IOE entrance Mathematics · Chapter 7
Statistics and Probability
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36 questions in 3 syllabus topics · 15 are 2-mark questions.
7.1 Measures of location and dispersion
10 questions
1. The arithmetic mean of the first n natural numbers is
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Mean = sum/number of terms.
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Answer: A. (n + 1)/2
Sum = n(n + 1)/2; dividing by n gives (n + 1)/2.
2. The median of the data 7, 3, 9, 5, 11, 2 is
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Arrange in order first; with an even count, average the two middle values.
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Answer: B. 6
Arranged: 2, 3, 5, 7, 9, 11. With 6 values the median is the average of the 3rd and 4th: (5 + 7)/2 = 6.
3. For a moderately skewed distribution the mean is 20 and the median is 18. Using the empirical relation, the mode is
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Mode = 3 Median − 2 Mean.
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Answer: A. 14
Mode = 3 Median − 2 Mean = 54 − 40 = 14.
4. The standard deviation of 2, 4, 6, 8, 10 is
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SD = √(∑(x − x̄)²/n).
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Answer: B. 2√2
Mean = 6; squared deviations 16, 4, 0, 4, 16 sum to 40; variance = 40/5 = 8; SD = √8 = 2√2.
5. If every observation of a data set is multiplied by 3, the variance becomes
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Variance scales with the square of the multiplier.
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Answer: A. 9 times the original
Var(3X) = 3² Var(X) = 9 Var(X).
6. A distribution has mean 40 and variance 64. Its coefficient of variation is
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CV uses the standard deviation, not the variance.
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Answer: C. 20%
SD = √64 = 8. CV = (SD/mean) × 100% = 8/40 × 100% = 20%.
7. For a distribution, the first quartile is 22 and the third quartile is 46. The quartile deviation is
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Quartile deviation is half the interquartile range.
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Answer: C. 12
QD = (Q₃ − Q₁)/2 = (46 − 22)/2 = 12.
8. The mean deviation about the mean of 3, 5, 7, 9, 11 is
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Take absolute values of deviations from the mean, then average.
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Answer: D. 2.4
Mean = 7; absolute deviations 4, 2, 0, 2, 4 sum to 12; MD = 12/5 = 2.4.
9. Which of the following is NOT changed when a constant is added to every observation?
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Distinguish measures of location from measures of dispersion.
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Answer: D. Standard deviation
Adding a constant shifts every measure of location by that constant, but the spread (SD) stays the same.
10. Section A has 30 students with mean mark 60 and section B has 20 students with mean mark 70. The combined mean mark is
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Weight each mean by its group size.
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Answer: A. 64
Combined mean = (30 × 60 + 20 × 70)/50 = (1800 + 1400)/50 = 3200/50 = 64.
7.2 Correlation and regression
10 questions
11. Karl Pearson's coefficient of correlation r always satisfies
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r is a bounded, unit-free measure that can be negative.
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Answer: A. −1 ≤ r ≤ 1
By the Cauchy–Schwarz inequality |r| ≤ 1, so −1 ≤ r ≤ 1.
12. If the regression coefficients are b_yx = 0.8 and b_xy = 0.2, the correlation coefficient is
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r is the geometric mean of the two regression coefficients.
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Answer: B. 0.4
r = +√(b_yx · b_xy) = √0.16 = 0.4 (positive because both coefficients are positive).
13. The two regression lines of a bivariate distribution are 3x + 2y = 26 and 6x + y = 31. The means (x̄, ȳ) are
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Regression lines intersect at the point of means.
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Answer: C. (4, 7)
Both lines pass through (x̄, ȳ). From the second, y = 31 − 6x; substituting: 3x + 62 − 12x = 26 ⇒ x = 4, y = 7.
14. The two regression lines of a bivariate distribution are 3x + 2y = 26 and 6x + y = 31. The correlation coefficient between x and y is
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Try both assignments; b_yx·b_xy must not exceed 1.
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Answer: D. −0.5
Take 3x + 2y = 26 as y on x: b_yx = −3/2; 6x + y = 31 as x on y: b_xy = −1/6. Product = 1/4 ≤ 1 (the other assignment gives 4 > 1, impossible). r = −√(1/4) = −0.5, negative since both slopes are negative.
15. If the correlation coefficient between x and y is zero, the two regression lines are
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Find the slopes when r = 0.
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Answer: C. perpendicular to each other
With r = 0, b_yx = b_xy = 0, so the lines are y = ȳ and x = x̄, which are perpendicular.
16. For 5 pairs of ranks, the sum of squares of rank differences is ∑d² = 10. Spearman's rank correlation coefficient is
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ρ = 1 − 6∑d²/(n(n² − 1)).
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Answer: A. 0.5
ρ = 1 − 6∑d²/(n(n² − 1)) = 1 − 60/(5 × 24) = 1 − 0.5 = 0.5.
17. Two judges rank five contestants. Judge X gives ranks 1, 2, 3, 4, 5 and judge Y gives ranks 2, 1, 4, 3, 5 to the same contestants. The rank correlation coefficient is
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Find each rank difference, square and add, then use Spearman's formula.
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Answer: B. 0.8
d = −1, 1, −1, 1, 0 so ∑d² = 4. ρ = 1 − 6 × 4/(5 × 24) = 1 − 24/120 = 0.8.
18. If r = 0.6, σx = 2 and σy = 5, the regression coefficient of y on x is
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b_yx = r·σy/σx.
