wet conditions with a probability of 0.35. If the probability of a thunderstorm given dry conditions is 0.3, given moist conditions is 0.6 and given wet conditions is 0.8, what is the probability of a thunderstorm? If we know the picnic was indeed cancelled, what is the probability moist conditions were in effect?

Finding the probability of a thunderstorm

We use the law of total probability:

\[ P(T) = P(T | D) P(D) + P(T | M) P(M) + P(T | W) P(W) \]

Where: - \(P(T)\) is the probability of a thunderstorm - \(P(D) = 0.2\) is the probability of dry conditions - \(P(M) = 0.45\) is the probability of moist conditions - \(P(W) = 0.35\) is the probability of wet conditions - \(P(T | D) = 0.3\) is the probability of a thunderstorm given dry conditions - \(P(T | M) = 0.6\) is the probability of a thunderstorm given moist conditions - \(P(T | W) = 0.8\) is the probability of a thunderstorm given wet conditions

Now calculating:

\[ P(T) = (0.3 \times 0.2) + (0.6 \times 0.45) + (0.8 \times 0.35) \]

\[ = 0.06 + 0.27 + 0.28 \]

\[ = 0.61 \]

So, the probability of a thunderstorm is 0.61 (or 61%).

Finding the probability that moist conditions were in effect given a thunderstorm (Bayes’ Theorem)

Applying Bayes’ theorem:

\[ P(M | T) = \frac{P(T | M) P(M)}{P(T)} \]

Substituting values:

\[ P(M | T) = \frac{0.6 \times 0.45}{0.61} \]

\[ = \frac{0.27}{0.61} \]

\[ ≈ 0.4426 \]

So, if the picnic was cancelled, the probability that moist conditions were in effect is approximately 44.26%.

---

AIOU 5412 Statistics for Management Solved Assignment 2 Spring 2025

---

AIOU 5412 Assignment 2

---

Q1. Distinguish among classical or a dprior probability, relative frequency or a posteriori probability, axiomatic probability and subjective or personalistic along with their advantages and disadvantages.

Classical (A Priori) Probability:

Classical probability is based on equally likely outcomes in a well-defined sample space. It is defined as:

$$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

For example, when rolling a fair six-sided die, the probability of getting a 3 is:

$$P(3) = \frac{1}{6}$$

Advantages:

• Simple to calculate when outcomes are equally likely.
• Provides precise probabilities without needing empirical data.
• Works well for games of chance and theoretical probability models.

Disadvantages:

• Not applicable when outcomes are not equally likely.
• Many real-world events do not fit this framework.
• Does not rely on observed frequencies.

Relative Frequency (A Posteriori) Probability:

Relative frequency probability is based on observed data rather than theoretical assumptions. It is defined as:

$$P(E) = \frac{\text{Number of times event E occurs}}{\text{Total number of trials}}$$

For instance, if a coin is flipped 1000 times and lands on heads 505 times, then:

$$P(\text{Heads}) = \frac{505}{1000} = 0.505$$

Advantages:

• Directly applicable to real-world scenarios.
• Provides probabilities based on actual observations.
• More trials lead to more precise estimations.

Disadvantages:

• Cannot determine probabilities of unique or one-time events.
• Requires large sample sizes for reliability.
• Lacks the mathematical rigor of other definitions.

Axiomatic Probability:

Axiomatic probability was formalized by Andrey Kolmogorov in 1933. It is based on a set of axioms:

• $$P(A) \geq 0$$ for any event \( A \).
• $$P(S) = 1$$, meaning the probability of the entire sample space is 1.
• If \( A \) and \( B \) are mutually exclusive, then $$P(A \cup B) = P(A) + P(B)$$.

Advantages:

• Provides a formal foundation for probability theory.
• Can accommodate classical, empirical, and subjective interpretations.
• Used in theoretical probability, machine learning, and complex systems.

Disadvantages:

• Does not directly explain how probability arises.
• Relies on predefined axioms that may not be intuitive.
• Not ideal for explaining single-instance events.

Subjective (Personalistic) Probability:

Subjective probability is based on an individual's personal belief or degree of confidence regarding an event.

