2nd PUC Statistics Previous Year Question Paper March 2018

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Karnataka 2nd PUC Statistics Previous Year Question Paper March 2018

Time: 3 Hours 15 Minutes
Max. Marks: 100

Section – A

I. Answer any Ten of the following question (10 × 1 = 10)

Question 1.
Define Cohort.
Answer:
Cohort is group of individuals who are born at the same time and who experience the same mortality conditions.

Question 2.
What is the value of Index Number for the base year?
Answer:
100

Question 3.
Which weights are used in the construction of Laspeyre’s Price Index Number?
Answer:
Base year quantities (q₀)

Question 4.
What is Time Series?
Answer:
Arrangement of statistical data in chronological order or in accordance with time is known as a Time Series.

Question 5.
Under what condition Poisson distribution tends to Normal distribution?
Answer:
When λ is large (λ → ∞)

Question 6.
Write the range of a Chi-square distribution.
Answer:
0 to ∞ OR (0, ∞)

Question 7.
Which is the best estimator of population mean?
Answer:
Sample Mean (X̄)

Question 8.
What is parameter space?
Answer:
The set of all the admissible values of the parameter is called Parameter space.

Question 9.
In Statistical Hypothesis, if H₁: µ < 50 kg then write H₀.
Answer:
H₀: µ = 50

Question 10.
What is meant by Statistical Quality Control?
Answer:
Statistical Quality Control (S.Q.C) is the method of controlling the quality of the products using statistical techniques.

Question 11.
What is feasible region (FR) in LPP?
Answer:
Feasible Region (FR) is the area which satisfies all the constraints simultaneously and also the non-negativity restriction.

Question 12.
When is a Transportation Problem (TP) balanced?
Answer:
Transportation problem is said to be balanced if total availability is equal to total requirement. (Σa_i = Σb_j)

Section – B

II. Answer any Ten of the following questions: (10 × 2 = 20)

Question 13.
Mention the methods of collecting Vital Statistics.
Answer:
Census Enumeration Method and Registration Method

Question 14.
Write any two limitations of Index Number.
Answer:
(a) Since index numbers are based on the sample data, they are only approximate indicators,
(b) There is a likelihood of error being introduced at each stage of construction of the index number,
(c) While constructing Index Numbers, the quality of product is not taken into consideration,
(d) Like other statistical techniques, Index Numbers can be misused,
(e) As customs, traditions and habits of people vary from time to time, it is difficult to assign proper weights to the various items,
(f) Many formulae are used in the construction of Index Numbers and different formulae give different answers.
(any two limitations from the above)

Question 15.
For the following data compute Value Index Number : Σp₀q₀ = 300 and Σp₁q₁ = 275.
Answer:

Question 16.
From the following Time Series data, calculate trends values by the method of Semi-Averages.

Answer:

Question 17.
Define Interpolation and Extrapolation.
Answer:
Interpolation is the technique of estimating the value of dependent variable (Y) for any intermediate value of independent variable (x).
Extrapolation is the technique of the estimating the value of dependent variable (Y) for any value of the independent variable (X) which is outside the range of the given series.

Question 18.
Write down the Bernoulli distribution with parameter p = 0.4.
Answer:
Bernoulli Distribution
X : 0 1 : Total
P(x) : 0.4 0.6 : 1(0R) P(X = x) = (0.4) × (0.6)^1-x, x = 0,1

Question 19.
State any two features of student t-distribution.
Answer:
(a) n is the parameter of the t distribution.
(b) Range of the t – distribution is -∞ to +∞
(c) It is symmetrical about t = 0 (β₁ = 0)
(d) It is bell shaped curve.
(e) Mean = Median = Mode =0
(f) It is asymptotic to the X – axis.
(g) It is leptokurtic (β₂ >3)
(h) For n → ∞, the t – variate tends to SNV
(i) Variance = \(\frac{n}{n-2}\) for n > 2
(any two features from the above)

Question 20.
Define point estimation and Interval estimation.
Answer:
If a single value is proposed as an estimate of the unknown parameter then it is Point Estimation.
If an interval is proposed as an estimate of the unknown parameter then it is Interval Estimation.

Question 21.
If n₁ = 40, n₂ = 60, p₁ = 0.02 and p₂ = 0.01 then find S.E (p₁ – p₂).
Answer:

Question 22.
What do you mean by Process control and Product control?
Answer:
Controlling the quality of the goods during the manufacturing process itself is called Process Control.
The process of inspection of manufactured lot for acceptability is called Product Control.

Question 23.
The objective function and two solutions of an LPP are Max Z = 5X + 4Y and A(12, 10); B(14, 4). Find the optimal solution.
Answer:

Value of x = 12, y = 10 and Maximum Value of z = 100

Question 24.
From the following Transportation Problem test whether solution is degenerate.

