ENSH 301Bachelor in Mechanical Engineering · Semester 54 Papers Available
Probability and Statistics
Past examination question papers and complete curriculum syllabus for Probability and Statistics (ENSH 301), Bachelor in Mechanical Engineering Semester 5 under Institute of Engineering (IOE), Tribhuvan University.
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Most Frequently Asked Questions
Top recurring IOE board exam questions for Probability and Statistics with verified mark schemes, formula notation, and recurrence frequency.
Showing 30 of 30 top repeated questions
Descriptive Statistics and Basic Probability
4 Questions
#1Repeated 4 Times[6 Marks]Descriptive Statistics and Basic Probability
State Bayes' Theorem. In a bolt factory, machines A, B, and C manufacture 50%, 30%, and 20% of the total output, and of their output 2%, 3%, and 4% respectively are defective. A bolt is drawn at random and found to be defective. What is the probability that it was manufactured by machine A?
Appeared in:2082 Bhadra2080 Bhadra2075 Chaitra2073 Shrawan
#2Repeated 3 Times[6 Marks]Descriptive Statistics and Basic Probability
List the Five-Number summary and prepare the Box-and-Whisker plot for the numbers of guests registered each day at a hotel. Calculate the mean, median, standard deviation, and inter-quartile range (IQR).
Appeared in:2082 Bhadra2080 Bhadra2075 Chaitra
#3Repeated 2 Times[6 Marks]Descriptive Statistics and Basic Probability
Define mean, median, mode, variance, and standard deviation. Explain Karl Pearson's and Bowley's coefficient of skewness with formulas and interpretation.
Appeared in:2082 Kartik2081 Chaitra
#4Repeated 2 Times[5 Marks]Descriptive Statistics and Basic Probability
State the axioms of probability and addition/multiplication theorems. An urn contains 5 red and 7 black balls. Two balls are drawn successively without replacement. Find the probability that both are black.
Appeared in:2082 Shrawan2079 Baishakh
Probability Distributions and Sampling Distribution
9 Questions
#1Repeated 5 Times[6 Marks]Probability Distributions and Sampling Distribution
State the Central Limit Theorem. An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a mean of 800 hours and a standard deviation of 40 hours. Find the probability that a random sample of 64 bulbs will have an average life between 790 and 810 hours.
Appeared in:2082 Bhadra2080 Baishakh2078 Kartik2075 Chaitra2074 Chaitra
#2Repeated 4 Times[8 Marks]Probability Distributions and Sampling Distribution
Define standard normal distribution and write down its importance in engineering. The breakdown voltage $X$ of a randomly chosen diode of a particular type is known to be normally distributed with mean $\mu = 40\text{ V}$ and standard deviation $\sigma = 1.5\text{ V}$. What is the probability that the breakdown voltage is between $38\text{ V}$ and $43\text{ V}$?
Appeared in:2082 Bhadra2080 Baishakh2078 Kartik2076 Ashwin
#3Repeated 4 Times[6 Marks]Probability Distributions and Sampling Distribution
A continuous random variable $X$ has the probability density function $f(x) = kx(2 - x)$ for $0 \le x \le 2$ and $0$ elsewhere. Find the value of constant $k$, the cumulative distribution function $F(x)$, the mathematical expectation $E(X)$, and the variance $\text{Var}(X)$.
Appeared in:2082 Bhadra2080 Bhadra2076 Ashwin2073 Shrawan
#4Repeated 4 Times[6 Marks]Probability Distributions and Sampling Distribution
Compare Binomial and Negative Binomial distributions. Define Poisson distribution and state the conditions under which the Poisson distribution serves as a limiting approximation to the Binomial distribution with examples.
Appeared in:2082 Bhadra2080 Bhadra2078 Kartik2074 Chaitra
#5Repeated 4 Times[8 Marks]Probability Distributions and Sampling Distribution
A population consists of five numbers: $2, 4, 6, 8, 10$. Consider all possible samples of size $n=2$ that can be drawn without replacement from this population. Show that the sample mean is an unbiased estimator of the population mean and verify that $\text{Var}(\bar{X}) = \frac{\sigma^2}{n}\left(\frac{N-n}{N-1}\right)$.
Appeared in:2082 Bhadra2080 Bhadra2078 Kartik2075 Chaitra
#6Repeated 2 Times[6 Marks]Probability Distributions and Sampling Distribution
Define Poisson distribution as a limiting case of Binomial distribution. If a random variable $X$ follows Poisson distribution such that $P(X = 1) = P(X = 2)$, find $P(X = 4)$ and $P(X \ge 1)$.
Appeared in:2082 Kartik2081 Chaitra
#7Repeated 2 Times[6 Marks]Probability Distributions and Sampling Distribution
Define exponential distribution and memoryless property. If the lifetime of an electronic component is exponentially distributed with mean $\theta = 1000\text{ hours}$, find the probability that it lasts more than $1500\text{ hours}$ given that it has already functioned for $500\text{ hours}$.
