SSC CGL Mock – Paper II Statistics Mock Test 04 – Paper II · Statistics | Online Quiz

1.

Before the main survey, a draft questionnaire is tried out on a few respondents to find faults in it. This trial is called

Correct Answer: C [a pilot survey]

Explanation:

A pilot (pre-test) survey tests the questionnaire and field work before the full survey.

2.

The 'less than' and 'more than' ogives of a distribution meet at the point (42, 60). What is the total frequency?

Correct Answer: B [120]

Explanation:

At the meeting point the cumulative frequency is N/2, so N = 2 × 60 = 120; the median is 42.

3.

Outputs of two fictional mills, 64,000 tonnes and 2,16,000 tonnes, are shown by cubes whose volumes are in proportion to output. If the first cube has side 4 cm, the side of the second cube is

Correct Answer: A [6 cm]

Explanation:

Side ∝ ∛value: 4 × ∛(216000/64000) = 4 × 1.5 = 6 cm.

4.

Statement-basedConsider the statements:
I. A false base line is used in a line graph when the smallest value is far above zero.
II. In a histogram, adjacent rectangles are separated by equal gaps.
Which statement(s) is/are correct?

Correct Answer: A [Only I]

Explanation:

I is correct. Histogram bars touch each other (continuous classes); gaps are used in bar diagrams.

5.

500 observations range from 20 to 200. Using Sturges' rule k = 1 + 3.322 log₁₀ N, the class width is about (log₁₀ 500 = 2.699)

Correct Answer: D [18]

Explanation:

k = 1 + 3.322 × 2.699 ≈ 9.97 ≈ 10 classes; width = 180/9.97 ≈ 18.

6.

Assertion–ReasonAssertion (A): A pie diagram is not suitable for showing the yearly sales of a firm over fifteen years.
Reason (R): A pie diagram shows the parts of one whole at a single point of time.

Correct Answer: A [Both A and R are true and R is the correct explanation of A]

Explanation:

Both true; since a pie shows the split of one total, it cannot show change over many years – R explains A.

7.

A fictional distribution has 'more than' cumulative frequencies: 0 or more – 80, 10 or more – 72, 20 or more – 55, 30 or more – 31, 40 or more – 12, 50 or more – 0. The modal class is

Correct Answer: C [20–30]

Explanation:

Class frequencies: 8, 17, 24, 19, 12; the highest, 24, is in 20–30.

8.

Workers of a factory are grouped at the same time by sex (male/female) and by skill (skilled/unskilled). This is

Correct Answer: B [manifold classification]

Explanation:

Grouping by two or more attributes at once is manifold classification.

9.

A fictional distribution 0–10: 6, 10–20: f₁, 20–30: 24, 30–40: f₂, 40–50: 7 has total frequency 70 and median 25. The values of f₁ and f₂ are

Correct Answer: D [f₁ = 17, f₂ = 16]

Explanation:

20 + (35 − 6 − f₁)/24 × 10 = 25 gives f₁ = 17; f₂ = 70 − 6 − 17 − 24 − 7 = 16.

10.

Waiting times at a fictional clinic form a moderately skewed distribution with modal time 60 minutes and median time 64 minutes. The mean waiting time (in minutes) is about

Correct Answer: A [66]

Explanation:

Mode = 3 Median − 2 Mean → Mean = (3 × 64 − 60)/2 = 66.

11.

Find the upper quartile of the fictional distribution 20–30: 8, 30–40: 15, 40–50: 22, 50–60: 18, 60–70: 7.

Correct Answer: C [54.17]

Explanation:

3N/4 = 52.5; class 50–60 (cf before 45): 50 + (52.5 − 45)/18 × 10 = 54.17.

12.

The mean of 25 observations is 36. It was found later that 42 had been taken as 24, and that one observation, 50, did not belong to the data and must be dropped. The correct mean is

Correct Answer: A [36.17]

Explanation:

Correct total = 900 + 18 − 50 = 868; n = 24; mean = 36.17.

13.