For example, a doctor estimating the likelihood of a patient responding to a new treatment might assign:

$$P(\text{Success}) = 0.8$$

Advantages:

• Useful in situations with little or no historical data.
• Reflects real-world uncertainties better than rigid models.
• Can be updated with new information using Bayes' theorem.

Disadvantages:

• Probabilities vary between individuals, leading to biases.
• No standard method to calculate subjective probabilities.
• Cannot be empirically tested without external verification.

Comparison:

Probability Type	Strengths	Weaknesses	Application Areas
Classical (A Priori)	Simple, exact, logical	Unrealistic assumptions, requires equally likely outcomes	Games of chance, theoretical models
Relative Frequency (A Posteriori)	Practical, improves with more data	Limited predictive power, requires large samples	Statistical inference, experimental data
Axiomatic	Mathematically rigorous, universal	Abstract, assumption-dependent	Theoretical probability, complex systems
Subjective (Personalistic)	Flexible, aligns with human intuition	Biased, lacks empirical validation	Bayesian probability, decision-making

Each probability approach serves a distinct purpose and is applicable in different contexts. Understanding these differences helps in making informed decisions across various fields.

---

Q2 a). Explain the concept of a random variable. What is the distribution function?

Understanding Random Variables and Distribution Functions

A random variable is a numerical representation of uncertain outcomes in a probability space. It allows us to analyze random processes mathematically.

Definition

Mathematically, a random variable is a function \( X: \Omega \to \mathbb{R} \), where \( \Omega \) represents the sample space of a random experiment. Each sample point is mapped to a real number.

Example: Consider a coin toss. If \( X \) represents the number of heads, then:

\( X(H) = 1, \quad X(T) = 0 \)

Types of Random Variables

Discrete Random Variables: These take on countable values, like the number of customers arriving at a store.

Continuous Random Variables: These take on values in a continuous range, such as height or temperature.

Probability Mass Function (PMF)

The PMF is used for discrete random variables, representing the probability of specific values:

\( P(X = x) = f(x) \)

Example: If \( X \) is the outcome of rolling a six-sided die:

\( P(X = k) = \frac{1}{6}, \quad k \in \{1, 2, 3, 4, 5, 6\} \)

Probability Density Function (PDF)

The PDF applies to continuous random variables:

\( P(a \leq X \leq b) = \int_a^b f(x) dx \)

Example: The normal distribution:

\( f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{\frac{-(x-\mu)^2}{2\sigma^2}} \)

Cumulative Distribution Function (CDF)

The CDF gives the probability that \( X \) is less than or equal to a given value:

\( F(x) = P(X \leq x) \)

For continuous cases:

\( F(x) = \int_{-\infty}^{x} f(t) dt \)

Properties of Distribution Functions

1. \( F(x) \) is non-decreasing: \( F(x_1) \leq F(x_2) \) for \( x_1 \leq x_2 \).

2. Limits: \( \lim_{x \to -\infty} F(x) = 0 \) and \( \lim_{x \to \infty} F(x) = 1 \).

Applications

Random variables and distribution functions are widely used in:

Finance: Stock price analysis, risk modeling.

Engineering: Signal processing, reliability analysis.

AI and Data Science: Bayesian inference, predictive modeling.

Conclusion

Random variables and distribution functions are essential tools for quantifying uncertainty in various domains. They enable mathematical modeling of randomness in practical applications.

---

Q2 b). An insurance salesman sells policies to 5 men, all of identical and good health. According to the actuarial tables, the probability that a man of this particular age will be alive 30 years hence is 2/3.
Find the probability that in 30 years.
i) all men
ii) at least 3 men
iii) only two men
iv) at most one man will be alive.

