Sample Quizzes For Preparation: The Normal Distribution
A2 Level Mathematics – Topic 5.5: The Normal Distribution
Question 1
What is the shape of the normal distribution curve?
A. Skewed left
B. Skewed right
C. Bell-shaped and symmetrical
D. Uniform
Question 2
If X ~ N(10, 4), what is the standard deviation?
A. 2
B. 4
C. 16
D. 8
Question 3
What percentage of values lie within one standard deviation of the mean in a normal distribution?
A. 50%
B. 68%
C. 95%
D. 99.7%
Question 4
Which of the following is not a property of the normal distribution?
A. Total area under the curve is 1
B. It is symmetric about the mean
C. Mean = Median = Mode
D. It is defined only for positive values
Question 5
If X ~ N(μ, σ²), what is the Z-score formula?
A. Z = X + μ
B. Z = (X + μ) / σ
C. Z = (X − μ) / σ
D. Z = (μ − X) / σ
Question 6
In a standard normal distribution, what is the value of μ and σ?
A. μ = 1, σ = 1
B. μ = 0, σ = 1
C. μ = 0, σ = 0
D. μ = 1, σ = 0
Question 7
What is P(Z > 1.64) to 2 decimal places?
A. 0.05
B. 0.10
C. 0.20
D. 0.30
Question 8
If P(Z < z) = 0.8413, what is z?
A. 0.50
B. 0.84
C. 1.00
D. 1.00
Question 9
If X ~ N(50, 25), what is P(X < 45)?
A. 0.16
B. 0.50
C. 0.84
D. 0.68
Question 10
Which value would you use to find the probability of P(X > 55) where X ~ N(50, 25)?
A. Z = 1
B. Z = -1
C. Z = 2
D. Z = -2
Question 11
What is the area under a normal curve between μ − σ and μ + σ?
A. 68%
B. 95%
C. 99.7%
D. 50%
Question 12
What does a Z-score of 0 represent?
A. The maximum value
B. The mode
C. The mean
D. One standard deviation above the mean
Question 13
Which of these is a valid condition for using the normal approximation to the binomial?
A. np < 5 and nq < 5
B. np > 5 and nq > 5
C. p = 0.5
D. n = 100
Question 14
What correction must be used when approximating a discrete distribution using the normal?
A. Z-correction
B. Continuity correction
C. Variance correction
D. Symmetry correction
Question 15
P(X = 10) for binomial approximation using normal distribution becomes:
A. P(9 < X < 11)
B. P(9.5 < X < 10.5)
C. P(10 < X < 11)
D. P(10.5 < X < 11.5)
Question 16
What is the Z-value corresponding to the 95th percentile?
A. 1.64
B. 1.28
C. 1.96
D. 2.58
Question 17
If X ~ N(100, 36), what value of X corresponds to Z = 1?
A. 94
B. 96
C. 106
D. 110
Question 18
Which of the following best describes the normal distribution?
A. Discrete and symmetrical
B. Continuous and symmetrical
C. Discrete and skewed
D. Continuous and skewed
Question 19
What is the mean of Z if Z ~ N(0,1)?
A. 0
B. 1
C. 2
D. Undefined
Question 20
What is the shape of the graph of y = f(X) for X ~ N(μ, σ²)?
A. Parabola
B. Straight line
C. Bell curve
D. Step function
Question 21
If Z = (X − μ) / σ and Z = 2, μ = 100, σ = 10, what is X?
A. 90
B. 110
C. 120
D. 130
Question 22
What is P(−1 < Z < 1)?
A. 34%
B. 68%
C. 95%
D. 99%
Question 23
If P(Z < a) = 0.025, then a ≈
A. −1.96
B. −2.33
C. 1.96
D. 2.33
Question 24
What is the variance of X ~ N(5, 4)?
A. 2
B. 4
C. 8
D. 16
Question 25
What is the standard deviation of X ~ N(5, 4)?
A. 2
B. 4
C. 8
D. 16
Question 26
If X ~ N(200, 100), what is the probability X > 210? (Use Z)
A. 0.16
B. 0.32
C. 0.84
D. 0.50
Question 27
What does a high standard deviation indicate about a normal distribution?
A. Narrow curve
B. Sharp peak
C. Wide spread
D. No change
Question 28
Which one is not a feature of the normal distribution?
A. Finite range
B. Continuous variable
C. Bell-shaped curve
D. Symmetry
Question 29
To approximate B(50, 0.4) using a normal distribution, the mean is:
A. 20
B. 40
C. 50
D. 60
Question 30
For B(50, 0.4), the variance for the normal approximation is:
A. 30
B. 20
C. 12
D. 5
Answer Key and Detailed Explanations – A2 Level Mathematics: The Normal Distribution (Topic 5.5)
Q1. C. Bell-shaped and symmetrical
The normal distribution is a continuous probability distribution that is symmetric about the mean.
Q2. A. 2
Standard deviation is the square root of variance. √4 = 2.
Q3. B. 68%
68% of the data lies within ±1 standard deviation from the mean in a normal distribution.
Q4. D. It is defined only for positive values
False — the normal distribution is defined for all real numbers.
Q5. C. Z = (X − μ) / σ
This is the standard Z-score formula used to standardize normal variables.
Q6. B. μ = 0, σ = 1
This defines the standard normal distribution, denoted by Z ~ N(0,1).
Q7. A. 0.05
From standard normal tables, P(Z > 1.64) ≈ 0.05.
Q8. D. 1.00
From tables, P(Z < 1.00) = 0.8413.
Q9. A. 0.16
Z = (45 − 50)/5 = −1 ⇒ P(Z < −1) ≈ 0.16.
Q10. A. Z = 1
(55 − 50)/5 = 1 ⇒ P(X > 55) uses Z = 1.
Q11. A. 68%
68% lies between μ ± σ.
Q12. C. The mean
A Z-score of 0 corresponds to the mean of the distribution.
Q13. B. np > 5 and nq > 5
This ensures the distribution is approximately symmetric.
Q14. B. Continuity correction
This adjusts for the fact that the binomial is discrete and the normal is continuous.
Q15. B. P(9.5 < X < 10.5)
This applies continuity correction to discrete value X = 10.
Q16. C. 1.96
This is the Z-value for 95% confidence level.
Q17. C. 106
Z = 1 ⇒ X = μ + Zσ = 100 + (1)(6) = 106.
Q18. B. Continuous and symmetrical
Normal distribution is a continuous, symmetrical probability distribution.
Q19. A. 0
Mean of the standard normal distribution is 0.
Q20. C. Bell curve
The normal distribution is famously bell-shaped.
Q21. C. 120
X = μ + Zσ = 100 + (2)(10) = 120.
Q22. B. 68%
P(−1 < Z < 1) ≈ 68%.
Q23. A. −1.96
P(Z < −1.96) ≈ 0.025.
Q24. B. 4
Variance is directly given as σ² = 4.
Q25. A. 2
Standard deviation is √variance = √4 = 2.
Q26. A. 0.16
Z = (210 − 200)/10 = 1 ⇒ P(Z > 1) = 0.16.
Q27. C. Wide spread
Larger standard deviation means the distribution is more spread out.
Q28. A. Finite range
Normal distribution has an infinite range (−∞, ∞).
Q29. A. 20
Mean = np = 50 × 0.4 = 20.
Q30. B. 12
Variance = npq = 50 × 0.4 × 0.6 = 12.