A clerk spends ₹ 1,200 on each of three kinds of pens priced ₹ 20, ₹ 30 and ₹ 40 a pen. The average price paid per pen is

Correct Answer: B [₹ 27.69]

Explanation:

Equal amounts spent → HM: 3/(1/20 + 1/30 + 1/40) = 27.69.

14.

By the step-deviation method (A = 125, h = 10), the mean of the fictional distribution 100–110: 7, 110–120: 12, 120–130: 18, 130–140: 9, 140–150: 4 is

Correct Answer: D [123.2]

Explanation:

Σfd = −14 − 12 + 0 + 9 + 8 = −9; mean = 125 + (−9/50) × 10 = 123.2.

15.

Statement-basedConsider the statements:
I. The geometric mean cannot be used if any observation is zero.
II. The harmonic mean gives more weight to the larger observations.
Which statement(s) is/are correct?

Correct Answer: A [Only I]

Explanation:

I is correct (the product becomes 0).

16.

In a fictional factory, the average daily wage of skilled workers is ₹ 900 and of unskilled workers ₹ 600. The average for all workers is ₹ 780. The ratio of skilled to unskilled workers is

Correct Answer: D [3 : 2]

Explanation:

n₁(900 − 780) = n₂(780 − 600) → n₁ : n₂ = 180 : 120 = 3 : 2.

17.

Which average can be found for a distribution with open-end classes at both ends without making any assumption?

Correct Answer: A [Median]

Explanation:

The median needs only the middle class; the means need the mid-values of the open-end classes.

18.

Odd one outWhich one of the following is NOT the same as the median of a distribution?

Correct Answer: B [P₂₅]

Explanation:

Median = Q₂ = D₅ = P₅₀; P₂₅ is the first quartile.

19.

Two fictional groups of 50 each have means 60 and 70. The first group has SD 8; the combined variance of all 100 is 107. The SD of the second group is

Correct Answer: C [10]

Explanation:

107 = [50(64 + 25) + 50(σ₂² + 25)]/100 → σ₂² = 100 → σ₂ = 10.

20.

Fifteen fictional runners wear bibs numbered 1 to 15. The variance of the bib numbers (divide by n) is

Correct Answer: B [18.67]

Explanation:

σ² = (n² − 1)/12 = 224/12 = 18.67.

21.

A sample of 8 items gives Σ(x − x̄)² = 56. The sample variance s² = Σ(x − x̄)²/(n − 1) is

Correct Answer: C [8]

Explanation:

s² = 56/7 = 8 (dividing by n would give 7).

22.

Ages (years) of 40 members of a fictional reading club: 0–10: 5, 10–20: 8, 20–30: 12, 30–40: 10, 40–50: 5. The semi-interquartile range (quartile deviation) of the ages is

Correct Answer: D [9.375]

Explanation:

Q₁ = 10 + (10 − 5)/8 × 10 = 16.25; Q₃ = 30 + (30 − 25)/10 × 10 = 35; QD = 9.375.

23.

In a normal distribution the mean deviation about the mean is 12. The SD is about (MD ≈ 4/5 σ)

Correct Answer: A [15]

Explanation:

Answer: 15. σ ≈ (5/4) × MD = 15.

24.

Which measure of dispersion uses every observation and is the least affected by sampling fluctuations?

Correct Answer: D [Standard deviation]

Explanation:

SD uses all values, is algebraically handy and is the most stable from sample to sample.

25.

For 10 observations with mean 20 and variance 9, find Σ(x − 25)².

Correct Answer: C [340]

Explanation:

Σ(x − A)² = Σ(x − x̄)² + n(x̄ − A)² = 90 + 10×25 = 340.

26.

Statement-basedConsider the statements:
I. If all the observations are equal, the standard deviation is zero.
II. The standard deviation is negative when all the observations are negative.
Which statement(s) is/are correct?

Correct Answer: A [Only I]

Explanation:

SD is the positive root of a sum of squares; it is never negative.

27.