Answer:
m + n = 5 and number of solutions = 4
Hence given solution is degenerate.

Section – C

III. Answer any Eight of the following questions (8 × 5 = 40)

Question 25.
From the following data calculate Crude Birth Rate (CBR) and General Fertility Rate (GFR).

Answer:

Question 26.
Calculate Un-weighted geometric mean price index number for the following data.

Answer:

Question 27.
What is Consumer Price Index Number (CPI)? Write its four uses.
Answer:
Consumer Price Index Number (CPI) is the index number of the cost met by a specified class of consumers in buying a ‘basket of goods and services’.
Uses of Consumer Price Index Numbers are
(a) It is commonly used in fixation of salary, dearness allowance or grant of bonus to the employees,
(b) They are used by the government for the formulation of price policy, wage policy and general economic policies.
(c) They are used for comparing changes in the cost of living of different classes of people.
(d) Cost of living index is used widely in wage negotiations and wage contracts.
(e) Cost of living Index Number is used to determine the purchasing power of money and computing the real wages.
(f) They are used for deflating money or income.
(any four uses from the above)

Question 28.
Compute the trend values by finding 3 years moving averages for the following data.

Answer:

Question 29.
Interpolate and Extrapolate the population of India for the census years 1981 and 2021 from the following data.

Answer:
Formula + Substitution + Ans (y₅ – 5y₄ + 10y₃ – 10y₂ + 5y₁ – y₀ = 0 ⇒ y₂ = 67.8)
Formula + Ans (y₆ – 5y₅ + 10y₃ + 5y₂ – y₁ = 0 ⇒ y₆ = 136)

Question 30.
The number of accidents in a year attributed to taxi drivers in a city follows Poisson distribution with mean 2.5. Out of 2000 taxi drivers, find approximately the number of drivers with a) one accident b) more than 2 accidents in a year.
Answer:
In P.D. Mean (λ) = 2.5, N = 2000, p.m.f. p(x = x) = \(\frac{e^{-\lambda} \lambda^{x}}{x !}\) x = 0,1,2, ………
(a) P(X = 1) = \(\frac{\mathrm{e}^{-25} 2.5^{1}}{1 !}\) = 0.2052 × 2000 = 410.4 = 410
(b) P(X > 2) = 1 – P(x ≤ 2) = 1 – (P(2) + p(1) + P(0)) = 0.4562 × 2000 = 912.4 a 912

Question 31.
In a Hyper Geometric distribution, if a = 6, b = 9 and n = 4 then find
a. P (X = 2)
b. Standard Deviation.
Answer:
In Hyper Geometric Distribution, a = 6, b = 9 and n = 4

Question 32.
A manufacturer claims that less than 2% of the products are defective. A retailer buys a batch of 400 products from the manufacturer and find that 12 are defectives. Test at 1% level of significance that whether the rnanuÍacturer claims is justifiable.
Answer:
Given P₀ = 0.02, Q₀ = 0.98, n = 400, x = 12. p = \(\frac{x}{n}-\frac{12}{400}\) = 0.03
H₀ = P = 0.02; H₁: P< 0.02

– K = -2.33
We accept H₀ (Manufacturer’s claim is not justifiable)

Question 33.
A random sample of size 16 has mean 53. The sum of the squared deviations taken from mean is 150. Can this sample be regarded as taken from the population having mean 56? ( K = ±2.13)
Answer:
H₀: The population mean is 56 (µ = 56) and H₁ : µ ≠ 56

d.f. = 15, ±k = ±2.13, Here, t_cal lies in rejection region.
∴ Reject H₀

Question 34.
Ten samples of size 50 each were inspected and the number of defectives in each of them were as follows.

Obtain the control limits for np-chart.
Answer:

Question 35.
Solve the following game by Maxi-min and Mini-max principle. Is game fair?

Answer:
Row minima ; 0, -3, -3 Column maxima; 3, 4, 0, 6
Maxi-min: 0 and Mini-max: 0 The value of the game zero.
∴ Best strategies are A₁, B₃. It is a fair game.

Question 36.
The capital cost of a machine is Rs. 10,500. Its resale value is Rs. 500. Its the maintenance costs are as follows.

When should the machine be replaced?
Answer:

Minimum annual average cost = Rs. 3,660, Optimal replacement period is 5^th year.

Section – D

IV. Answer any Two 0f the following questions: (2 × 10 = 20)

Question 37.
(a) From the following data show that Town – A is healthier.

Answer:
PA: 3,50,000, 2,70,000, 4,00,000, 4,50,000, 14,70,000
PB : 4,20,000, 3,00,000, 4,80,000, 5,40,000, 17,40,000
Σp = 1,00,000, S.T.D R(A) = \(\frac{\Sigma \mathrm{PA}}{\Sigma \mathrm{P}}\) ; S.T.D. R(B) = \(\frac{\Sigma P B}{\Sigma P}\) ;
STDR (A) = 14.7, STDR (B) = 17.4. Town A is healthier.