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Answer: B. 1.5
b_yx = r σy/σx = 0.6 × 5/2 = 1.5.
19. For a bivariate data set, Cov(x, y) = 12, Var(x) = 16 and Var(y) = 25. The correlation coefficient is
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Use standard deviations, not variances, in the denominator.
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Answer: B. 0.6
σx = 4, σy = 5, so r = Cov/(σx σy) = 12/20 = 0.6.
20. Which of the following statements about the regression coefficients b_yx and b_xy is correct?
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Multiply the formulas for the two coefficients.
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Answer: B. r is the geometric mean of b_yx and b_xy
b_yx·b_xy = (r σy/σx)(r σx/σy) = r², so |r| = √(b_yx b_xy), the geometric mean.
7.3 Probability, Bayes’ theorem and binomial distribution
16 questions
21. If a lottery ticket is drawn at random from a lottery box containing 10 prizes and 25 blanks, what is the probability of getting a prize?
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Favourable over total tickets.
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Answer: C. 2/7
There are 10 + 25 = 35 tickets, 10 of them prizes: P = 10/35 = 2/7.
22. Two fair dice are thrown. The probability that the sum of the numbers is 8 is
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List ordered pairs adding to 8.
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Answer: C. 5/36
Favourable outcomes: (2,6), (3,5), (4,4), (5,3), (6,2) — 5 out of 36.
23. A card is drawn from a well-shuffled pack of 52. The probability that it is a king or a heart is
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Use the addition rule and subtract the overlap.
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Answer: B. 4/13
P(K ∪ H) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13 (the king of hearts is counted once).
24. A and B are mutually exclusive events with P(A) = 0.3 and P(B) = 0.4. Then P(A ∪ B) is
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Mutually exclusive means they cannot occur together.
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Answer: A. 0.7
For mutually exclusive events P(A ∩ B) = 0, so P(A ∪ B) = 0.3 + 0.4 = 0.7.
25. If P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2, then the probability that neither A nor B occurs is
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Neither A nor B is the complement of A ∪ B.
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Answer: D. 0.3
P(A ∪ B) = 0.5 + 0.4 − 0.2 = 0.7. P(neither) = P(A′ ∩ B′) = 1 − 0.7 = 0.3.
26. If P(A ∩ B) = 0.15 and P(B) = 0.5, then P(A | B) is
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Conditional probability divides by the probability of the given event.
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Answer: D. 0.3
P(A | B) = P(A ∩ B)/P(B) = 0.15/0.5 = 0.3.
27. A bag contains 5 red and 3 black balls. Two balls are drawn one after another without replacement. The probability that both are red is
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After the first red is removed, only 4 red remain among 7 balls.
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Answer: B. 5/14
P = (5/8) × (4/7) = 20/56 = 5/14.
28. A and B are independent events with P(A) = 1/2 and P(B) = 1/3. Then P(A ∪ B) is
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For independent events, P(A ∩ B) = P(A)P(B).
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Answer: B. 2/3
P(A ∩ B) = (1/2)(1/3) = 1/6. P(A ∪ B) = 1/2 + 1/3 − 1/6 = 2/3.
29. Bag I contains 3 white and 2 black balls; Bag II contains 2 white and 3 black balls. A bag is chosen at random and a ball drawn from it is white. The probability that it came from Bag I is
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Bayes: P(Bag I | white) = P(Bag I)P(white | I)/P(white).
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Answer: D. 3/5
By Bayes' theorem: (½ × 3/5)/(½ × 3/5 + ½ × 2/5) = (3/10)/(5/10) = 3/5.
30. Machines A and B produce 60% and 40% of a factory's output, with 2% and 3% of their items defective respectively. An item picked at random is found defective. The probability that it was made by B is
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Weight each defect rate by the machine's share of output, then apply Bayes' theorem.
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Answer: B. 1/2
P(def from A) = 0.6 × 0.02 = 0.012; P(def from B) = 0.4 × 0.03 = 0.012. P(B | def) = 0.012/0.024 = 1/2.
31. For a binomial distribution with n = 10 and p = 0.4, the variance is
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Variance of a binomial distribution is npq.
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Answer: A. 2.4
Variance = npq = 10 × 0.4 × 0.6 = 2.4.
32. A binomial distribution has mean 4 and variance 3. The number of trials n is
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Divide variance by mean to get q.
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Answer: D. 16
npq/np = q = 3/4, so p = 1/4 and n = 4/(1/4) = 16.
33. A fair coin is tossed 4 times. The probability of getting exactly 2 heads is
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Use ⁿCᵣ pʳ qⁿ⁻ʳ.
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Answer: C. 3/8
P(X = 2) = ⁴C₂ (½)⁴ = 6/16 = 3/8.
34. A fair die is thrown 3 times. The probability of getting at least one six is
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Use the complement: 1 − P(no six).
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Answer: C. 91/216
P(no six) = (5/6)³ = 125/216, so P(at least one six) = 1 − 125/216 = 91/216.
35. Two events A and B are said to be independent if
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Do not confuse independent with mutually exclusive.
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Answer: A. P(A ∩ B) = P(A)·P(B)
Independence means the occurrence of one does not affect the other: P(A | B) = P(A), equivalently P(A ∩ B) = P(A)P(B). P(A ∩ B) = 0 describes mutually exclusive events.
36. The odds in favour of an event are 3 : 5. The probability of the event is
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Odds compare favourable to unfavourable cases, not to the total.
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Answer: A. 3/8
Odds in favour a : b ⇒ P = a/(a + b) = 3/8.