Appeared in:2081 Chaitra2080 Chaitra
#8Repeated 2 Times[8 Marks]Probability Distributions and Sampling Distribution
The marks obtained by students in an examination are normally distributed with mean $\mu = 55$ and standard deviation $\sigma = 10$. If $1000$ students appeared, find: (i) number of students scoring above $75$, (ii) number of students scoring between $40$ and $70$, and (iii) minimum marks required for top $10\%$ distinction.
Appeared in:2082 Kartik2081 Chaitra
#9Repeated 2 Times[6 Marks]Probability Distributions and Sampling Distribution
State and prove the Central Limit Theorem (CLT) for independent and identically distributed random variables. Explain its role in large sample hypothesis testing.
Appeared in:2082 Kartik2080 Chaitra
Statistical Inference
14 Questions
#1Repeated 4 Times[6 Marks]Statistical Inference
Define hypothesis. What are Type I and Type II errors? Describe the decision criteria and procedure of the test of significance for the difference of two population means for large samples ($Z$-test).
Appeared in:2082 Bhadra2080 Baishakh2078 Kartik2075 Chaitra
#2Repeated 4 Times[8 Marks]Statistical Inference
Define Chi-square distribution. Bricks made in four kilns have been graded as high, medium, and low quality. From the contingency data, test at $\alpha = 0.05$ whether the quality of bricks is independent of the kilns in which they were produced.
Appeared in:2082 Bhadra2080 Bhadra2078 Kartik2075 Chaitra
#3Repeated 3 Times[6 Marks]Statistical Inference
What are estimator and estimates? Describe the criteria for a good estimator (unbiasedness, consistency, efficiency, and sufficiency). A random sample of size 16 showed a mean of 52 with a standard deviation of 4. Obtain a 95% confidence interval for the population mean $\mu$.
Appeared in:2080 Baishakh2078 Kartik2076 Chaitra
#4Repeated 3 Times[6 Marks]Statistical Inference
Ten objects were chosen at random from a large population and their weights were found to be: $63, 63, 64, 65, 66, 69, 69, 70, 70, 71\text{ kg}$. In the light of these data, test the hypothesis that the mean weight in the population is $65\text{ kg}$ at $5\%$ level of significance ($t_{0.05, 9} = 2.262$).
Appeared in:2080 Baishakh2078 Kartik2075 Chaitra
#5Repeated 3 Times[6 Marks]Statistical Inference
Explain the procedure for the test of significance of paired data (paired $t$-test). In trying to evaluate the effectiveness of a new antihypertensive drug, blood pressure was measured in 10 patients before and after treatment. Test whether the drug is effective at $\alpha = 0.05$.
Appeared in:2080 Baishakh2076 Ashwin2074 Chaitra
#6Repeated 3 Times[8 Marks]Statistical Inference
Three trained operators work on the production of a new product. Given productivity outputs across treatments, perform a One-Way Analysis of Variance (ANOVA) test at $\alpha = 0.05$ to conclude whether there is a significant difference among the average outputs of the three operators.
Appeared in:2082 Bhadra2080 Baishakh2078 Kartik
#7Repeated 3 Times[6 Marks]Statistical Inference
In a random sample of 500 households, 330 said they owned a solar water heater. Test the hypothesis that at least 60% of all households have solar water heaters at $\alpha = 0.05$ significance level.
Appeared in:2080 Baishakh2078 Kartik2074 Chaitra
#8Repeated 2 Times[6 Marks]Statistical Inference
Differentiate between point estimation and interval estimation. Derive the $100(1-\alpha)\%$ confidence interval for the population mean $\mu$ when population variance $\sigma^2$ is unknown (small sample $n < 30$ using Student's $t$-distribution).
Appeared in:2081 Chaitra2079 Chaitra
#9Repeated 2 Times[6 Marks]Statistical Inference
A random sample of 64 bulb lifetimes gave a mean of 1200 hours and a standard deviation of 80 hours. Construct a $95\%$ and $99\%$ confidence interval for the true average lifetime of all bulbs.
Appeared in:2082 Kartik2078 Bhadra
#10Repeated 2 Times[6 Marks]Statistical Inference
Explain Type I error ($\alpha$), Type II error ($\beta$), and power of a statistical test ($1-\beta$). How are the null hypothesis $H_0$ and alternative hypothesis $H_1$ formulated?
Appeared in:2082 Kartik2081 Chaitra
#11Repeated 2 Times[8 Marks]Statistical Inference
A manufacturer claims that the average tensile strength of a synthetic fiber exceeds $85\text{ kg/cm}^2$. A sample of 16 specimens tested gave a mean tensile strength of $87.5\text{ kg/cm}^2$ with standard deviation $4\text{ kg/cm}^2$. Test the claim at $\alpha = 0.05$ significance level using Student's $t$-test.