This problem follows a binomial probability distribution, where:

\( n = 5 \) (total number of men)

\( p = \frac{2}{3} \) (probability that a man survives 30 years)

\( q = 1 - p = \frac{1}{3} \) (probability that a man does not survive)

The probability mass function is:

\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]

i) Probability that all 5 men survive

\[ P(X = 5) = \binom{5}{5} \left(\frac{2}{3}\right)^5 \left(\frac{1}{3}\right)^0 \]

\[ P(X = 5) = 1 \times \left(\frac{32}{243}\right) \times 1 = \frac{32}{243} \approx 0.1317 \]

ii) Probability that at least 3 men survive

This includes cases \( X = 3, 4, \) and \( 5 \).

\[ P(X = 3) = \binom{5}{3} \left(\frac{2}{3}\right)^3 \left(\frac{1}{3}\right)^2 \]

\[ = 10 \times \left(\frac{8}{27}\right) \times \left(\frac{1}{9}\right) = 10 \times \frac{8}{243} = \frac{80}{243} \]

\[ P(X = 4) = \binom{5}{4} \left(\frac{2}{3}\right)^4 \left(\frac{1}{3}\right)^1 \]

\[ = 5 \times \left(\frac{16}{81}\right) \times \left(\frac{1}{3}\right) = 5 \times \frac{16}{243} = \frac{80}{243} \]

Summing for \( X \geq 3 \):

\[ P(X \geq 3) = P(3) + P(4) + P(5) = \frac{80}{243} + \frac{80}{243} + \frac{32}{243} = \frac{192}{243} \approx 0.7901 \]

iii) Probability that only two men survive

\[ P(X = 2) = \binom{5}{2} \left(\frac{2}{3}\right)^2 \left(\frac{1}{3}\right)^3 \]

\[ = 10 \times \left(\frac{4}{9}\right) \times \left(\frac{1}{27}\right) = 10 \times \frac{4}{243} = \frac{40}{243} \approx 0.1646 \]

iv) Probability that at most one man survives

Includes \( X = 0 \) and \( X = 1 \).

\[ P(X = 0) = \binom{5}{0} \left(\frac{2}{3}\right)^0 \left(\frac{1}{3}\right)^5 \]

\[ = 1 \times 1 \times \left(\frac{1}{243}\right) = \frac{1}{243} \]

\[ P(X = 1) = \binom{5}{1} \left(\frac{2}{3}\right)^1 \left(\frac{1}{3}\right)^4 \]

\[ = 5 \times \left(\frac{2}{3}\right) \times \left(\frac{1}{81}\right) = 5 \times \frac{2}{243} = \frac{10}{243} \]

Summing for \( X \leq 1 \):

\[ P(X \leq 1) = P(0) + P(1) = \frac{1}{243} + \frac{10}{243} = \frac{11}{243} \approx 0.0453 \]

Final probabilities:

All 5 men survive: \( \frac{32}{243} \approx 0.1317 \)

At least 3 men survive: \( \frac{192}{243} \approx 0.7901 \)

Only 2 men survive: \( \frac{40}{243} \approx 0.1646 \)

At most 1 man survives: \( \frac{11}{243} \approx 0.0453 \)

---

Q3. Suppose we want to predict the job performance of chevy mechanics based on mechanical aptitude test scores and test scores from personality tests that measure conscientiousness.

Person	1	2	3	4	5	6	7	8	9
Mechanical aptitude test score	40	45	38	50	48	55	53	55	58
Job performance	1	2	1	3	2	2	3	4	3

i) State the dependent and independent variables.
ii) Draw a scatter diagram of the above data.
iii) Does the Mechanical Aptitude test score affect Job Performance?
iv) Find the correlation between two variables.
v) Interpret both the results.

Step 1: Define Variables

Independent variable (X): Mechanical aptitude test score

Dependent variable (Y): Job performance

Step 2: Scatter Diagram

You can create a scatter plot by plotting Mechanical Aptitude Test Score (X) on the X-axis and Job Performance (Y) on the Y-axis. If you use Excel or Google Sheets, simply enter the values and generate a scatter plot.

Step 3: Identify the Effect

Observing the scatter plot will help determine whether higher mechanical aptitude test scores correspond to higher job performance ratings. If the points show an upward trend, it suggests a positive relationship between the two variables.