For 5 observations, Σx = 50 and Σx² = 580. The coefficient of variation is

Correct Answer: A [40%]

Explanation:

x̄ = 10; σ² = 116 − 100 = 16, σ = 4; CV = 4/10 × 100 = 40%.

28.

For the values 1, 2, 3, 4, 5, the coefficient of kurtosis β₂ = μ₄/μ₂² is

Correct Answer: B [1.7]

Explanation:

μ₂ = 10/5 = 2; μ₄ = (16 + 1 + 0 + 1 + 16)/5 = 6.8; β₂ = 6.8/4 = 1.7 (platykurtic).

29.

Hourly pay of fictional farm workers: mean ₹ 70, median ₹ 66, SD ₹ 12. Taking the mode from Mode = 3 Median − 2 Mean, the coefficient (mean − mode)/σ is

Correct Answer: C [1]

Explanation:

Mode = 3 × 66 − 2 × 70 = 58; (70 − 58)/12 = 1.

30.

The first four moments of a fictional distribution about zero are 2, 8, 32 and 160. The distribution is

Correct Answer: A [mesokurtic (β₂ = 3)]

Explanation:

μ₂ = 8 − 4 = 4; μ₄ = 160 − 256 + 192 − 48 = 48; β₂ = 48/16 = 3.

31.

Statement-basedConsider the statements:
I. If β₁ = 0, the distribution must be symmetrical.
II. Kurtosis describes how peaked or flat-topped a curve is compared with the normal curve.
Which statement(s) is/are correct?

Correct Answer: B [Only II]

Explanation:

A symmetric curve gives β₁ = 0, but β₁ = 0 (μ₃ = 0) can also occur for a non-symmetric one; II is correct.

32.

For the fictional distribution 0–10: 6, 10–20: 14, 20–30: 20, 30–40: 8, 40–50: 2, Bowley's coefficient of skewness is

Correct Answer: D [−0.11]

Explanation:

Q₁ = 14.6429, Md = 22.5, Q₃ = 28.75; Sk = (Q₃ + Q₁ − 2Md)/(Q₃ − Q₁) ≈ −0.11.

33.

For a distribution μ₂ = 9 and μ₃ = 13.5. The moment coefficient of skewness γ₁ = μ₃/σ³ is

Correct Answer: B [0.5]

Explanation:

σ³ = 3³ = 27; γ₁ = 13.5/27 = 0.5.

34.

For a moderately skewed distribution, which of the following gives the correct link between Karl Pearson's two coefficients of skewness?

Correct Answer: D [They are equal when Mode = 3 Median − 2 Mean]

Explanation:

(x̄ − Mo)/σ = 3(x̄ − Md)/σ exactly when x̄ − Mo = 3(x̄ − Md), i.e.

35.

For 8 pairs of fictional values: Σx = 40, Σy = 48, Σx² = 240, Σy² = 320 and Σxy = 260. The correlation coefficient is about

Correct Answer: C [0.56]

Explanation:

Cov = 32.5 − 30 = 2.5; σₓ² = 5, σᵧ² = 4; r = 2.5/√20 ≈ 0.56.

36.

Spearman's rank correlation for 10 pairs of ranks is 0.6. The sum of the squares of the rank differences Σd² is

Correct Answer: A [66]

Explanation:

1 − 6Σd²/(10 × 99) = 0.6 → Σd² = 0.4 × 990/6 = 66.

37.

The rank correlation of 10 fictional pairs was found to be 0.2. Later it was found that one difference of ranks had been taken as 7 instead of 3. The correct rank correlation is

Correct Answer: B [0.44]

Explanation:

Σd² = 0.8 × 990/6 = 132; corrected = 132 − 49 + 9 = 92; ρ = 1 − 6 × 92/990 ≈ 0.44.

38.

Marks in Statistics (1), Mathematics (2) and Economics (3) of fictional students give r₁₂ = 0.8, r₁₃ = 0.6, r₂₃ = 0.5. With Mathematics held constant, the correlation between Statistics and Economics, r₁₃.₂, is about

Correct Answer: B [0.38]

Explanation:

(r₁₃ − r₁₂r₂₃)/√((1 − r₁₂²)(1 − r₂₃²)) = (0.6 − 0.4)/√(0.36 × 0.75) ≈ 0.38.