(b) Compute the Gross Reproduction Rate (GRR) from the following.

Answer:
W.S.F. R(t) = \(\frac{F B(t)}{F p(t)}\) × 1000 Fp(t)
or \(\frac{20}{1600}\) × ; 12.5, 70, 60, 43, 75, 10, 10 ;225 1600
GRR = i ΣWFR = 5 × 225 = 1125

Question 38.
Compute Fisher’s Price Index Number. Show that Fisher’s index number satisfies TRT and FRT.

Answer:
p₀q₀ : 112, 180, 120, 125 : Σp₀q₀ = 537
p₀q₁ : 140, 225,180, 200 : Σp₀q₁ = 745
p₁q₀ : 120,160, 132, 150 : Σp₁q₀ = 562
p₁q₁ : 150, 200, 198, 240 : Σp₁q₁ = 788

P₀₁ (F) × P₁₀ (F) = 1 (Condition), Substitution and showing.
P₀₁ (F) × Q₀₁ (F) = \(\frac{\Sigma p_{1} q_{1}}{\Sigma p_{0} q_{0}}\) (Condition), Substitution and showing.

Question 39.
Fit a parabolic trend of the form Y = a + bX + cX² for the following time series.

Answer:

Σy = 115, Σx = 0, Σxy = -25, Σx² = 10, Σx²y = 237, Σx⁴ = 34 and n = 5
b = \(\frac{\Sigma x y}{\Sigma x^{2}}=\frac{-25}{10}\) = – 2.5
Σy = na + cΣx² and Σx²y = aΣx² + cΣx⁴ ⇒ c = 0.5 and a = 22
The trend line is, Y = 22 – 2.5 x + 0.5 x²

Question 40.
Fit a binomial distribution for the following data and test at 5% level of significance that binomial distribution is a good fit.

Answer:
N = 200 , n = 5, np = \(\frac{\Sigma f x}{N}=\frac{600}{200}\) = 3, p = \(\frac{n p}{n}=\frac{3}{5}\) = 0.6 ⇒ q = 0.4
P (x) = nC_x (p)^x (q)^n-x,T(0) = N × P(0) = 200 × qⁿ = 200 x (0.4)⁵ = 2.048 = 2
Theoretical Frequencies : 2,15, 46, 69, 52,16
H₀ : B.D is good fit and H₁ : B.D is not good fit.
Test Statistic, x² = \(\Sigma \frac{\left(0_{i}-E_{i}\right)^{2}}{E_{i}}\) = 0.1207
d.f = 3, k₂ =7.81 here, x² < k₂ ∴ Accept H₀ i.e., B.D. is good fit

Section – E

V. Answer any Two of the following. (2 × 10 = 20)

Question 41.
X is a normal variate with parameters mean (µ) = 50 and variance (σ²) = 16. Find the probability that
a. X < 56
b. 52 < X < 55.
Answer:
a. P(X < 56) = p\(\left(\frac{x-\mu}{\sigma}<\frac{56-50}{4}\right)\) = p(Z < 1.5) = 1 – 0.0668 = 0.9332
b. P (0.5 < Z < 1.25) = 0.3085 – 0.1057 = 0.2028

Question 42.
From the following data regarding average weight of boys and girls. Test whether the mean weight of boys
are greater than meañ weight of girls.


Answer:
H₀ : The Mean weight of boys and girls are equal (µ₁ = µ₂) and H₁ : µ₁ > µ₂

K = 1.65 Here, Z_cal lies in acceptance region. ∴ Accept H₀

Question 43.
To test the effectiveness of vaccination against Tuberculosis (TB) and following data was obtained.

Test at 1 % level of significance whether vaccination and attack of TB are independent.
Answer:
H₀ : Vaccination and attack of TB are independent.
H₁ : Vaccination and attack of TB are not independent.

K₂ = 6.65 χ²_cal > k₂: reject H₀, Vaccination and attack of TB are not independent

Question 44.
The annual demand for an item is 3000 units. Capital cost is Rs. 7 per unit. Inventory carrying cost is 20% of capital cost per, annum. If setup cost is Rs. 150 then find
(a) Economic Order Quantity and
(b) Optimum number of orders.
Answer:
R = 3000/units, P = Rs. 7/units, I = 20%, C₃ = Rs. 150.
C₁ = PI = Rs. 1.4/unit
Formula and Ans Q⁰ = \(\sqrt{\frac{2 \mathrm{RC}_{3}}{\mathrm{C}_{1}}}\) = 801.78 = 802 units/cycle
Formula and Ans n⁰ = \(\frac{R}{Q_{0}}\) = 3.75 orders