IGNOU MSCDSA MCS 66 Mathematical Foundations - II Solved Assignment ENGLISH

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MCS 066 (July 2026 - January 2027) - ENGLISH

Course Code : MCS-066
Course Title : Mathematical Foundations for Data Science-II
Assignment Number : MSCDSA (II)/066/Assign/2026-27
Maximum Marks : 100
Weightage : 30%
Last Date of Submission : 31st October, 2026 (for July session)
30th April, 2027 (for January session)
There are four questions in this assignment, which carry 80 marks. Each question carries 20
marks. Rest 20 marks are for viva voce. You may use illustrations and diagrams to enhance
the explanations, if necessary.
Q1: (Covers Block 1)
(a) A retail company wants to study customer purchasing behaviour in its stores. What would be the
population for this study be? How will you sample the data and draw inferences? Also, differentiate
between descriptive and inferential statistics with suitable examples. (2 Marks)
(b) A researcher collects information on the following variables from 200 university students: (2 Marks)
 Gender
 Academic programme
 Annual family income
 Student satisfaction level (Very Low to Very High)
 CGPA
Identify the appropriate measurement scale (nominal, ordinal, interval or ratio) for each variable. Also,
explain the importance of selecting the correct measurement scale during statistical analysis.
(c) The following marks (out of 100) were obtained by 30 students: (5 Marks)
42, 55, 67, 72, 48, 63, 59, 74, 81, 65, 58, 69, 77, 84, 71, 62, 57, 66, 73, 60, 75, 79, 68, 64, 53, 70, 61, 56,
76, 82
Construct:
 a frequency distribution,
 relative frequency table,
 histogram,
 ogive, and
 stem-and-leaf plot.
(d) Using the data given in Question 3, calculate: (5 Marks)
 Mean
 Median
 First Quartile (Q1)
 Third Quartile (Q3)
Draw a box plot and comment on the presence of any outliers.

(e) The monthly sales (in lakh rupees) of a company for twelve months are: (4 Marks)
35, 42, 38, 41, 45, 39, 47, 50, 43, 44, 46, 40
Calculate:
 Range
 Variance
 Standard deviation
 Coefficient of variation
Comment on the sales pattern.
(f) Explain the concepts of skewness and kurtosis. A data scientist observes two datasets with the same mean
and variance but different skewness and kurtosis. Explain how these measures help in understanding the
nature of the distributions. Also, explain the characteristics of a normally distributed dataset. (2 Marks)
Q2: (Covers Block 2)
(a) If the letters of the word “Probability” are arranged randomly, what is the probability that all the letter ‘b’
are together in the random arrangement. (2 Marks)
(b) A factory produces bulbs using two machines: (3 Marks)
 Machine A manufactures 60% of the bulbs, and 2% of the bulbs it produces are defective.
 Machine B manufactures 40% of the bulbs, and 5% of the bulbs produced by it are defective.
A randomly selected bulb is found to be defective. Using Bayes' theorem, determine the probability that
the defective bulb was manufactured by Machine B.

(d) Explain the concepts of joint probability mass function and marginal probability function for two discrete
random variables with the help of an example. (2 Marks)
(e) A manufacturing process produces 2% defective products. A random sample of 25 products is selected.
Perform the following tasks for this process: (2 Marks)
(i) Calculate the probability of obtaining:
5
 exactly one defective product,
 at most two defective products.
(ii) State the assumptions under which the binomial distribution is applicable.
(f) Explain the Bernoulli, Binomial, Poisson and Hypergeometric distributions. Compare their assumptions,
parameters and applications. (3 Marks)
(g) The lifetime (in hours) of a sensor follows a normal distribution with a mean of 800 hours and a standard
deviation of 60 hours. Compute the following for this sensor: (3 Marks)
 Probability that a sensor lasts more than 850 hours.
 Probability that its lifetime lies between 760 and 900 hours.
(h) Explain the Uniform, Normal, Chi-square, t and F distributions. Highlight the relationships among these
distributions. (3 Marks)
Q3: (Covers Block 3)
(a) Explain the Central Limit Theorem and discuss its significance in statistical inference. Illustrate how the
theorem helps in estimating population parameters when the population distribution is unknown.
 (3 Marks)
(b) A population has a mean of 80 and a standard deviation of 18. Random samples of size 64 are repeatedly
selected and reported. Compute the following for this: (3 Marks)
 Mean of the sampling distribution of the sample mean.
 Standard error.
 Probability that the sample mean exceeds 84.
(c) A random sample of 64 observations has a sample mean of 48 and a sample standard deviation of 10.
Find the following: (4 Marks)
 A 95% confidence interval for the population mean.
 Distinguish between point estimation and interval estimation.
(d) Explain the principles of Maximum Likelihood Estimation (MLE). Also, explain the characteristics of an
estimator. (3 Marks)
(e) A company claims that the average battery life of its rechargeable battery is 18 hours. A random sample of
40 batteries gives a mean life of 17.2 hours with a standard deviation of 2.5 hours. Using a 5%
significance level, test the manufacturer's claim. (4 Marks)
Clearly state the following while answering:
 Null and alternative hypotheses,
 Test statistic,
 Conclusion.
(f) Explain the concepts of significance level, Type-I error, Type-II error and p-value. Also, compare the ztest and t-test, and discuss the situations in which each test should be applied.

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