Appeared in:2082 Kartik2080 Chaitra
#12Repeated 2 Times[8 Marks]Statistical Inference
Explain the Paired $t$-test for dependent samples. Ten participants were tested on memory scores before and after a training program. Test whether the training program significantly improved memory scores at $\alpha = 0.05$.
Appeared in:2081 Chaitra2079 Baishakh
#13Repeated 2 Times[6 Marks]Statistical Inference
Explain the Chi-Square ($\chi^2$) test for independence of attributes. State the conditions required for the validity of the $\chi^2$ test.
Appeared in:2082 Kartik2081 Chaitra
#14Repeated 2 Times[8 Marks]Statistical Inference
Explain One-Way Analysis of Variance (ANOVA) technique. Derive the formulas for Between-Group Sum of Squares (SSB), Within-Group Sum of Squares (SSW), and the $F$-statistic test.
Appeared in:2082 Kartik2080 Chaitra
Correlation and Regression
3 Questions
#1Repeated 4 Times[8 Marks]Correlation and Regression
Differentiate between correlation and regression analysis. From a sample of bivariate observations $(X, Y)$, calculate Karl Pearson's correlation coefficient $r$, fit the sample linear regression equation of $Y$ on $X$ using the method of least squares, and state properties of regression coefficients.
Appeared in:2082 Bhadra2080 Bhadra2078 Kartik2075 Chaitra
#2Repeated 4 Times[6 Marks]Correlation and Regression
Define partial and multiple correlation with suitable examples. Write down the properties of multiple correlation coefficients. If the simple correlation coefficients between fertilizer ($X_1$), seeds ($X_2$), and productivity ($X_3$) are $r_{12} = 0.8$, $r_{13} = 0.6$, and $r_{23} = 0.5$, calculate $r_{12.3}$ and $R_{1.23}$.
Appeared in:2082 Bhadra2080 Baishakh2078 Kartik2076 Ashwin
#3Repeated 2 Times[8 Marks]Correlation and Regression
Derive the two linear regression lines: $Y$ on $X$ and $X$ on $Y$. Prove that the correlation coefficient $r = \pm\sqrt{b_{yx} \cdot b_{xy}}$ and lies between $-1$ and $+1$.
Chapter-wise Units & Micro-Syllabus Topics (5 Units)
1. Descriptive Statistics and Basic Probability
1.1Introduction to statistics and its importance in engineering
1.2Measure of central tendency and measure of variation
1.3Graphical representation of data: Histograms, box plots and scatter plots
1.4Basic probability concepts, additive law, multiplicative law
1.5Conditional probability and Bayes’ theorem
2. Probability Distributions and Sampling Distribution
2.1Random variables: Discrete and continuous
2.2Expectation and variance of discrete and continuous random variables
2.3Discrete probability distributions: Binomial, Poisson, negative Binomial
2.4Continuous probability distributions: Normal, Gamma, Chi-Square
2.5Population and sample
2.6Sampling distribution of mean and proportion
2.7Central limit theorem
3. Statistical Inference
3.1Point estimations and properties of estimators
3.2Confidence intervals for mean and proportions
3.3Hypothesis testing, parametric and non-parametric tests, procedure of hypothesis
3.4Hypothesis testing of mean (Single mean, two means, paired t-test and one-way)
3.5Goodness of fit tests and independence of attributes (Chi-square and Kolmogorov–Smirnov test)
4. Correlation and Regression
4.1Correlation analysis and test of linear correlation
4.2Simple regression analysis, the concept of explained, unexplained, and total
4.3Multiple regression analysis
5. Statistical Quality Control
5.1Quality control and its importance in engineering
5.2Control charts for variables (X-bar, R-chart, P-chat)
5.3Six sigma concepts
Examination Scheme & Marks Distribution
Evaluation Structure
Final Board Theory Exam: 60 Marks (Pass mark: 24)
Internal Assessment: 40 Marks (Pass mark: 16)
Practical / Lab Exam: 25 or 50 Marks (Continuous lab evaluation + viva, where applicable)
* This is the general current IOE 60/40 scheme; verify course-specific details in the syllabus above.
Exam Preparation Guidelines
Review the 4 available past examination papers to identify recurring patterns, core problem types, and chapter weightage.
Practice numerical problems step-by-step with clean formula derivations, clear units, and standard assumptions.
Structure answers with labeled diagrams, concise bullet points, and highlight final answers in numerical solutions.
Frequently Asked Questions (Probability and Statistics)
Q: How can I download Probability and Statistics past question papers?
You can preview or download the Probability and Statistics question papers (PDF) directly using the built-in viewer on this page with zero redirects or paywalls.
Q: What is the pass mark for Probability and Statistics?
The general current scheme is a 60-mark final theory exam and a 40-mark internal assessment, with pass marks of 24 and 16. Verify the course-specific syllabus above.
Q: Where can I find the complete syllabus for this subject?
The available chapter-wise syllabus and topic breakdown is indexed in the Syllabus section above, with links to the curriculum PDF source.