Step 4: Compute Correlation Coefficient (Pearson's r)

The correlation coefficient formula:

$$r = \frac{n\sum XY - (\sum X)(\sum Y)}{\sqrt{[n\sum X^2 - (\sum X)^2][n\sum Y^2 - (\sum Y)^2]}}$$

Step 4.1: Compute Summations

Given data:

Person	X (Aptitude Score)	Y (Job Performance)	X²	Y²	XY
1	40	1	1600	1	40
2	45	2	2025	4	90
3	38	1	1444	1	38
4	50	3	2500	9	150
5	48	2	2304	4	96
6	55	2	3025	4	110
7	53	3	2809	9	159
8	55	4	3025	16	220
9	58	3	3364	9	174

Now sum up all the values:

$$\sum X = 442, \quad \sum Y = 21, \quad \sum X^2 = 22096, \quad \sum Y^2 = 57, \quad \sum XY = 1077$$

Step 4.2: Apply to Formula

$$r = \frac{9(1077) - (442)(21)}{\sqrt{[9(22096) - (442)^2][9(57) - (21)^2]}}$$

$$r = \frac{9693 - 9282}{\sqrt{[198864 - 195364][513 - 441]}}$$

$$r = \frac{411}{\sqrt{3500 \times 72}}$$

$$r = \frac{411}{\sqrt{252000}}$$

$$r = \frac{411}{502} \approx 0.82$$

Step 5: Interpretation

Since \( r = 0.82 \), this indicates a strong positive correlation between mechanical aptitude test scores and job performance. The higher the test score, the better the predicted job performance.

---

Q4 a). Explain the relationship between confidence estimates and p-values in drawing inferences.

Confidence estimates and p-values are fundamental concepts in statistical inference, helping researchers draw conclusions from data while quantifying uncertainty. They are closely related, yet serve different purposes in hypothesis testing and estimation. Below is a comprehensive discussion of their relationship, differences, and significance in statistical analysis.

Introduction to Statistical Inference

Statistical inference involves drawing conclusions about a population based on sample data. Since sample data is subject to variability, confidence estimates and p-values help quantify uncertainty and assess the strength of evidence.

Two primary approaches to statistical inference are:

• Estimation – Determining the value of a population parameter (e.g., mean, proportion) using sample data.
• Hypothesis Testing – Assessing whether sample data provides sufficient evidence to reject a null hypothesis.

Confidence Estimates: A Measure of Precision

A confidence estimate, often represented by a confidence interval (CI), is a range within which the true population parameter is expected to fall with a certain level of confidence. The most common confidence levels are 90%, 95%, and 99%.

A 95% confidence interval means that if we were to repeatedly sample the population and compute intervals, 95% of those intervals would contain the true population parameter.

For example, if the mean height of a sample is 170 cm with a 95% confidence interval of (168 cm, 172 cm), it implies that we are 95% confident that the true mean height of the population lies within this range.

The confidence interval is calculated as:

$$CI = \bar{x} \pm Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$

where:

• \(\bar{x}\) is the sample mean
• \(Z_{\alpha/2}\) is the critical value from the standard normal distribution
• \(\sigma\) is the population standard deviation (or sample standard deviation)
• \(n\) is the sample size

The width of the confidence interval reflects the precision of our estimate. A narrower interval suggests greater certainty, while a wider interval indicates more uncertainty.

P-Values: A Measure of Evidence Against the Null Hypothesis

A p-value quantifies the probability of observing the sample data (or more extreme data) under the assumption that the null hypothesis is true.

A small p-value (e.g., < 0.05) suggests that the observed data is unlikely under the null hypothesis, leading to rejection of the null.

A large p-value (e.g., > 0.05) suggests that the observed data could reasonably occur under the null, failing to reject the null hypothesis.

For hypothesis testing, the test statistic is:

$$Z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$$

The p-value is computed based on the probability of obtaining a test statistic at least as extreme as the observed \(Z\) under the assumption that \(H_0\) is true.

The Connection Between Confidence Estimates and P-Values

Confidence intervals and p-values are complementary:

• A 95% confidence interval excludes values that would have resulted in a p-value < 0.05.
• If a hypothesis test yields a p-value < 0.05, the hypothesized value is outside the 95% confidence interval.

For example:

• Suppose we test whether the population mean is 175 cm, and our 95% confidence interval for the sample mean is (168 cm, 172 cm).
• Since 175 cm is outside this range, we reject \(H_0\) at α = 0.05 (yielding p < 0.05).