39.

Rainfall x (cm) and rice output y (quintal per hectare) of fictional farms satisfy the regression lines 2y − x = 16 and 5x − 4y = 28. The value of r is about

Correct Answer: A [0.63]

Explanation:

y on x: y = x/2 + 8 (bᵧₓ = 0.5); x on y: x = (4y + 28)/5 (bₓᵧ = 0.8); product 0.4 ≤ 1; r = +√0.4 ≈ 0.63.

40.

Of 500 fictional persons, 200 were vaccinated. 50 of the vaccinated and 150 of the others caught a fever. Yule's Q between vaccination and catching fever is

Correct Answer: C [−0.5]

Explanation:

(AB) = 50, (Aβ) = 150, (αB) = 150, (αβ) = 150; Q = (7500 − 22500)/(7500 + 22500) = −0.5 – vaccination is negatively associated with fever.

41.

In the regression equation X₁ = a + b₁₂.₃X₂ + b₁₃.₂X₃, the coefficient b₁₂.₃ is called the

Correct Answer: D [partial regression coefficient of X₁ on X₂]

Explanation:

b₁₂.₃ measures the change in X₁ per unit change in X₂ with X₃ held fixed.

42.

Statement-basedConsider the statements:
I. A high correlation between two variables proves that one causes the other.
II. The correlation coefficient is a pure number, free of the units of the variables.
Which statement(s) is/are correct?

Correct Answer: B [Only II]

Explanation:

Correlation shows association only (it may be spurious); r has no units.

43.

For bus fare (x) and number of passengers (y) on fictional routes, bᵧₓ = −0.4 while bₓᵧ = −1.6. What is the correlation between fare and passengers?

Correct Answer: D [−0.8]

Explanation:

r = −√(0.4 × 1.6) = −√0.64 = −0.8 (both coefficients negative → r negative).

44.

The regression line of y on x for fictional data is y = 0.9x + 3, with σₓ = 4 and σᵧ = 6. The correlation coefficient is

Correct Answer: A [0.6]

Explanation:

bᵧₓ = r σᵧ/σₓ → r = 0.9 × 4/6 = 0.6.

45.

σᵧ = 10 and r = 0.8. The standard error of estimate of y on x, σᵧ√(1 − r²), is

Correct Answer: C [6]

Explanation:

10 × √(1 − 0.64) = 10 × 0.6 = 6.

46.

2% of the people of a fictional district have a certain disease. A test detects it in 95% of the sick, but also wrongly shows 'positive' for 10% of the healthy. A person tests positive. The probability that the person has the disease is about

Correct Answer: B [0.1624]

Explanation:

Bayes: 0.019/(0.019 + 0.098) = 0.019/0.117 ≈ 0.1624.

47.

P(A) = P(B) = P(C) = 1/4, P(A∩B) = P(B∩C) = 0 and P(A∩C) = 1/8. The probability that at least one of A, B, C occurs is

Correct Answer: C [5/8]

Explanation:

P(A∩B∩C) = 0 (since P(A∩B) = 0); 3/4 − 1/8 + 0 = 5/8.

48.

A fictional family has two children, each equally likely to be a boy or a girl. Given that at least one is a boy, the probability that both are boys is

Correct Answer: B [1/3]

Explanation:

Possible: BB, BG, GB (equally likely) → 1/3.

49.

A fictional archer hits the target with probability 0.3 on each shot, independently. The least number of shots needed so that the chance of at least one hit is more than 0.9 is

Correct Answer: A [7]

Explanation:

1 − 0.7ⁿ > 0.9 → 0.7ⁿ < 0.1; 0.7⁶ ≈ 0.118, 0.7⁷ ≈ 0.082 → n = 7.

50.

