MASTER STATISTICS EXAM: DIVE INTO
PROBABILITY AND INFERENCE (ADVANCED
STATISTICAL ANALYSIS)
## Master Statistics Exam Paper: Probability and Inference
**Course Content:** Probability Theory, Statistical Inference
**Exam Name:** Advanced Statistical Analysis Test
**Date:** 12th December 2024
**Time:** 9:00 AM - 12:00 PM
**Total Marks:** 120
**Total Questions:**
35 ### Instructions:
- Answer all questions.
- The exam is closed book and closed notes. No electronic devices are
permitted.
- Calculators are allowed.
- Please provide detailed steps for all calculations.- Indicate your answers
clearly.
---
### Question 1: Probability Theory (10 marks)
, Suppose you have a fair 6-sided die and you roll it 100 times. What is the
probability that you will roll a "6" exactly 10 times?
**Answer:**
To find the probability of rolling a "6" exactly 10 times out of 100 rolls, we
can use the binomial probability formula: P(X = k) = (n choose k) * p^k * (1 -
p)^(n - k). Here, n = 100, k = 10, and p = 1/6 since the die is fair. Calculate
the probability.
---
### Question 2: Probability Theory (10 marks)
Consider a random variable X that follows a Poisson distribution with a
mean of 5. Find P(X ≤ 3).
**Answer:**
To calculate P(X ≤ 3) for a Poisson distribution with mean = 5, we use the
Poisson λ probability mass function: P(X = k) = ( ^k * e^(- )) / k!. Sum the
probabilities from k = 0 to λ λ k = 3. Provide the detailed calculation
steps.
---
### Question 3: Statistical Inference (12 marks)
Suppose you have a sample of size n = 50 drawn from a normal population
with an unknown mean and a known standard deviation of σ = 15. You find
that the sample mean is 100.
1. Calculate a 95% confidence interval for the population mean.
2. What does this confidence interval tell us about the true population
mean?
**Answer:**
PROBABILITY AND INFERENCE (ADVANCED
STATISTICAL ANALYSIS)
## Master Statistics Exam Paper: Probability and Inference
**Course Content:** Probability Theory, Statistical Inference
**Exam Name:** Advanced Statistical Analysis Test
**Date:** 12th December 2024
**Time:** 9:00 AM - 12:00 PM
**Total Marks:** 120
**Total Questions:**
35 ### Instructions:
- Answer all questions.
- The exam is closed book and closed notes. No electronic devices are
permitted.
- Calculators are allowed.
- Please provide detailed steps for all calculations.- Indicate your answers
clearly.
---
### Question 1: Probability Theory (10 marks)
, Suppose you have a fair 6-sided die and you roll it 100 times. What is the
probability that you will roll a "6" exactly 10 times?
**Answer:**
To find the probability of rolling a "6" exactly 10 times out of 100 rolls, we
can use the binomial probability formula: P(X = k) = (n choose k) * p^k * (1 -
p)^(n - k). Here, n = 100, k = 10, and p = 1/6 since the die is fair. Calculate
the probability.
---
### Question 2: Probability Theory (10 marks)
Consider a random variable X that follows a Poisson distribution with a
mean of 5. Find P(X ≤ 3).
**Answer:**
To calculate P(X ≤ 3) for a Poisson distribution with mean = 5, we use the
Poisson λ probability mass function: P(X = k) = ( ^k * e^(- )) / k!. Sum the
probabilities from k = 0 to λ λ k = 3. Provide the detailed calculation
steps.
---
### Question 3: Statistical Inference (12 marks)
Suppose you have a sample of size n = 50 drawn from a normal population
with an unknown mean and a known standard deviation of σ = 15. You find
that the sample mean is 100.
1. Calculate a 95% confidence interval for the population mean.
2. What does this confidence interval tell us about the true population
mean?
**Answer:**