Limitations and Misinterpretations

• P-Values Do Not Indicate Strength of Effect – A small p-value suggests that the observed data is unlikely under the null, but does not measure the size or importance of the effect.
• Confidence Intervals Are More Informative – A confidence interval shows the range within which the true effect size likely lies, allowing for better decision-making.
• Sample Size Dependence – Both p-values and confidence intervals are affected by sample size, influencing statistical and practical significance.

Conclusion

Confidence estimates and p-values play crucial roles in statistical inference. Confidence intervals provide estimates of population parameters with quantified uncertainty, while p-values assess the evidence against a null hypothesis.

Understanding these tools helps researchers interpret data properly, avoid misinterpretations, and make sound conclusions in scientific and applied research.

---

Q4 b). An inventor has developed a new, energy -efficient lawn mower engine. He claims that the engine will run continuously for 5 hours (300 minutes) on a single gallon of regular gasoline. From his stock of 2000 engines, the inventor selects a simple random sample of 50 engines for testing. The engines run for an average of 295 minutes, with a standard deviation of 20 minutes. Test the null hypothesis tht the mean run time is 300 minutes against the alternative hypothesis that the mean run time is not 300 minutes. Use a 0.05 level of significance. Assume the run times for the population of the engine are normally distributed).

Step 1: Define the Hypotheses

Null hypothesis (H₀): The mean run time is 300 minutes. \( \mu = 300 \)

Alternative hypothesis (Hₐ): The mean run time is not 300 minutes. \( \mu \neq 300 \)

This is a two-tailed test.

Step 2: Determine the Test Statistic

Since the population standard deviation is unknown, we use the t-test formula:

\[ t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \]

where:

• \(\bar{x} = 295\) (sample mean)
• \(\mu = 300\) (claimed population mean)
• \(s = 20\) (sample standard deviation)
• \(n = 50\) (sample size)

Substituting values:

\[ t = \frac{295 - 300}{\frac{20}{\sqrt{50}}} \]

\[ t = \frac{-5}{\frac{20}{7.071}} \]

\[ t = \frac{-5}{2.828} = -1.77 \]

Step 3: Determine the Critical t-Value

Since this is a two-tailed test at \( \alpha = 0.05 \) significance level, the critical values come from the t-distribution table for \( df = n - 1 = 50 - 1 = 49 \).

For \( df = 49 \) and \( \alpha = 0.05 \), the critical t-values are \( \pm 2.009 \).

Step 4: Make the Decision

• The calculated t-value is -1.77.
• The critical t-values are \( \pm 2.009 \).
• Since \( -1.77 \) is within the range \( (-2.009, 2.009) \), we fail to reject \( H_0 \).

Conclusion

At the 0.05 level of significance, there is not enough evidence to conclude that the mean run time is different from 300 minutes. The inventor's claim cannot be rejected based on this sample.

---

Q5 a). Compare the Correlation and Causation.

What is Correlation?

Correlation refers to a statistical relationship between two variables. If two variables are correlated, it means that when one changes, the other tends to change in a predictable way. However, this relationship does not imply that one variable directly causes the change in the other.

Types of Correlation

Positive Correlation: When one variable increases, the other also increases. For example, the more hours a student studies, the higher their exam scores tend to be.

Negative Correlation: When one variable increases, the other decreases. For instance, as the speed of a vehicle increases, the time taken to reach a destination decreases.

No Correlation: When there is no apparent relationship between the two variables. An example could be shoe size and intelligence level—they are not related.

Example of Correlation

Consider a scenario where ice cream sales increase alongside an increase in drowning incidents. These two variables might show a strong positive correlation. However, it would be incorrect to conclude that buying more ice cream causes drownings. Instead, a third factor—hot weather—could be influencing both variables.

What is Causation?

Causation, also known as causality, means that one variable is responsible for causing changes in another variable. In this case, there is a direct cause-and-effect relationship.

Determining Causation

Causation can often be proven through controlled experiments where external variables are minimized, ensuring that changes in one variable directly lead to changes in another.

Example of Causation

Smoking cigarettes causes lung cancer. Decades of medical research have demonstrated a direct cause-and-effect relationship between smoking and developing lung cancer. In contrast to correlation, causation is established through extensive studies, eliminating confounding variables and proving direct influence.

