Data Analytics and Data Analysis Question Bank with Solutions

Short Answers

1. Explain the importance of data in modern decision-making processes.

Answer:
Data is vital in today’s decision-making because it provides a factual basis for analysis, rather than relying on intuition or guesswork. Key reasons include:

Evidence-Based Decisions: Data enables organizations to make informed choices.
Trend Analysis: Patterns in data help predict future outcomes.
Improved Efficiency: Data-driven insights optimize processes and reduce costs.
Personalization: Businesses use data to offer tailored services (e.g., Netflix or Amazon recommendations).
Performance Measurement: Data helps track progress via KPIs and metrics.

2. Define random variable and differentiate between discrete and continuous random variables.

Answer:
A random variable is a variable whose outcome is determined by a random process.

Discrete Random Variable:
- Takes countable values (e.g., number of students in a class).
- Example: Number of heads in 10 coin tosses.
Continuous Random Variable:
- Takes infinite values within a range (e.g., weight, height, temperature).
- Example: Weight of a randomly selected person.

3. Compare the roles of a data analyst and a data scientist.

Answer:

Feature	Data Analyst	Data Scientist
Focus	Descriptive and diagnostic analytics	Predictive and prescriptive analytics
Tools	Excel, SQL, Tableau	Python, R, Machine Learning
Goal	Identify trends and patterns	Build models, make predictions
Skills	Data cleaning, visualization	Statistics, Machine Learning, Programming
Example	Create sales dashboards	Predict customer churn rate

4. Define central tendency and explain its types.

Answer:
Central tendency refers to the central or typical value of a dataset.
Types include:

Mean: Arithmetic average.
Median: Middle value when data is sorted.
Mode: Most frequent value in the dataset.

5. Explain variance and standard deviation. How do they differ?

Answer:

Variance (σ²): Measures how far each data point is from the mean, on average. $\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}$
Standard Deviation (σ): Square root of variance.
$\sigma = \sqrt{\text{Variance}}$
Difference: Variance gives result in squared units; standard deviation brings it to the original units.

6. Define probability and explain its role in data analytics.

Answer:
Probability is a measure of the likelihood that an event will occur (between 0 and 1).
In data analytics, probability helps:

Model Uncertainty: Predict outcomes in uncertain conditions.
Statistical Inference: Estimate population parameters using sample data.
Risk Analysis: Assess likelihood of possible risks.

7. How is the confidence interval estimated, and why is it important?

Answer:
Confidence Interval (CI) provides a range of values within which the true population parameter lies with a certain confidence level (e.g., 95%).

Formula: $CI = \bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right)$

$\bar{x}$: Sample mean
$Z$: Z-score from normal distribution
$\sigma$: Population standard deviation
$n$: Sample size

Importance:
Gives an estimate with a degree of certainty, accounts for sample variability.

8. Describe the concept of the distribution of sample means.

Answer:
The distribution of sample means (also called the sampling distribution) is the distribution formed by means of all possible samples from a population.

By the Central Limit Theorem:

The sampling distribution of the mean will be approximately normal if the sample size is large enough.
Mean of sampling distribution = population mean (μ)
Standard deviation = population standard deviation / √n (called standard error)

9. List three real-world applications of Python and explain their significance.

Answer:

Web Development – Using Django, Flask: Build dynamic websites (e.g., Instagram backend).
Data Science & AI – With libraries like NumPy, Pandas, scikit-learn: Analyze data, build ML models.
Automation/Scripting – Automating tasks like file handling, reporting (e.g., writing a Python script to send automated emails).

These applications make Python versatile and highly valuable across industries due to its simplicity, broad libraries, and community support.

10. Explain the relationship between confidence level, sample size, and margin of error.

Answer:

Confidence Level ↑ → Margin of Error ↑ (wider interval for more certainty)
Sample Size ↑ → Margin of Error ↓ (more precise estimate)
Margin of Error: Indicates the level of uncertainty around the sample estimate.

\[\text{Margin of Error} = Z \left(\frac{\sigma}{\sqrt{n}}\right)\]

Larger samples give more accurate population estimates.

Long Answers ( 4 Marks)

1. Find the population variance for the following set of data representing tree heights in feet: 3, 21, 98, 203, 17, 9.

Solution:

Let the data be:
x = {3, 21, 98, 203, 17, 9}

Step 1: Find the mean (μ)
$\mu = \frac{3 + 21 + 98 + 203 + 17 + 9}{6} = \frac{351}{6} = 58.5$

Step 2: Find the squared differences from the mean
$(3 - 58.5)^2 = 3080.25 \\ (21 - 58.5)^2 = 1416.25 \\ (98 - 58.5)^2 = 1580.25 \\ (203 - 58.5)^2 = 20880.25 \\ (17 - 58.5)^2 = 1722.25 \\ (9 - 58.5)^2 = 2450.25$

Step 3: Compute variance $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N} = \frac{3080.25 + 1416.25 + 1580.25 + 20880.25 + 1722.25 + 2450.25}{6} \\ = \frac{31129.5}{6} = 5188.25$

Answer: Population variance = 5188.25 ft²

2. There are 45 students in a class. 5 students were randomly selected and their weights (in kg) were: 51, 38, 79, 46, 57.

