Federal Public Service Commission (FPSC) Competitive Examination for Recruitment to BPS-17 Posts under the Federal Government
Time Allowed: 3 Hours
PAPER: STATISTICS
MAXIMUM MARKS: 100 Marks
Attempt FIVE questions in all, including Question No.1 which is COMPULSORY. All questions carry EQUAL marks.
PART – II
Q.2. A candy company distributes boxes of chocolates with a mixture of creams, toffees and nuts coated in both light and dark chocolate. For a randomly selected box, let X and Y, respectively, be the proportion of the light and dark chocolates that are creams and suppose that the joint density function is: f(x, y) = (2/3)(2x + 3y), for 0 < x < 1, 0 < y < 1 and 0 elsewhere. (a) Verify that joint integration with respect to x and y is one. (05) (b) Let ‘A’ be defined as the region {(x, y) | 0 < x < 1/2, 0 < y < 1/4}. Find P[(X, Y) ∈ A]. (06) (c) Find g(x) and h(y). (05)
Q.3. (a) In how many ways can 8 people be lined up to get on a bus? (04) (b) If three specific persons insist on following each other? (04) (c) If two specific persons refuse to follow each other? (04) (d) If 4 persons are male and 4 are females, in how many ways can they line up? (04)
Q.4. Determine if the use of z-test or t-test is appropriate, giving reasons, for the following hypotheses. Also find the critical region for the test: (a) n = 19, σ is unknown and the population distribution is normal, left-tailed test, α = 0.05. (04) (b) n = 11, σ is known and the population distribution is normal, right-tailed test, α = 0.01. (04) (c) n = 56, σ is unknown, two-tailed test, α = 0.10. (04) (d) n = 12, σ is unknown and the population distribution is normal, left-tailed test, α = 0.05. (04)
Q.5. (a) Show that the sample mean X̄ of a random sample of size ‘n’ from a distribution having p.d.f. f(x; θ) = (1/θ) e^(-x/θ), for 0 < x < ∞, 0 < θ < ∞, and zero elsewhere, is an unbiased estimator of θ. (10) (b) Let X₁, X₂, …, Xₙ be a random sample from a Bernoulli distribution. Find the maximum likelihood estimator of the probability of success. (06)
Q.6. (a) For the following 2×2 table, compute the Chi-square value for the test of independence: (10)
Attribute A / Attribute B
+
−
+
n₊₊
n₊₋
−
n₋₊
n₋₋
(b) A die is tossed 180 times with the following results:
x
1
2
3
4
5
6
f
28
36
36
30
27
23
Is this a balanced die? Use 0.05 level of significance. (06)
Q.7. (a) Describe and explain the “Principle of Least Squares”. Also obtain the least squares estimates of the slope and y-intercept of a simple linear regression model. (08) (b) The following are 15 readings of traffic volume (X cars/hour) and carbon monoxide concentration (Y, PPM) taken at a metropolitan air quality sampling site:
X
100
110
125
150
175
190
200
225
250
275
300
325
350
375
400
Y
8.8
9.5
10.0
10.5
10.5
10.5
10.6
11.0
12.1
12.1
12.5
13.0
13.2
14.0
14.5
Fit a linear Regression model of Y on X. Also plot error vs X. (08)
Q.8. (a) Describe the situation where one-way ANOVA can be applied. Also state the relevant hypotheses. (06) (b) Researchers wish to know if two populations differ with respect to the mean value of total serum complement activity (CH₅₀). Samples of size n₁ = 10 and n₂ = 20 are taken from diseased and normal subjects. The sample means and standard deviations are: Sample 1 (Diseased): x̄₁ = 62.6, s₁ = 33.8 Sample 2 (Normal): x̄₂ = 47.2, s₂ = 10.1 Using an appropriate test, give your opinion on what the researchers wish to know. (10)
Q.9. Write short notes on ANY FOUR of the following: (04 + 04 + 04 + 04 = 16) (i) Difference between simple and partial correlation. (ii) Multiple regression. (iii) Use of statistics in electoral politics. (iv) Test for equality of two variances. (v) Joint probability distribution. (vi) Mathematical expectation.