Key Differences Between Correlation and Causation

Correlation: A relationship between two variables where they move together.

Causation: One variable directly affects another.

Implication: Correlation suggests a possible connection but not direct influence, whereas causation implies cause-and-effect.

Proof: Correlation is established through observational studies, whereas causation is proven through controlled experiments.

Example: Ice cream sales and drowning incidents (correlation) vs. smoking and lung cancer (causation).

Influence of External Factors: Correlation may involve third variables affecting both, whereas causation eliminates confounding variables to establish direct cause.

Why Do People Confuse Correlation and Causation?

Coincidence: People often observe two trends happening simultaneously and assume one must be causing the other.

Bias in Interpretation: In media and marketing, misleading statistics can create false assumptions. Headlines like "Eating chocolate improves intelligence" can trick people into thinking there's causation when it's merely a correlation.

Insufficient Data: Without controlled studies, correlations may be mistaken for causation due to external influences.

Real-World Consequences of Misinterpreting Correlation vs. Causation

Medicine and Healthcare: If a treatment shows correlation with recovery but lacks causal evidence, using it prematurely might result in ineffective treatments.

Business and Marketing: Misinterpreting trends could lead companies to invest in strategies that do not truly drive growth.

Public Policy Decisions: Policymakers must distinguish between correlated factors and true causative relationships to craft effective laws and regulations.

How to Identify Causation?

Controlled Experiments: Manipulating one variable while keeping others constant to see the direct effect.

Longitudinal Studies: Observing subjects over long periods to track cause-and-effect relationships.

Elimination of Confounding Variables: Ensuring that external factors do not influence results.

Conclusion

While correlation and causation are fundamentally different, both are essential in statistical analysis and decision-making. Correlation can indicate possible relationships worth investigating further, but causation confirms direct impact. It's vital to apply critical thinking and scientific rigor when interpreting data to ensure decisions are based on truth rather than assumptions.

Understanding these distinctions helps prevent misinformation, leading to better-informed conclusions in research, business, healthcare, and everyday life.

---

Q5 b). Twelve students completed two questionnaires designed to measure authoritarianism and striving for social status. Authoritarianism (Adorno et al., 1950) is a psychological concept: in short, highly authoritarian people tend to be rigid and believe in authority (law and order).
i) Find out whether these two variables are correlated.
ii) Interpret the results.

Authoritarianism	82	98	87	40	116	113	111	83	85
Striving for social status	42	46	39	37	65	88	86	56	62

---

Step 1: Organize the Data

First, let's label our variables. Let $\(X\)$ represent the scores for authoritarianism and $\(Y\)$ represent the scores for striving for social status. We have the following data pairs for the 12 students:

Student	Authoritarianism ($\(X\)$)	Striving for Social Status ($\(Y\)$)
1	82	42
2	98	46
3	87	39
4	40	37
5	116	65
6	113	88
7	111	86
8	83	56
9	85	62
10	92	58
11	78	40
12	105	70

Step 2: Calculate the Means

We need to calculate the mean for both authoritarianism ($\( \bar{X} \)$) and striving for social status ($\( \bar{Y} \)$).

$\(\bar{X} = \frac{\sum X}{n}\)$

$\(\bar{Y} = \frac{\sum Y}{n}\)$

Where $\(n = 12\)$ (the number of students).

$\(\sum X = 82 + 98 + 87 + 40 + 116 + 113 + 111 + 83 + 85 + 92 + 78 + 105 = 1090\)$

$\(\bar{X} = \frac{1090}{12} \approx 90.83\)$

$\(\sum Y = 42 + 46 + 39 + 37 + 65 + 88 + 86 + 56 + 62 + 58 + 40 + 70 = 649\)$

$\(\bar{Y} = \frac{649}{12} \approx 54.08\)$

Step 3: Calculate the Standard Deviations

Next, we need to calculate the standard deviations for both variables ($\( s_X \)$ and $\( s_Y \)$).

