AD-302 Probability and Statistics for Data Science - Dec 2023 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
   a

   Calculate the mean and standard deviation for the following table given the age distribution of 542 members.

   Age in year	20-30	30-40	40-50	50-60	60-70	70-80	80-90
   No. of members	3	61	132	153	140	51	2

   How to answer

   • Find mid-values (x) for each class interval (25, 35, 45, 55, 65, 75, 85)
   • Choose an assumed mean A and compute deviations u = (x - A)/h
   • Calculate products f*u and f*u^2 and their sums
   • Apply formula to calculate mean
   • Apply formula to calculate standard deviation
2. 7 Marks
   bThe first of the two samples has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation $\sqrt{13.44}$, find the standard deviation of the second group.
   How to answer

   • Identify n2 = 250 - 100 = 150
   • Use combined mean formula to find x̄2
   • Calculate deviations d1 = x̄1 - x̄ and d2 = x̄2 - x̄
   • Substitute into combined variance formula to solve for σ2
3. 14 Marks
   The first four moments of the distribution about the value for 4 of the variable are $-1.5, 17, -30$ and $108$, find the moments about mean $\beta_1$ and $\beta_2$.
   How to answer

   • State formulas relating central moments (μ) to raw moments (μ')
   • Calculate μ2, μ3, and μ4
   • Calculate skewness measure β1 = μ3^2 / μ2^3
   • Calculate kurtosis measure β2 = μ4 / μ2^2
4. 7 Marks
   aA continuous random variable $X$ has a p.d.f. $f(x) = 3x^2, 0 \le x \le 1$. Find $a$ and $b$ such that i) $P(X < a) = P(X > a)$, ii) $P(X > b) = 0.05$.
   How to answer

   • Integrate f(x) to obtain CDF F(x) = x^3
   • For (i), set F(a) = 0.5 => a^3 = 0.5 => a = (0.5)^(1/3)
   • For (ii), set 1 - F(b) = 0.05 => 1 - b^3 = 0.05 => b^3 = 0.95 => b = (0.95)^(1/3)
5. 7 Marks
   bTen coins are thrown simultaneously. Find the probability of getting at least seven heads.
   How to answer

   • Define parameters n=10, p=1/2, q=1/2
   • Write formula for binomial probability P(X = r) = 10Cr * (1/2)^10
   • Compute 10C7, 10C8, 10C9, 10C10
   • Sum the probabilities: (120 + 45 + 10 + 1) / 1024 = 176 / 1024 = 11 / 64
6. 7 Marks
   a

   Fit a straight line to the following data:

   X:	1	2	3	4	6	8
   Y:	2.4	3	3.6	4	5	6

   How to answer

   • Form table for X, Y, X^2, and XY (noting Y=2.4 as likely intended for printed '204')
   • Calculate sums: ΣX, ΣY, ΣX^2, ΣXY
   • Set up normal equations for Y = a + bX
   • Solve the simultaneous equations for a and b
   • Write the fitted equation
7. 7 Marks
   b

   Ten competitors in a musical test were ranked by the three judges A, B and C following order

   Rank by A	1	6	5	10	3	2	4	9	7	8
   Rank by B	3	5	8	4	7	10	2	1	6	9
   Rank by C	6	4	9	8	1	2	3	10	5	7

   Using Rank Correlation method, discuss which pair of judges has the nearest approach to common likings in music.

   How to answer

   • Calculate differences of ranks d_AB, d_BC, and d_AC for each competitor
   • Compute sums: Σd_AB^2, Σd_BC^2, and Σd_AC^2
   • Calculate Spearman's rank correlation coefficient for each pair: ρ(A, B), ρ(B, C), ρ(A, C)
   • Identify the highest positive correlation coefficient to determine the pair with the nearest approach
8. 7 Marks
   aVariable X and Y have the joint p.d.f. $$f(x, y) = \frac{1}{3}(x + y), 0 \le x \le 1, 0 \le y \le 2$$ Find: i) $r(X, Y)$ ii) The two lines of regression iii) The two regression curve for the means
   How to answer

   • Determine marginal distributions f_X(x) and f_Y(y)
   • Compute E(X), E(Y), E(X^2), E(Y^2), and Var(X), Var(Y)
   • Compute E(XY) and Cov(X, Y), then correlation coefficient r(X, Y)
   • Derive equations for lines of regression of Y on X and X on Y
   • Find regression curves of the means using conditional expectations E(Y|X=x) and E(X|Y=y)
9. 7 Marks
   b

   Find the most likely price in Bombay corresponding to the price of Rs. 70 at Calcutta from the following:

   	Calcutta	Bombay
   Average price	65	67
   Standard deviation	2.5	3.5

   Correlation coefficient between the prices of commodities in the two cities is 0.8.