In a fictional dice game, your score is the larger of the two numbers on two fair dice. The chance that you score exactly 4 is

Correct Answer: D [7/36]

Explanation:

max ≤ 4: 16 outcomes; max ≤ 3: 9; so max = 4: 7/36.

51.

A fictional warehouse has two independent fire alarms. Alarm A works with probability 0.3, and at least one alarm works with probability 0.65. The probability that alarm B works is

Correct Answer: C [0.5]

Explanation:

0.3 + p − 0.3p = 0.65 → 0.7p = 0.35 → p = 0.5.

52.

Three cards are drawn together at random from a pack of 52. The probability that they are of three different suits is about

Correct Answer: A [0.3976]

Explanation:

Choose 3 suits (4 ways) and one card of each (13³): 4 × 2197/22100 ≈ 0.3976.

53.

Statement-basedConsider the statements:
I. If event A is a subset of event B, then P(A) ≤ P(B).
II. P(A ∪ B) = P(A) + P(B) for any two events A and B.
Which statement(s) is/are correct?

Correct Answer: A [Only I]

Explanation:

II holds only for mutually exclusive events; in general subtract P(A ∩ B).

54.

The odds in favour of event A are 2 : 3 and the odds against an independent event B are 3 : 1. The probability that both A and B happen is

Correct Answer: D [1/10]

Explanation:

P(A) = 2/5; P(B) = 1/(3 + 1) = 1/4; P(A ∩ B) = 2/5 × 1/4 = 1/10.

55.

A bag has 4 red and 6 green balls. A ball is drawn, its colour noted and it is put back; this is done three times. The probability of getting at least one red ball is

Correct Answer: B [0.784]

Explanation:

1 − (0.6)³ = 1 − 0.216 = 0.784.

56.

Each of 5 fictional machines needs repair in a week with probability 0.1, independently. The probability that at least two need repair in a week is about

Correct Answer: A [0.0815]

Explanation:

1 − (0.9)⁵ − 5(0.1)(0.9)⁴ = 1 − 0.59049 − 0.32805 = 0.08146 ≈ 0.0815.

57.

Accidents at a fictional junction occur at a Poisson rate of 4 a month. Using e⁻⁴ = 0.0183, the probability of at least two accidents in a month is

Correct Answer: D [0.9085]

Explanation:

1 − e⁻⁴(1 + 4) = 1 − 0.0183 × 5 = 0.9085.

58.

Monthly bills of fictional customers are normal with mean ₹ 800 and SD ₹ 120. Given Φ(0.5) = 0.6915 and Φ(1.5) = 0.9332, the proportion of bills between ₹ 860 and ₹ 980 is

Correct Answer: C [0.2417]

Explanation:

z = 0.5 and 1.5; 0.9332 − 0.6915 = 0.2417.

59.

X is binomial with n = 10, p = 0.5 and Y is Poisson with mean 3; X and Y are independent. Var(X + Y) is

Correct Answer: B [5.5]

Explanation:

Var X = npq = 2.5; Var Y = 3; independent → add: 5.5.

60.

For students of a fictional college, X = 1 if a student passes Mathematics and Y = 1 if she passes Physics (0 = fails), with P(0, 0) = 0.2, P(0, 1) = 0.3, P(1, 0) = 0.1, P(1, 1) = 0.4. The probability of passing Physics given failure in Mathematics is

Correct Answer: D [0.6]

Explanation:

P(0, 1)/P(X = 0) = 0.3/0.5 = 0.6.

61.

Repair time (hours) of a fictional appliance is exponential with λ = 0.1 per hour. Using ln 2 = 0.6931, the median repair time is

Correct Answer: C [6.931 hours]

Explanation:

Median: e −λM = 1/2 → M = ln 2/λ = 6.931 hours.

62.

100 fictional families with 4 children each are studied, assuming boys and girls are equally likely. The expected number of families with exactly 2 boys is

Correct Answer: B [37.5]

Explanation:

100 × C(4,2)(1/2)⁴ = 100 × 6/16 = 37.5.

63.

