وَأَحْصَىٰ كُلَّ شَيْءٍ عَدَدًا
"And He has enumerated everything in numbers." (Allah perfectly counts and knows the exact measure of all creations).
Q1. The normal probability distribution is:
A. Continuous ✓ Correct Answer
B. Discontinuous
C. Discrete
D. None
Explanation: A normal random variable can take any value on a
continuous scale, so the normal distribution is continuous.
Q2. The normal density function is called:
A. Abnormal curve
B. Skewed curve
C. Asymmetrical curve
D. Normal curve ✓ Correct Answer
Explanation: The graph of the normal density function is simply called
the normal curve (bell-shaped curve).
Q3. Normal curve always:
A. Stays above x-axis ✓ Correct Answer
B. Touches x-axis
C. Go below the x-axis
D. None
Explanation: The normal curve is asymptotic to the x-axis — it
approaches but never touches it, so it always stays above.
Q4. Normal probability density function is:
A. Unimodal
B. Symmetrical
C. Bell shaped
D. All ✓ Correct Answer
Explanation: The normal curve is unimodal, perfectly symmetrical about
the mean, and bell shaped — all three describe it.
Q5. A normal distribution is characterized by:
A. Three parameters
B. Two parameter ✓ Correct Answer
C. One parameter
D. None
Explanation: A normal distribution N(μ, σ²) is fully defined by
exactly two parameters: the mean μ and variance σ².
Q6. The parameter σ controls the:
A. Flatness ✓ Correct Answer
B. Skewness
C. Symmetry
D. All
Explanation: σ (the standard deviation) controls the spread/flatness
of the curve; the normal curve is always symmetric regardless of σ.
Q7. Keeping μ constant and decreasing σ causes the
density function:
A. Unchanged
B. Flatten
C. Sharply peaked ✓ Correct Answer
D. None
Explanation: A smaller σ concentrates the distribution closer to the
mean, making the curve taller and sharply peaked.
Q8. Keeping μ constant and increasing σ causes the
density function:
A. Sharply peaked
B. Flatten ✓ Correct Answer
C. Unchanged
D. None
Explanation: A larger σ spreads the distribution out wider, so the
curve becomes flatter and shorter.
Q9. If σ held constant and μ is varied the normal
curve would change its:
A. Peak
B. Shape
C. Dispersion
D. Location ✓ Correct Answer
Explanation: Changing μ while σ stays fixed just shifts the
same-shaped curve left or right — i.e. it changes its location.
Q10. The value of X being close to μ has higher
probability as variance:
A. Decreases ✓ Correct Answer
B. Increases
C. Doesn't change
D. None
Explanation: A smaller variance concentrates probability tightly
around μ, so values near μ become more likely.
Q11. The value of X being close to μ has lower
probability as variance:
A. Decreases
B. Increases ✓ Correct Answer
C. Doesn't change
D. None
Explanation: As variance increases, the distribution spreads out,
reducing the probability concentrated right at μ.
Q12. The total area under the normal curve is OR If X
~ N(μ,σ²) then P(-∞ < X < ∞):
A. More than one
B. Less than one
C. One (unity) ✓ Correct Answer
D. None
Explanation: Being a valid probability distribution, the total area
under the normal curve always equals 1 (unity).
Q13. The maximum ordinate of normal density function:
A. 1 / (σ√2π) ✓ Correct Answer
B. 1 / √2π
C. 1 / √2σ
D. 1 / √2σ²
Explanation: The peak height of a general normal density (at x = μ) is
1/(σ√2π).
Q14. The two points containing the middle 68.27%
area:
A. μ ± σ ✓ Correct Answer
B. μ ± 2σ
C. μ ± 3σ
D. None
Explanation: About 68.27% of the area under the normal curve lies
within one standard deviation of the mean, i.e. μ ± σ.
