AD-302 Probability and Statistics for Data Science - Nov 2022 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aWhat is Probability? Explain the calculation of probability under the classical approach.
   How to answer

   • Define mathematical/statistical probability
   • Explain classical/Laplacian definition: $P(A) = m/n$
   • Assumptions: equally likely, mutually exclusive, exhaustive outcomes
   • Give a simple example (e.g., rolling a die or tossing a coin)
2. 7 Marks
   bWhat do you understand by regression? What role does it play in business and economic analysis?
   How to answer

   • Define regression analysis and regression equations
   • Distinction between dependent and independent variables
   • Role in business: demand estimation, sales forecasting, pricing decisions
   • Role in economics: econometric modeling, policy impact analysis
3. 7 Marks
   aA machine fills coffee powder in pouches, with an average of $200\text{ gm}$ and a standard deviation of $4\text{ gm}$. Assuming that the coffee weight is normally distributed. Find the probability that a coffee pouch selected at random will contain the following quantity of a coffee: i) Atleast $200\text{ gm}$. ii) Between $200$ to $206\text{ gm}$.
   How to answer

   • State $Z = (X - \mu)/\sigma$
   • For (i): $X=200 \implies Z = (200-200)/4 = 0$. $P(X \ge 200) = P(Z \ge 0) = 0.5$
   • For (ii): $X=206 \implies Z = (206-200)/4 = 1.5$. $P(0 \le Z \le 1.5) = \Phi(1.5) - 0.5 = 0.4332$
4. 7 Marks
   bList some of the important areas where Poisson distribution is used. Also state the important properties of a Poisson distribution.
   How to answer

   • Areas of application: call arrival at call centers, radioactive decay, traffic arrivals, defect count in manufacturing
   • Probability mass function: $P(X=x) = \frac{e^{-\lambda}\lambda^x}{x!}$
   • Properties: discrete distribution, $\text{Mean} = \text{Variance} = \lambda$, unimodal/bimodal conditions, additive property
5. 7 Marks
   aABC Tool Company makes Laser XR; a special engine used in speedboats. The company's engineers believe that the engine delivers an average power of $220\text{ horsepower}$, and that the standard deviation of power delivered is $15\text{ horsepower}$. A potential buyer intends to sample $100$ engines (each engine to be run a single time). What is the probability that the sample mean X will be less than $217\text{ horsepower}$?
   How to answer

   • Compute standard error: $\sigma_{\bar{X}} = 15/\sqrt{100} = 1.5$
   • Calculate Z-score: $Z = (217 - 220)/1.5 = -3/1.5 = -2.0$
   • Find probability: $P(\bar{X} < 217) = P(Z < -2.0) = 0.5 - 0.4772 = 0.0228$
6. 7 Marks
   b

   The data on price and quantity purchased relating to a commodity for 5 months is given below:

   Month :	January	February	March	April	May
   Prices(Rs):	10	10	11	12	12
   Quantity(Kg):	5	6	4	3	3

   Find the Karl Pearson's Coefficient of Correlation between prices and quantity and comment on its sign and magnitude.

   How to answer

   • Form a calculation table with $X, Y, X-\bar{X}, Y-\bar{Y}, (X-\bar{X})^2, (Y-\bar{Y})^2, (X-\bar{X})(Y-\bar{Y})$
   • Calculate means: $\bar{X} = 55/5 = 11$, $\bar{Y} = 21/5 = 4.2$
   • Compute deviations, sum of squares, and sum of cross-products
   • Evaluate $r = \frac{-3.6}{\sqrt{4 \times 6.8}} \approx -0.69$
   • Comment: Negative sign indicates inverse relationship; moderate to strong degree of correlation
7. 7 Marks
   aWhat is a sampling distribution, and what are the uses of sampling distributions?
   How to answer

   • Define sampling distribution as the probability distribution of a given statistic obtained from random samples
   • Concept of standard error
   • Uses: hypothesis testing, constructing confidence intervals, determining sample size, evaluating estimators
8. 7 Marks
   bThe average monthly electricity consumption for a sample of $100$ families is $1250\text{ units}$. Assuming the standard deviation of electric consumption of all families is $150\text{ units}$, construct a $95\text{ percent}$ confidence interval estimate of the actual mean electric consumption.
   How to answer

   • Identify critical value $Z_{0.025} = 1.96$ for $95\%$ confidence
   • Compute standard error: $\sigma/\sqrt{n} = 150/\sqrt{100} = 15$
   • Margin of error: $1.96 \times 15 = 29.4$
   • Confidence Interval: $1250 \pm 29.4 = [1220.6, 1279.4]\text{ units}$
9. 7 Marks
   a

   Fit a straight line trend by the method of least squares to the following data: -

   Year :	2012	2013	2014	2015	2016	2017
   Sales of T.V. sets (in '000):	7	10	12	14	17	24