X ~ N(10, 9) and Y ~ N(20, 16) are independent (second figure = variance). Given Φ(2) = 0.9772, P(X + Y > 40) is

Correct Answer: A [0.0228]

Explanation:

X + Y ~ N(30, 25), σ = 5; z = (40 − 30)/5 = 2; 1 − 0.9772 = 0.0228.

64.

f(x) = k(1 − x²) for −1 ≤ x ≤ 1 (zero elsewhere) is a probability density. The value of k is

Correct Answer: B [3/4]

Explanation:

∫₋₁¹ (1 − x²) dx = 2 − 2/3 = 4/3 → k = 3/4.

65.

Statement-basedConsider the statements:
I. In a Poisson distribution the mean and the variance are equal.
II. In a binomial distribution (0 < p < 1) the mean is less than the variance.
Which statement(s) is/are correct?

Correct Answer: A [Only I]

Explanation:

Binomial: variance = npq < np = mean, so II is wrong.

66.

Which distribution has the moment generating function M(t) = exp[m(eᵗ − 1)]?

Correct Answer: C [Poisson with mean m]

Explanation:

Poisson: E(e tX ) = Σ e tx e⁻ᵐ mˣ/x!

67.

To cut the standard error of the sample mean to half its present value, the sample size must be

Correct Answer: D [made 4 times as large]

Explanation:

SE = σ/√n; halving it needs √n to double, i.e.

68.

Three fictional shops earn daily profits of 2, 4 and 6 (₹ thousand). Every sample of two shops drawn WITH replacement is listed (9 samples). The variance of the 9 sample means is

Correct Answer: A [4/3]

Explanation:

9 samples; σ² = 8/3 for the population; Var(x̄) = σ²/n = (8/3)/2 = 4/3.

69.

In a systematic sample from a numbered list, the random start is 7 and the interval is 15. The number of the 4th unit selected is

Correct Answer: B [52]

Explanation:

Answer: 52. Units: 7, 22, 37, 52 → 52.

70.

Statement-basedConsider the statements:
I. Under proportional allocation the sampling fraction nₕ/Nₕ is the same in every stratum.
II. Under Neyman allocation, a stratum with more variability gets a larger sample, other things equal.
Which statement(s) is/are correct?

Correct Answer: C [Both I and II]

Explanation:

nₕ ∝ Nₕ makes nₕ/Nₕ constant; nₕ ∝ NₕSₕ favours strata with larger Sₕ.

71.

Before drawing a sample of voters, an agency needs the electoral roll – the list of every voter it can pick from. In sampling, such a list is called the

Correct Answer: A [sampling frame]

Explanation:

The frame is the list; a sampling unit is one element of it; n/N is the sampling fraction.

72.

A sample size of n₀ = 400 is needed for a very large population. For a fictional population of only N = 2000, using n = n₀/(1 + n₀/N), the sample size needed (rounded up to a whole number) is

Correct Answer: C [334]

Explanation:

Answer: 334. 400/(1 + 0.2) = 333.33 → 334.

73.

A fictional telephone survey of household incomes leaves out every home without a telephone. The resulting error is mainly a

Correct Answer: D [selection (coverage) bias – a non-sampling error]

Explanation:

Part of the population has no chance of selection; a bigger sample does not remove this bias.

74.

x₁, x₂, x₃ is a random sample from a population with mean μ. Which of these is an unbiased estimator of μ?

Correct Answer: B [(x₁ + 2x₂ + x₃)/4]

Explanation:

E[(x₁ + 2x₂ + x₃)/4] = 4μ/4 = μ; the others give 2μ/3, 3μ/2 and 3μ/2.

75.

In fictional town A, 60 of 200 sampled homes use solar water heaters; in town B, 40 of 200. Using the pooled proportion, the z statistic for the difference is about

Correct Answer: C [2.31]

Explanation:

p = 100/400 = 0.25; SE = √(0.25 × 0.75 × 2/200) ≈ 0.0433; z = 0.1/0.0433 ≈ 2.31.

76.