Q15. The two points containing the middle 95.45%
area:
A. μ ± σ
B. μ ± 2σ ✓ Correct Answer
C. μ ± 3σ
D. None
Explanation: About 95.45% of the area lies within two standard
deviations of the mean, i.e. μ ± 2σ.
Q16. The two points containing the middle 99.73%
area:
A. μ ± σ
B. μ ± 2σ
C. μ ± 3σ ✓ Correct Answer
D. None
Explanation: About 99.73% of the area lies within three standard
deviations of the mean, i.e. μ ± 3σ.
Q17. In a normal distribution area between μ-σ, μ+σ
OR If X ~ N(μ,σ²) then P(μ-σ < X < μ+σ):
A. 0.6827 ✓ Correct Answer
B. 0.9545
C. 0.9973
D. 0
Explanation: The area within one standard deviation of the mean is
approximately 0.6827.
Q18. In a normal distribution area between μ-2σ, μ+2σ
OR If X ~ N(μ,σ²) then P(μ-2σ < X < μ+2σ):
A. 0.6827
B. 0.9545 ✓ Correct Answer
C. 0.9973
D. 0
Explanation: The area within two standard deviations of the mean is
approximately 0.9545.
Q19. In a normal distribution area between μ-3σ, μ+3σ
OR If X ~ N(μ,σ²) then P(μ-3σ < X < μ+3σ):
A. 0.6827
B. 0.9545
C. 0.9973 ✓ Correct Answer
D. 0
Explanation: The area within three standard deviations of the mean is
approximately 0.9973.
Q20. The Quartile deviation of normal distribution
is:
A. (x₀.₇₅ - x₀.₂₅) / 2
B. 0.6745σ
C. 2/3 σ
D. All ✓ Correct Answer
Explanation: All three represent the same quantity: QD = (Q3-Q1)/2 =
0.6745σ ≈ (2/3)σ, so 'all' is correct.
Q21. The Mean deviation of normal distribution is:
A. σ√(2/π)
B. 0.7979σ
C. 4/5 σ
D. All ✓ Correct Answer
Explanation: MD = σ√(2/π) = 0.7979σ ≈ (4/5)σ, so all three
descriptions are equivalent.
Q22. In normal distribution β₁ = 0 and β₂ =
A. Equal to 3 ✓ Correct Answer
B. Less than 3
C. Greater than 3
D. None
Explanation: For a normal distribution, skewness β₁ = 0 and kurtosis
β₂ = 3 (mesokurtic).
Q23. Normal distribution if:
A. Platykurtic
B. Leptokurtic
C. Mesokurtic ✓ Correct Answer
D. None
Explanation: The normal distribution's kurtosis β₂ = 3, which defines
the mesokurtic (normal-peaked) shape.
Q24. In normal distribution all odd order moments
are:
A. Zero ✓ Correct Answer
B. Negative
C. Positive
D. None
Explanation: Because the normal distribution is symmetric about its
mean, all odd central moments (μ₁, μ₃, μ₅...) equal zero.
Q25. In normal distribution μ₄
A. Negative
B. 0
C. 3σ⁴ ✓ Correct Answer
D. All
Explanation: The fourth central moment of a normal distribution equals
3σ⁴.
Q26. In a normal distribution σ² = 5 then μ₄ =
A. 25
B. 75 ✓ Correct Answer
C. 0
D. √5
Explanation: μ₄ = 3σ⁴ = 3 × 5² = 3 × 25 = 75.
Q27. In normal distribution x₀.₂₅
A. μ ÷ 0.647σ
B. μ - 0.647σ ✓ Correct Answer
C. μ × 0.647σ
D. μ + 0.647σ
Explanation: The 25th percentile (first quartile) of a normal
distribution lies below the mean: x₀.₂₅ = μ - 0.6745σ.
Q28. In normal distribution x₀.₇₅
A. μ + 0.647σ ✓ Correct Answer
B. μ - 0.647σ
C. μ × 0.647σ
D. None
Explanation: The 75th percentile (third quartile) of a normal
distribution lies above the mean: x₀.₇₅ = μ + 0.6745σ.
