AD-302 Probability and Statistics for Data Science - Dec 2024 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
   aA survey of the number of customer used PCO/STD both located at a college gate to make telephone calls last week revealed the following information 52 43 30 38 30 42 12 46 39 37 34 46 32 18 41 5
   How to answer

   • List raw observations 52,43,30,38,30,42,12,46,39,37,34,46,32,18,41,5
   • Arrange / tabulate frequency if required
   • Apply mean / median / dispersion formulas
   • Interpret result for customer usage
2. 7 Marks
   bAn air-charter company has been requested to quote a realistic turn-round time for a contract to handle certain imports and exports of a fragile nature. The contract manager has provided the following analysis of turn-round times for similar goods over a given twelve-monthly period.

   1. i)Calculate mean and standard deviation.
   2. ii)Advice the contract manager about the turn-round time to be quoted using I) Mean plus one standard deviation; II) Mean plus two standard deviations.

   

   How to answer

   • Compute mid-points and $f\cdot x$, $f\cdot x^2$ for grouped frequency
   • Calculate mean $\bar{x}=\frac{\sum fx}{N}$ and standard deviation $\sigma=\sqrt{\frac{\sum fx^2}{N}-\bar{x}^2}$
   • Compute mean+SD and mean+2SD values
   • Advise quoted turn-round time with interpretation
3. 7 Marks
   a

   From the given data determine the lines of regression.

   $x$	2	4	6	8	10
   $y$	5	7	9	8	11

   How to answer

   • Compute means $\bar{x}, \bar{y}$
   • Compute regression coefficients $b_{yx}, b_{xy}$
   • Write lines $y-\bar{y}=b_{yx}(x-\bar{x})$ and $x-\bar{x}=b_{xy}(y-\bar{y})$
   • Interpret slopes
4. 7 Marks
   bProve the Poisson distribution is as a limiting case of binomial distribution.
   How to answer

   • Write binomial pmf $^nC_x p^x q^{n-x}$
   • Take limit $n\to\infty$, $p\to 0$ with $np=\lambda$ fixed
   • Show $(1-p)^n \to e^{-\lambda}$ and $^nC_x p^x \to \lambda^x/x!$
   • Obtain Poisson pmf $e^{-\lambda}\lambda^x/x!$
5. 7 Marks
   aThe income of a group of 10,000 persons was found to be normally distributed with mean 750 Rs. Per month and standard deviation of 50 Rs. Show that, of this group, about 95% had income exceeding 688 Rs and only 5% had income exceeding 832 Rs. Also Determine the lowest income among the richest 100.
   How to answer

   • Standardize with $Z=(X-750)/50$
   • Find $P(X>688)$ using $Z=-1.24$ ~ 89-95% area
   • Find $P(X>832)$ using $Z=1.64$ ~ 5% upper tail
   • Find $x_0$ for top 100/10000=1% using $Z=2.33$
6. 7 Marks
   b

   Calculate the first four moments about the mean of the following distribution:

   $x$	2	2.5	3	3.5	4	4.5	5
   $f$	5	38	65	92	70	40	10

   Also find the measures of skewness and kurtosis.

   How to answer

   • Compute mean $\bar{x}=\sum fx/\sum f$
   • Compute central moments $\mu_r=\sum f(x-\bar{x})^r/\sum f$ for r=1 to 4
   • Compute skewness $\gamma_1=\mu_3/\mu_2^{3/2}$
   • Compute kurtosis $\gamma_2=\mu_4/\mu_2^2$ and comment
7. 7 Marks
   a

   Ten students got the following percentage of marks in Chemistry and Physics:

   Students:	1	2	3	4	5	6	7	8	9	10
   Marks in Chemistry:	78	36	98	25	75	82	90	62	65	39
   Marks in Physics:	84	51	91	60	68	62	86	58	63	47

   Calculate the rank correlation co-efficient.