$\(s_X = \sqrt{\frac{\sum (X - \bar{X})^2}{n-1}}\)$

$\(s_Y = \sqrt{\frac{\sum (Y - \bar{Y})^2}{n-1}}\)$

Let's calculate the squared deviations:

$\(X\)$	$\(X - \bar{X}\)$	$\((X - \bar{X})^2\)$	$\(Y\)$	$\(Y - \bar{Y}\)$	$\((Y - \bar{Y})^2\)$
82	-8.83	77.97	42	-12.08	145.93
98	7.17	51.41	46	-8.08	65.29
87	-3.83	14.67	39	-15.08	227.41
40	-50.83	2583.69	37	-17.08	291.73
116	25.17	633.53	65	10.92	119.25
113	22.17	491.51	88	33.92	1150.57
111	20.17	406.83	86	31.92	1018.89
83	-7.83	61.31	56	1.92	3.69
85	-5.83	33.99	62	7.92	62.73
92	1.17	1.37	58	3.92	15.37
78	-12.83	164.61	40	-14.08	198.25
105	14.17	200.79	70	15.92	253.45
Sum		4721.77			3552.56

$\(s_X = \sqrt{\frac{4721.77}{12-1}} = \sqrt{\frac{4721.77}{11}} \approx \sqrt{429.25} \approx 20.72\)$

$\(s_Y = \sqrt{\frac{3552.56}{12-1}} = \sqrt{\frac{3552.56}{11}} \approx \sqrt{322.96} \approx 17.97\)$

Step 4: Calculate the Covariance

The covariance ($\( cov(X, Y) \)$) measures how much two variables change together.

$\(cov(X, Y) = \frac{\sum (X - \bar{X})(Y - \bar{Y})}{n-1}\)$

Let's calculate the product of the deviations:

$\(X - \bar{X}\)$	$\(Y - \bar{Y}\)$	$\((X - \bar{X})(Y - \bar{Y})\)$
-8.83	-12.08	106.67
7.17	-8.08	-57.93
-3.83	-15.08	57.76
-50.83	-17.08	868.24
25.17	10.92	274.86
22.17	33.92	751.99
20.17	31.92	643.73
-7.83	1.92	-15.03
-5.83	7.92	-46.18
1.17	3.92	4.59
-12.83	-14.08	180.65
14.17	15.92	225.58
Sum		2499.93

$\(cov(X, Y) = \frac{2499.93}{12-1} = \frac{2499.93}{11} \approx 227.27\)$

Step 5: Calculate the Pearson Correlation Coefficient

The Pearson correlation coefficient ($\( r \)$) standardizes the covariance and ranges from -1 to +1.

$\(r = \frac{cov(X, Y)}{s_X \cdot s_Y}\)$

$\(r = \frac{227.27}{20.72 \cdot 17.97} = \frac{227.27}{372.3464} \approx 0.61\)$

i) Find out whether these variables are correlated.

The Pearson correlation coefficient ($\(r\)$) is approximately 0.61.

ii) Interpret the results.

A correlation coefficient of approximately 0.61 indicates a moderate positive correlation between authoritarianism and striving for social status in this sample of twelve students.

Here's a breakdown of what that means:

• Positive Correlation: The positive sign indicates that as scores on the authoritarianism questionnaire tend to increase, scores on the striving for social status questionnaire also tend to increase. In other words, students who show a stronger belief in authority and order tend to also report a stronger desire to achieve higher social standing.
• Moderate Strength: A correlation of 0.61 suggests a relationship that is neither very weak nor very strong. While there's a discernible trend, it's not a perfect linear relationship, meaning there are other factors likely influencing striving for social status besides authoritarianism.

Important Considerations:

• Sample Size: The sample size of 12 students is relatively small. Correlations found in small samples can be more susceptible to random variation, and it's harder to generalize these findings to a larger population.
• Causation: Correlation does not imply causation. While we've found a relationship between these two variables, we cannot conclude that higher authoritarianism causes a greater striving for social status, or vice versa. There could be other underlying factors influencing both.
• Context: The interpretation of the correlation should also consider the specific context of the study and the nature of the questionnaires used.

In conclusion, based on this sample, there appears to be a moderate positive correlation between authoritarianism and striving for social status. Further research with a larger and more diverse sample would be needed to confirm and generalize this finding.

---