Solution:

This is sample data, so use sample mean and sample variance.

Step 1: Find mean $\bar{x} = \frac{51 + 38 + 79 + 46 + 57}{5} = \frac{271}{5} = 54.2 \text{ kg}$

Step 2: Find squared differences $(51 - 54.2)^2 = 10.24 \\ (38 - 54.2)^2 = 262.44 \\ (79 - 54.2)^2 = 615.04 \\ (46 - 54.2)^2 = 67.24 \\ (57 - 54.2)^2 = 7.84$

Step 3: Calculate sample variance and standard deviation $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{962.8}{4} = 240.7 \\ s = \sqrt{240.7} \approx 15.5 \text{ kg}$

Answer:

Sample mean = 54.2 kg
Sample variance = 240.7 kg²
Sample standard deviation ≈ 15.5 kg

3. A population has mean 75 and standard deviation 12. Random samples of size 121 are taken.

(a) Find mean and standard deviation of the sample mean.
$\mu_{\bar{x}} = \mu = 75 \\ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{12}{\sqrt{121}} = \frac{12}{11} = 1.09$

(b) If sample size = 400?
$\sigma_{\bar{x}} = \frac{12}{\sqrt{400}} = \frac{12}{20} = 0.6$

Answer:

(a) Sample mean = 75, Sample std. dev. = 1.09
(b) If n=400, Sample std. dev. = 0.6

4. What are the measures of dispersion? How are they useful in data analysis?

Answer:

Measures of dispersion describe the spread or variability in a dataset. Key types include:

Range: Difference between highest and lowest value.
Variance: Average squared deviation from the mean.
Standard Deviation: Square root of variance; interpretable in original units.
Interquartile Range (IQR): Difference between 75th and 25th percentile.

Importance in Data Analysis:

Identify consistency and variability.
Understand reliability of mean.
Detect outliers and anomalies.
Compare spread between different datasets.

5. A machine fills bottles with soda. A sample of 100 bottles showed mean = 2.01L, population σ = 0.05L. Find 90% confidence interval.

Solution:

Given:

\[\bar{x} = 2.01 \text{ L},\ \sigma = 0.05 \text{ L},\ n = 100\]
For 90% CI, Z = 1.645

CI formula: $CI = \bar{x} \pm Z \left( \frac{\sigma}{\sqrt{n}} \right) = 2.01 \pm 1.645 \times \frac{0.05}{10} \\ = 2.01 \pm 0.008225$

\[\text{CI} = [2.0018,\ 2.0182] \text{ L}\]

Answer: 90% Confidence Interval = [2.0018 L, 2.0182 L]

6. Survey of 81 households: mean bill = $110, σ = $15. Find 95% CI.

Given:

\[\bar{x} = 110, \ \sigma = 15, \ n = 81\]
Z = 1.96 for 95%

\[CI = 110 \pm 1.96 \times \frac{15}{\sqrt{81}} = 110 \pm 1.96 \times \frac{15}{9} \\ = 110 \pm 3.27\]

Answer: 95% Confidence Interval = [$106.73,\ 113.27$]

7. Differentiate between discrete and continuous probability distributions. Give one example of each.

Answer:

Aspect	Discrete	Continuous
Takes values	Countable	Infinite within a range
Distribution	Probability Mass Function (PMF)	Probability Density Function (PDF)
Example	Binomial Distribution	Normal Distribution

Example:

Discrete: Number of customers entering a store.
Continuous: Heights of students in a class.

8. Random samples of size 225 drawn from population with μ = 100, σ = 20. Find mean and standard deviation of sample mean.

\[\mu_{\bar{x}} = \mu = 100 \\ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{225}} = \frac{20}{15} = 1.33\]

Answer:

Mean of sample means = 100
Std. dev. of sample mean = 1.33

9. Define range, variance, and standard deviation. How are they useful in data analysis?

Definitions:

Range: Max value – Min value.
Variance: Average of squared differences from mean.
Standard Deviation: Square root of variance.

Use in analysis:

Shows how data is spread.
High standard deviation means more variability.
Critical for comparing datasets, risk assessment, and anomaly detection.

Long Answers (5 Marks)

1. Define probability distribution. Explain its types with example.

Answer:

A probability distribution is a function that provides the probabilities of occurrence of different possible outcomes in an experiment.

Types of Probability Distributions:

Discrete Probability Distribution:
- Deals with discrete random variables (finite/countable outcomes).
- Represented by Probability Mass Function (PMF).
- Example: Binomial Distribution
  - Tossing a coin 3 times: P(X = number of heads)
Continuous Probability Distribution:
- Deals with continuous random variables (infinite outcomes in an interval).
- Represented by Probability Density Function (PDF).
- Example: Normal Distribution
  - Heights of students follow a bell-shaped curve.

Key Differences:

Feature	Discrete	Continuous
Variable type	Countable	Uncountably infinite
Probability Function	PMF	PDF
Example	Poisson, Binomial	Normal, Exponential

2. A sample of 49 students had a mean course completion time of 38 hours. Population standard deviation is 5 hours. Find 99% confidence interval for the mean.