   How to answer

   • Rank Chemistry and Physics marks separately
   • Compute differences $d$ and $d^2$
   • Apply $r_s=1-6\sum d^2/n(n^2-1)$
   • Interpret degree of association
8. 7 Marks
   b

   Find the least square line that fits the following data, assuming that x-values are free from error:

   $x$	1	2	3	4	5	6
   $y$	5.04	8.12	10.64	13.18	16.20	20.04

   How to answer

   • Assume model $y=a+bx$ with x error-free
   • Form normal equations $\sum y=na+b\sum x$, $\sum xy=a\sum x+b\sum x^2$
   • Solve for $a,b$ using given 6 pairs
   • Write fitted least square line
9. 7 Marks
   aFind the moment generating function of the discrete binomial distribution given by $P(x) = {^nC_x} p^x q^{n-x}$ where $q = 1-p$. Also find the first and second moments about the mean.
   How to answer

   • Write MGF definition $M(t)=E(e^{tX})=\sum e^{tx} {^nC_x}p^x q^{n-x}$
   • Simplify to $(q+pe^t)^n$
   • Differentiate or expand to get mean $np$
   • Obtain first and second central moments $0$ and $npq$
10. 7 Marks
    bIn a bolt factory, machines A, B and C manufacture respectively 25%, 35% and 40% of the total. Of their output 5, 4 and 2 percent are defective bolts. A bolt is drawn at random from the product and is found and is found to be defective. What is probability that it was manufactured by machine B?
    How to answer

    • Define priors $P(A)=0.25, P(B)=0.35, P(C)=0.40$
    • Likelihoods of defective 0.05,0.04,0.02
    • Apply Bayes theorem $P(B|D)=P(B)P(D|B)/\sum P(.)P(D|.)$
    • Compute numerical probability
11. 7 Marks
    aA manufacturer knows from experience that the resistance of resistors he produces is normal with mean $\mu = 100$ ohms and standard deviation $\sigma = 2$ ohms. What percentage of resistors will have resistance between 98 ohms and 102 ohms?
    How to answer

    • State $\mu=100$, $\sigma=2$
    • Standardize limits 98 and 102 to $Z=\pm1$
    • Find area $P(-1<Z<1)\approx 0.6826$
    • Convert to percentage ~68.26%
12. 7 Marks
    bThe 9 items of a sample have the following values: 45, 47, 50, 52, 48, 47, 49, 53, 51. Does the mean of these values differ significantly from the assumed mean 47.5? At 5% level of significance is ($t_{0.05} = 2.31$)
    How to answer

    • Compute sample mean $\bar{x}$ and $s$ from 9 values
    • Form $t=(\bar{x}-47.5)/(s/\sqrt{n})$ with $df=8$
    • Compare with critical $t_{0.05}=2.31$
    • Decide significant difference or not
13. 7 Marks
    aA manufacturer knows that the condensers he makes contain on an average 1% of defectives. He packed them in boxes of 100. What is the probability that a box picked at random will contain 4 or more faulty condensers?
    How to answer

    • Mean defectives $\lambda=np=1$
    • Use Poisson $P(X=x)=e^{-\lambda}\lambda^x/x!$
    • Compute $P(X\ge 4)=1-P(0)-P(1)-P(2)-P(3)$
    • Interpret probability
14. 7 Marks
    bThe average income of persons was ₹210 with a S.D. of ₹10 in sample of 100 people of a city. For another sample of 150 persons, the average income was ₹220 with S.D. of ₹12. The S.D. of incomes of the people of the city was ₹11. Test whether there is any significant difference between average incomes of the localities.
    How to answer

    • State $n_1=100, \bar{x}_1=210$ and $n_2=150, \bar{x}_2=220$, $\sigma=11$
    • Compute $Z=(\bar{x}_1-\bar{x}_2)/\sigma\sqrt{1/n_1+1/n_2}$
    • Compare with 1.96 at 5% level
    • Conclude significant difference or not
15. 7 Marks
    a

    Calculate the coefficient of correlation for the following data:

    Height of father (in inches )	65	66	67	67	68	69	70	72
    Height of son (in inches)	67	68	65	68	72	72	69	71

    How to answer

    • Compute means of father and son heights
    • Compute $r=\frac{\sum (x-\bar{x})(y-\bar{y})}{\sqrt{\sum (x-\bar{x})^2\sum (y-\bar{y})^2}}$
    • Substitute 8 pairs values
    • Interpret correlation
16. 7 Marks
    bThe first four moments of a distribution about the value 5 of the variables are 2, 20, 40 and 50. Calculate the moments about the mean and comment upon the skewness and kurtosis of the distribution.
    How to answer

    • Mean $=5+\mu_1'=7$
    • Convert raw moments about 5 to central moments $\mu_2,\mu_3,\mu_4$
    • Compute $\beta_1=\mu_3^2/\mu_2^3$, $\beta_2=\mu_4/\mu_2^2$
    • Comment skewness ($\mu_3$) and kurtosis (lepto/platy/meso)