Q29. In normal distribution:
A. Mean < median < mode
B. Mean ≠ median ≠ mode
C. Mean > median > mode
D. Mean = median = mode ✓ Correct Answer
Explanation: For a symmetric normal distribution, the mean, median,
and mode all coincide.
Q30. Points of inflexion of normal probability
density function are:
A. μ ± 3σ
B. μ ± 2σ
C. μ ± σ ✓ Correct Answer
D. None
Explanation: The normal curve changes curvature (points of inflexion)
exactly at one standard deviation from the mean, μ ± σ.
Q31. If z₀.₂₅ = -0.6745 and z₀.₇₅ = 0.6745 then Q.D =
A. 1.349
B. 0.6745 ✓ Correct Answer
C. 0.975
D. None
Explanation: QD = (z₀.₇₅ - z₀.₂₅)/2 = (0.6745 - (-0.6745))/2 = 1.349/2
= 0.6745.
Q32. Sum of two normal variables is also a normal
variable is:
A. Reproductive
property ✓ Correct Answer
B. Asymptotic property
C. Dispersion
D. Location
Explanation: This closure property — that combinations of normal
variables remain normal — is called the reproductive property.
Q33. The mode of standard normal distribution is:
A. Zero ✓ Correct Answer
B. Less than zero
C. Greater than zero
D. None
Explanation: The standard normal distribution N(0,1) has mean = median
= mode = 0.
Q34. The maximum ordinates of standard normal density
function:
A. 1 / (σ√2π)
B. 1 / √2π ✓ Correct Answer
C. 1 / √2σ
D. 1 / √2σ²
Explanation: Since σ = 1 for the standard normal, the peak height
simplifies to 1/√2π.
Q35. The area to the right of z = 1 is 0.1587 then
area to the left of z = 1 is:
A. Zero
B. 1
C. 0.8413 ✓ Correct Answer
D. None
Explanation: Total area is 1, so area to the left = 1 - 0.1587 =
0.8413.
Q36. The mean of standard normal distribution is:
A. Zero ✓ Correct Answer
B. Negative
C. Positive
D. None
Explanation: By definition, the standard normal distribution N(0,1)
has mean = 0.
Q37. If x₀.₂₅ = 2 and x₀.₇₅ = 4 then μ =
A. 6
B. 3 ✓ Correct Answer
C. 4
D. 2
Explanation: For a symmetric distribution, μ is the midpoint of the
two quartiles: (2+4)/2 = 3.
Q38. If x₀.₂₅ = 2 and x₀.₇₅ = 4 then σ =
A. 4.0
B. 1.48 ✓ Correct Answer
C. 2.5
D. 6.0
Explanation: QD = (4-2)/2 = 1, and QD = 0.6745σ, so σ = 1/0.6745 ≈
1.48.
Q39. In a normal distribution μ = 3 and σ = 1.48 then
x₀.₂₅ =
A. 1.48
B. 3
C. 3.0
D. 2 ✓ Correct Answer
Explanation: x₀.₂₅ = μ - 0.6745σ = 3 - 0.6745(1.48) ≈ 3 - 1.0 = 2.
Q40. In a normal distribution μ = 10 and σ² = 25 then
x₀.₇₅ =
A. 13.37 ✓ Correct Answer
B. 10
C. 25
D. 5
Explanation: σ = √25 = 5, so x₀.₇₅ = μ + 0.6745σ = 10 + 0.6745(5) ≈
13.37.
Q41. In a normal distribution σ² = 25 then m.d. =
A. 4 ✓ Correct Answer
B. 3
C. 2
D. 1
Explanation: σ = √25 = 5, and m.d. = 0.7979σ = 0.7979(5) ≈ 4.