Fictional machine P: n = 11, s² = 50; machine Q: n = 9, s² = 20. With F₀.₀₅(10, 8) = 3.35, the F-test for equal variances gives

Correct Answer: A [F = 2.5; variances not significantly different]

Explanation:

F = 50/20 = 2.5 < 3.35 → do not reject H₀.

77.

A fictional survey of 200 people by area (urban/rural) and preferred news source (TV / paper / internet) gives urban: 20, 30, 50; rural: 30, 20, 50. With χ²₀.₀₅,₂ = 5.991, the χ² test of independence gives

Correct Answer: D [χ² = 4; no significant association]

Explanation:

Expected 25, 25, 50 in each row; χ² = 4 × 25/25 + 0 + 0 = 4 < 5.991; df = (2 − 1)(3 − 1) = 2.

78.

A fictional sample of 9 gives x̄ = 30 and s = 6. With t₀.₀₂₅,₈ = 2.306, the 95% confidence interval for μ is

Correct Answer: A [25.388 to 34.612]

Explanation:

x̄ ± t s/√n = 30 ± 2.306 × 6/3 = 30 ± 4.612.

79.

For a two-tailed z-test at the 1% level of significance, H₀ is rejected when

Correct Answer: C [|z| > 2.576]

Explanation:

Two tails of 0.005 each → z₀.₀₀₅ = 2.576.

80.

A fictional trial gives a 2 × 2 table: treated – 18 improved, 12 not; untreated – 12 improved, 18 not. The χ² statistic with Yates' correction is

Correct Answer: D [1.67]

Explanation:

χ² = N(|ad − bc| − N/2)²/(R₁R₂C₁C₂) = 60(180 − 30)²/30⁴ = 1.67 (without correction 2.4).

81.

A random sample 2, 4, 9, 13 comes from a normal population. The maximum likelihood estimate of σ² is

Correct Answer: B [18.5]

Explanation:

Mean 7; MLE of σ² = Σ(x − x̄)²/n = 74/4 = 18.5 (it divides by n, not n − 1).

82.

Choosing the estimates that make the sum of squared differences between the observed and the fitted values as small as possible is the method of

Correct Answer: D [least squares]

Explanation:

Least squares minimises Σ(observed − fitted)²; it gives the regression and trend lines.

83.

A fictional factory claims mean net weight 500 g. A sample of 100 packets gives x̄ = 498 g, s = 10 g. At the 5% level (two-sided, z = 1.96) the claim is

Correct Answer: C [rejected, since z = −2]

Explanation:

z = (498 − 500)/(10/10) = −2; |z| = 2 > 1.96 → reject.

84.

Fictional sales (units) of three shops on different days: P – 10, 12, 14; Q – 15, 17, 16, 20; R – 8, 9, 10. With F₀.₀₅(2, 7) = 4.74, the one-way ANOVA gives

Correct Answer: A [F ≈ 16.76; the shops differ significantly]

Explanation:

SSB = 114.9, SSW = 24; F = (114.9/2)/(24/7) ≈ 16.76 > 4.74.

85.

Output of three fictional operators (rows) on four machines (columns), one shift each: 20, 22, 24, 26 / 18, 21, 22, 23 / 25, 26, 29, 32. In the two-way ANOVA, the error mean square is

Correct Answer: B [0.67]

Explanation:

SS operators = 104, SS machines = 60, SST = 168; SSE = 4 with (3 − 1)(4 − 1) = 6 df; MSE = 0.67.

86.

When only two groups are compared, the one-way ANOVA F statistic and the pooled two-sample t statistic are related by

Correct Answer: A [F = t²]

Explanation:

With k = 2, F(1, ν) = [t(ν)]², ν = n₁ + n₂ − 2: both tests give the same decision.

87.

When the treatment means really differ, the expected value of the between-treatments mean square is

Correct Answer: C [greater than σ²]

Explanation:

E(MSB) = σ² + Σnᵢαᵢ²/(k − 1) > σ² when some αᵢ ≠ 0; this pushes F above 1.

