Normal distribution

Question 1

Features of Normal distribution (how the graph looks, type of data)

Answer 1

• Normal distribution graphs are symmetrical
• They show continuous data

Question 2

What adds to 1 in Normal distribution

Answer 2

In normal distribution, area under the graph = 1

Question 3

Normal distribution: p(x=n) ?

Answer 3

For continuous values, p(x=n)=0, because the value would be infinitesimally small

Question 4

Normal distribution: Where are the points of inflection on the graph?

Answer 4

Points of inflection = μ ± σ (mean ± one standard deviation)

Question 5

Normal distribution: How much data is within 1, 2, and 3 standard deviations of the mean? (percentage rule)

Answer 5

1σ = 68%
2σ = 95%
3σ = 99.7%

Question 6

The normal distribution model format

Answer 6

X ~ N ( μ, σ² )

where μ = mean
and σ = standard deviation

Question 7

What is the standard normal distribution model?

Answer 7

Z ~ N (0 , 1²)

• Used when μ (mean) or σ (standard deviation) are unknown
• Followed by the coding formula

Question 8

What is the coding formula for standard normal distribution?

Answer 8

Z = (X - μ) / σ

where Z is number of standard deviations from mean (the value gained from the standard normal model [Z ~ N (0 , 1²)]

μ is the mean

σ is th standard deviation

X is the variable that we want to find the Z value of

Question 9

What is a Z value in standard normal distribution?

Answer 9

• Z is the number of standard deviations away from the mean
• It tells you how extreme a value is

Question 10

Using phi (Φ) to represent standard normal probabilities

Answer 10

Φ (a) = p (Z < a)

Question 11

Normal distribution: How to find μ and σ when given probabilities are given

Answer 11

• Use standard normal distribution [ Z ~ N (0 , 1²) ] to find Z values that correspond to given probabilities
• Insert into coding equation [ Z = (X - μ) / σ ] (in terms of mu and sigma)
• solve simulataneous equations

Question 12

What are the conditions that allow binomial distributions to be approximated as normal distributions?

Answer 12

• P value is close to 0.5 (means the bd is symmetrical)
• n is large (means the bd is smoother curve, closer to continuous nd)

Question 13

Approximating binomial distributions as normal distributions: how to find mean (μ) and standard deviation (σ)

Answer 13

X~B (n, p) —-> Y~N (μ, σ²)

μ = n x p n=trials p=probability
σ² = np(1-p)

Question 14

Approximating binomial distributions as normal distributions: continuity corrections

Answer 14

1. if > or < change to ≥ or ≤
2. enlarge range by 0.5

ex. p(x>5) = p(x≥6) = p(x≥5.5)