Chapter 21: The Normal Distribution

Question 1

Distinguish between the probabilities of discrete random variables and continuous random variables

Answer 1

DISCRETE RANDOM VARIABLES:
- X: any non-negative integer value
- Probability of X: P(X=x)

CONTINUOUS RANDOM VARIABLE:
- X: any real value within reasonable domain
- Probability of X: P(X=x)=0
[measurements of any 2 members of a population will never be identical and only the probability that the value lies in an interval can be measured)

Question 2

Distinguish between the probability functions of discrete and continuous random variables

Answer 2

DISCRETE RANDOM VARIABLE:
- Probability function: probability mass function

CONTINUOUS RANDOM VARIABLE:
- Probability function: probability density function/distribution curve
[Value of function is not probability, and probability is the area under the curve for a particular interval]

• f(x)>0 for entire domain
• P(c≤x≤d) = ∫꜀ᵈ f(x) dx
• If domain is a≤x≤b, then ∫ₐᵇ f(x) dx=1

Question 3

Describe the normal distribution and the normal distribution curve

Answer 3

Symmetrical bell-shaped curve
- Mean (μ): centre of curve/line of symmetry
- Standard deviation (σ): distance between μ and point of inflection
- Most members of the population will have measurements distributed around the mean, and few members will have measurements significantly different from the mean

Question 4

Use normal distribution cure (given μ and σ) to calculate probabilites

Answer 4

• 68% of the population will lie within 1σ
• 95% of the population will lie within 2σ
• 99.7% of the population will lie between 3σ

GDC: Menu,5,5,3 (normal CDF)
- Key in lower & upper bound (key in -9E999/9E999 for unlimited lower/upper bound)

Question 5

Calculate the z-score of a normal distribution

Answer 5

If X~N(μ,σ²):
- Z=(x-μ)/σ
[no. of σ x is from the mean]
- Z~N(0,1²)

Question 6

Find quantiles (k)

Answer 6

Finding quantiles:
- When given probability and asked to find corresponding measurement
- GDC: Menu, 5,5,3 (inverse normal)

*Only areas to the left of k can be used
- To find P(X>K) = 1-P(X<K)

Question 7

Find an unknown μ and/or σ

Answer 7

• Convert to z-score
• Equate z-score to invnorm
• Solve simultaneous equations if both μ and σ are unknown