88.

Compared with a completely randomised design, a randomised block design reduces the error sum of squares by

Correct Answer: B [removing the variation between blocks]

Explanation:

Block differences are taken out as a separate source in the two-way analysis.

89.

Diagram-basedThe line graph shows the yearly sales of a fictional company.

Using semi-averages (2016–19 and 2020–23), the annual increase in the trend is

Correct Answer: D [3]

Explanation:

Semi-averages 35 (centred mid-2017/2018) and 47; they are 4 years apart → (47 − 35)/4 = 3.

90.

A fictional series has the value 75 in quarter 1 and 96 in quarter 2. The link relative of quarter 2 is

Correct Answer: A [128]

Explanation:

Link relative = current/previous × 100 = 96/75 × 100 = 128.

91.

Fictional yearly data for six years are 20, 24, 27, 31, 35, 37. A least-squares line y = a + bx is fitted with x = −5, −3, −1, 1, 3, 5 (x in half-years). The trend rises per YEAR by

Correct Answer: D [3.49]

Explanation:

Σxy = 122, Σx² = 70 → b = 1.743 per half-year; per year 2b = 3.49.

92.

Statement-basedConsider the statements:
I. A moving average with a period equal to that of a regular fluctuation removes that fluctuation.
II. The moving-average method gives trend values for every year, including the first and the last.
Which statement(s) is/are correct?

Correct Answer: A [Only I]

Explanation:

Moving averages lose values at both ends of the series, so II is false.

93.

Under the additive model, a fictional month has actual sales 340 and a seasonal effect of +25. The seasonally adjusted value is

Correct Answer: C [315]

Explanation:

Answer: 315. Y − S = 340 − 25 = 315.

94.

The trend value of a fictional product's sales for the coming third quarter is 200 and the seasonal index of that quarter is 115. The forecast (multiplicative model) is

Correct Answer: B [230]

Explanation:

Answer: 230. 200 × 115/100 = 230.

95.

Fictional prices (₹ per unit) and quantities:
Wheat: base price ₹ 20, base quantity 10, current price ₹ 25, current quantity 8
Pulses: base price ₹ 60, base quantity 5, current price ₹ 80, current quantity 6
Ghee: base price ₹ 400, base quantity 1, current price ₹ 440, current quantity 1
Fisher's ideal price index is about

Correct Answer: B [121.42]

Explanation:

L = 121.111, P = 121.739; Fisher = √(L × P) ≈ 121.42.

96.

Price relatives of three fictional items are 125, 140 and 110 with weights 5, 3 and 2. The weighted arithmetic mean of the relatives is

Correct Answer: A [126.5]

Explanation:

Answer: 126.5. (625 + 420 + 220)/10 = 126.5.

97.

A fictional family consumes base quantities 50, 20, 10 of three goods whose prices changed from ₹ 8, 30, 25 to ₹ 10, 33, 30. The cost of living index by the aggregate expenditure method is

Correct Answer: D [116.8]

Explanation:

Σp₁q₀/Σp₀q₀ × 100 = 1460/1250 × 100 = 116.8.

98.

A fictional clerk's monthly income rose from ₹ 20,000 to ₹ 30,000 while the CPI rose from 125 to 175. The percentage increase in his real income is about

Correct Answer: C [7.14%]

Explanation:

Real incomes 16,000 and 17,142.86; increase ≈ 7.14%.

99.

Statement-basedConsider the statements:
I. Laspeyres' price index uses base-year quantities as weights.
II. Paasche's price index uses current-year quantities as weights.
Which statement(s) is/are correct?

Correct Answer: C [Both I and II]

Explanation:

L = Σp₁q₀/Σp₀q₀; P = Σp₁q₁/Σp₀q₁.

100.

Which of the following index numbers satisfies the circular test?

Correct Answer: C [Simple geometric mean of price relatives]

Explanation:

For the simple GM of relatives (and fixed-weight aggregatives) P₀₁ × P₁₂ × P₂₀ = 1; L, P and Fisher fail.

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