   How to answer

   • Set origin at the middle of 2014 and 2015 (or $u = 2(X - 2014.5)$ so $\sum u = 0$)
   • Construct table with values of $X, Y, u, u^2, uY$
   • Calculate $a = \sum Y / n = 84 / 6 = 14$
   • Calculate $b = \sum uY / \sum u^2$
   • Formulate the straight-line equation $Y = a + bX$
10. 7 Marks
    bAssuming the probability of male birth as $\frac{1}{2}$, find the probability distribution of number of boys out of $5$ births. i) Find the probability that a family of $5$ children have I) atleast one boy II) atmost 3 boys ii) Out of 960 families with 5 children each find the expected number of families with I) and II) above
    How to answer

    • Binomial parameters: $n=5$, $p=1/2$, $q=1/2$
    • Distribution: $P(X=r) = \binom{5}{r}(1/2)^5 = \binom{5}{r}/32$
    • i) I) $P(X \ge 1) = 1 - P(X=0) = 1 - 1/32 = 31/32$
    • i) II) $P(X \le 3) = 1 - [P(X=4) + P(X=5)] = 1 - [5/32 + 1/32] = 26/32 = 13/16$
    • ii) Expected families: I) $960 \times (31/32) = 930$, II) $960 \times (26/32) = 780$
11. 7 Marks
    aWhat are the different components of statistics? How is statistics used in everyday life? Explain with suitable examples.
    How to answer

    • Components: Collection, Presentation/Organization, Analysis, and Interpretation of data
    • Branches: Descriptive statistics vs. Inferential statistics
    • Real-life applications: Weather forecasting, medical research, market surveys, finance, quality control
12. 7 Marks
    b"Correlation and Regression are two sides of the same coin". Explain.
    How to answer

    • Explain similarities: both study relationship between variables
    • State the mathematical link: $r = \pm \sqrt{b_{yx} b_{xy}}$
    • Explain differences: correlation measures degree/direction of association, regression predicts dependent from independent variable
    • Conclude why they complement each other
13. 7 Marks
    a"Every average has its own peculiar characteristics. It is difficult to say which average is the best." Explain with examples.
    How to answer

    • Briefly outline different averages (AM, Median, Mode, GM, HM)
    • Characteristics and ideal use case for each (e.g., AM for symmetric continuous data, Median for skewed/income data, Mode for popular sizes, GM for ratios/growth, HM for rates/speed)
    • Explain why no single average is universally 'best'
    • Provide illustrative examples
14. 7 Marks
    b

    In a hotel a total of $500$ bulbs were installed simultaneously and their failure over time was observed as detailed below. You are required to calculate the mean life of bulbs.

    End of week	1	2	3	4	5	6	7
    No. of failures	12	40	108	242	346	428	500

    How to answer

    • Recognize given data as cumulative frequencies ($cf$)
    • Obtain weekly failure frequencies ($f$): $f_1=12, f_2=28, f_3=68, f_4=134, f_5=104, f_6=82, f_7=72$
    • Determine midpoints or week number as $x_i$ (e.g. interval midpoints $0.5, 1.5, \dots, 6.5$ or week indices $1, 2, \dots, 7$)
    • Compute $\sum f_i x_i$
    • Calculate mean life $\bar{X} = \sum f_i x_i / 500$
15. 7 Marks
    aFor a distribution, Bowley's coefficient of skewness is $-0.56$, $Q_1 = 16.4$ and $\text{Median} = 24.2$. What is the coefficient of quartile deviation?
    How to answer

    • Use Bowley's formula: $S_k = \frac{Q_3 + Q_1 - 2M}{Q_3 - Q_1}$
    • Substitute given values: $-0.56 = \frac{Q_3 + 16.4 - 2(24.2)}{Q_3 - 16.4} = \frac{Q_3 - 32}{Q_3 - 16.4}$
    • Solve for $Q_3$: $-0.56(Q_3 - 16.4) = Q_3 - 32 \implies Q_3 \approx 26.4$
    • Calculate coefficient of quartile deviation: $\frac{Q_3 - Q_1}{Q_3 + Q_1} = \frac{26.4 - 16.4}{26.4 + 16.4} = \frac{10}{42.8} \approx 0.2336$
16. 7 Marks
    bDifferentiate between Binomial distribution, Poisson distribution and Normal distribution.
    How to answer

    • Formulate a comparative table with columns: Property, Binomial, Poisson, Normal
    • Type of variable: Discrete (Binomial, Poisson) vs Continuous (Normal)
    • Parameters: Binomial $(n, p)$, Poisson $(\lambda)$, Normal $(\mu, \sigma^2)$
    • Relation of Mean and Variance: $\text{Mean} > \text{Variance}$ (Binomial), $\text{Mean} = \text{Variance}$ (Poisson), Mean and Variance are independent (Normal)
    • Shapes and limiting conditions connecting them