   How to answer

   • Let x denote Calcutta price and y denote Bombay price
   • Write down the regression line equation of y on x: (y - ȳ) = r * (σy / σx) * (x - x̄)
   • Substitute the given values: x̄ = 65, ȳ = 67, σx = 2.5, σy = 3.5, r = 0.8
   • Substitute x = 70 and calculate y
10. 7 Marks
    aTest the hypothesis data $\sigma = 10$ given data $s = 15$ for a random sample of the size 50 from a normal population.
    How to answer

    • Formulate null hypothesis H0: σ = 10 against H1: σ ≠ 10
    • Since n = 50 is large, use chi-square test or normal approximation for s
    • Calculate test statistic: χ^2 = (n-1)s^2/σ^2 or z = (s - σ) / (σ / √(2n))
    • Compare with critical value at 5% significance level and state conclusion
11. 7 Marks
    bThe height of 10 males of a given locality are found to be $70, 67, 62, 68, 61, 68, 70, 64, 64, 60$ inches. Is it responsible to believe that the average height is greater than 64 inches? Test at 5% significant level, assuming that for 9 degree of freedom $P(t > 1.833) = 0.05$.
    How to answer

    • Set up H0: μ = 64 versus H1: μ > 64 (one-tailed test)
    • Calculate sample mean x̄ and unbiased sample standard deviation s
    • Compute test statistic t = (x̄ - 64) / (s / √10)
    • Compare computed t with critical value 1.833 at df = 9, α = 0.05
    • State acceptance or rejection of H0
12. 7 Marks
    aThe contents of urns I, II and III are as follows: 1 white, 2 black and 3 red balls 2 white, 1 black and 1 red ball 4 white, 5 black and 3 red balls One urn is chosen at random and two balls drawn. They happens to be white and red. What is the probability that they come from urns I, II and III?
    How to answer

    • Assign prior probabilities: P(Urn I) = P(Urn II) = P(Urn III) = 1/3
    • Calculate likelihood of drawing 1 white and 1 red ball from each urn: P(E|Urn I), P(E|Urn II), P(E|Urn III)
    • Apply total probability formula to find P(E)
    • Apply Bayes' theorem to calculate posterior probabilities P(Urn I|E), P(Urn II|E), and P(Urn III|E)
13. 7 Marks
    bSix coins are tossed 6,400 times. Using the Poisson distribution, find the approximate probability of getting six heads $r$ times.
    How to answer

    • Find single trial probability of 6 heads: p = (1/2)^6 = 1/64
    • Calculate Poisson parameter λ = N * p = 6400 * (1/64) = 100
    • Write the Poisson probability mass function formula: P(X = r) = e^(-λ) * λ^r / r!
    • Substitute λ = 100 to get P(X = r) = e^(-100) * 100^r / r!
14. 7 Marks
    aProve that for the normal distribution, the quartile derivation, the mean deviation and standard deviation are approximately $10:12:15$.
    How to answer

    • State QD for a normal distribution: QD ≈ 0.6745 σ ≈ (2/3) σ
    • State MD for a normal distribution: MD = √(2/π) σ ≈ 0.7979 σ ≈ (4/5) σ
    • Take SD = σ
    • Form ratio QD : MD : SD = 2/3 : 4/5 : 1
    • Multiply by 15: 10 : 12 : 15
15. 7 Marks
    b

    By the method of least squares fit a parabola of the form $y = a + bx + cx^2$ for the following data.

    X	1	2	3	4	5	6	7
    Y	2.3	5.2	9.7	16.5	29.4	35.5	54.4

    How to answer

    • Create calculation table for x, y, x^2, x^3, x^4, xy, x^2 y (or shift origin u = x - 4 to simplify)
    • Write normal equations for second-degree polynomial
    • Substitute the summation values into the normal equations
    • Solve the linear system for coefficients a, b, and c
    • Write down the final equation of the fitted parabola