Q42. If X ~ N(50, 100) then σ =
A. 100
B. 50
C. 10 ✓ Correct Answer
D. None
Explanation: In N(μ, σ²) notation, the second value is the variance:
σ² = 100, so σ = √100 = 10.
Q43. If Z ~ N(0,1) then P(Z ≤ a) =
A. 2Φ(-a)
B. 1 - Φ(a)
C. 2Φ(a) - 1
D. Φ(a) ✓ Correct Answer
Explanation: By definition, the cumulative distribution function Φ(a)
directly gives P(Z ≤ a).
Q44. If Z ~ N(0,1) then P(Z ≥ a) =
A. Φ(-a)
B. 1 - Φ(a)
C. a & b ✓ Correct Answer
D. 2Φ(-a)
Explanation: P(Z ≥ a) = 1 - Φ(a), and by symmetry this also equals
Φ(-a) — so both (a) and (b) are correct.
Q45. If Z ~ N(0,1) then P(a ≤ Z ≤ b) =
A. Φ(a)
B. Φ(b) - Φ(a) ✓ Correct Answer
C. 2Φ(a) - 1
D. 2Φ(-a)
Explanation: The probability between two points is the difference of
their cumulative probabilities: Φ(b) - Φ(a).
Q46. Φ(-a) =
A. Φ(a)
B. Φ(a) - Φ(b)
C. 1 - Φ(a) ✓ Correct Answer
D. 2Φ(-a)
Explanation: By the symmetry of the standard normal distribution,
Φ(-a) = 1 - Φ(a).
Q47. P(|Z| ≤ a) =
A. 1 - Φ(a)
B. Φ(a)
C. 2Φ(a) - 1 ✓ Correct Answer
D. 2Φ(-a)
Explanation: P(-a ≤ Z ≤ a) = Φ(a) - Φ(-a) = Φ(a) - (1-Φ(a)) = 2Φ(a) -
1.
Q48. P(|Z| ≥ a) =
A. 1 - Φ(a)
B. Φ(a)
C. 2Φ(a) - 1
D. 2Φ(-a) ✓ Correct Answer
Explanation: P(|Z| ≥ a) = 2P(Z ≥ a) = 2(1-Φ(a)) = 2Φ(-a).
Q49. If Z ~ N(0,1) then P(Z < 0) =
A. 0.75
B. 0.50 ✓ Correct Answer
C. 0.25
D. 0.05
Explanation: The standard normal curve is symmetric about 0, so
exactly half the area lies below 0.
Q50. Z ~ N(0,1) then P(Z > 0) =
A. 0.05
B. 0.25
C. 0.5 ✓ Correct Answer
D. 1.00
Explanation: By symmetry about zero, exactly half the area lies above
0, so P(Z>0) = 0.5.
Q51. If Z ~ N(0,1) then P(Z < -0.6745) =
A. 0.05
B. 0.25 ✓ Correct Answer
C. 0.5
D. 1.00
Explanation: Since z = 0.6745 marks the 75th percentile
(P(Z<0.6745)=0.75), by symmetry P(Z<-0.6745) = 1 - 0.75 = 0.25.
Q52. If Z ~ N(0,1) then P(Z > 0.6745) =
A. 0.05
B. 0.25 ✓ Correct Answer
C. 0.5
D. 1.00
Explanation: Since z = 0.6745 is the 75th percentile, the area beyond
it is 1 - 0.75 = 0.25.
Q53. If Z ~ N(0,1) the first quartile i.e. q₁ =
A. 0.05
B. 0.25
C. 0.5
D. -0.6745 ✓ Correct Answer
Explanation: The first quartile (25th percentile) of the standard
normal occurs at z = -0.6745.
Q54. If Z ~ N(0,1) the third quartile i.e. q₃ =
A. 0.05
B. 0.25
C. 0.6745 ✓ Correct Answer
D. 1.00
Explanation: The third quartile (75th percentile) of the standard
normal occurs at z = 0.6745.
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