Solution:

Given:

Sample mean, $\bar{x} = 38$
Population standard deviation, $\sigma = 5$
Sample size, $n = 49$
Z-value for 99% CI = 2.576

Step 1: Standard error: $SE = \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{49}} = \frac{5}{7} = 0.714$

Step 2: Margin of error: $ME = Z \times SE = 2.576 \times 0.714 = 1.839$

Step 3: Confidence Interval: $CI = \bar{x} \pm ME = 38 \pm 1.839 = [36.161,\ 39.839]$

Answer: 99% Confidence Interval = [36.161, 39.839] hours

3. Explain how Python is used in Data Science, Artificial Intelligence, and Web Development. Discuss examples and libraries.

Answer:

Python’s simplicity, rich libraries, and active community make it ideal for various tech domains:

1. Data Science

Used for data processing, analysis, visualization.
Libraries:

Pandas – For data manipulation
NumPy – Numeric computations
Matplotlib / Seaborn – Data visualization
Example: Analyze sales data and generate graphs.

2. Artificial Intelligence

Used to build ML and deep learning models.
Libraries:

Scikit-learn – ML algorithms (classification, regression)
TensorFlow / PyTorch – Deep learning
Example: Predict house prices, classify emails as spam.

3. Web Development

Used to develop backends and dynamic websites.
Libraries/Frameworks:

Flask and Django – Web frameworks
Example: Build a blog website or e-commerce platform backend.

Conclusion: Python’s versatility enables integration between data handling, AI logic, and deployment via web interfaces.

4. Differentiate between population and sample statistics. Explain formulas for population mean and variance with example.

Answer:

Feature	Population	Sample
Data	Entire group	Subset of the population
Size notation	$N$	$n$
Mean notation	$\mu$ (mu)	$\bar{x}$
Variance	$\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$	$s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$

Example:

Population data: {2, 5, 7, 9, 10}

Mean (μ): $\mu = \frac{2 + 5 + 7 + 9 + 10}{5} = \frac{33}{5} = 6.6$
Variance (σ²): $\sigma^2 = \frac{(2 - 6.6)^2 + (5 - 6.6)^2 + \dots + (10 - 6.6)^2}{5} \\ = \frac{21.16 + 2.56 + 0.16 + 5.76 + 11.56}{5} = \frac{41.2}{5} = 8.24$

Answer:

Population Mean = 6.6
Population Variance = 8.24

5. Explain in detail how confidence intervals are constructed. Derive the formula and interpret a numerical result.

Answer:

A confidence interval (CI) gives a range of values within which the true population parameter is likely to fall, with a certain confidence level (e.g., 95%).

Confidence Interval Formula (for large sample or known σ):

$CI = \bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right)$

Where:

$\bar{x}$ = sample mean
$Z$ = Z-score based on confidence level
$\sigma$ = population standard deviation
$n$ = sample size

Steps:

Compute standard error:
$SE = \frac{\sigma}{\sqrt{n}}$
Find margin of error (ME):
$ME = Z \cdot SE$
Apply the CI formula.

Numerical Example:

Sample mean $\bar{x} = 80$, σ = 10, n = 64, 95% CI (Z = 1.96)

\[SE = \frac{10}{8} = 1.25\]
\[ME = 1.96 \cdot 1.25 = 2.45\]
\[CI = 80 \pm 2.45 = [77.55,\ 82.45]\]

Interpretation: We are 95% confident that the true population mean lies between 77.55 and 82.45.

6. Why is data considered a valuable asset for organizations?

Answer:

Data is often called the “new oil” due to its importance in driving modern business decisions.

Reasons why data is a valuable asset:

Informed Decision-Making: Data supports evidence-based strategies.
- E.g., Amazon uses customer behavior data to improve suggestions.
Efficiency & Cost Reduction: Helps automate processes and avoid waste.
- E.g., Predictive maintenance in manufacturing.
Customer Insights & Personalization:
- Tailors services to individual preferences (Netflix, Spotify).
Competitive Advantage:
- Data-driven firms outperform others.
Risk Management:
- Fraud detection, credit scoring using data analytics.

Conclusion: Proper use of data empowers smarter, faster, and more accurate decisions across all sectors.

7. What are the measures of dispersion? How are they useful in data analysis?

Answer:

Measures of dispersion indicate how data points in a dataset are spread or scattered around the central value.

Main Types:

Range = Max – Min
Variance (σ² or s²):
- Shows average of squared deviations from mean.
Standard Deviation (σ or s):
- Square root of variance; used widely due to interpretable units.
Interquartile Range (IQR):
- Spread of middle 50% data = Q3 – Q1

Importance in Data Analysis:

Detect Variability: High dispersion → low consistency
Compare Datasets: One dataset may have the same mean but higher variance.
Identify Outliers: Dispersion helps flag anomalies.
Improve Predictions: Tight dispersion leads to better models.

Conclusion: Dispersion measures are essential for determining data reliability, variability, and comparisons.