Normal Distribution

Question 1

Normal Distribution Notation

Answer 1

X ∼ N(μ, σ^2)

Remember to square root for σ before putting in calculator

Question 2

Normal curve shape

Answer 2

Bell shaped
Peak at the mean μ
Symmetrical about the mean
Mean = median = mode as no skew
The x-axis is an asymptote

Question 3

Area under a normal curve

Answer 3

Shows probabilities, total area = 1

Question 4

P(X = x) for normal distribution

Answer 4

0 so P(X <= x) = P(X < x)

Question 5

Normal distribution calculator

Answer 5

Use Ncd, with a comically large upper or lower if it is only a less than or greater than an X

Question 6

Z

Answer 6

Z ∼ N(0, 1)

Question 7

X to Z

Answer 7

Z = (X - μ)/σ and plug that value of Z into the standard normal distribution

Question 8

Z to X

Answer 8

X = σZ + μ

Question 9

Inverse normal calc

Answer 9

< means left, > means right

Question 10

Finding an unknown σ and/or μ from a known probability

Answer 10

Find the Z value that would give it in the standard distribution and use X = σZ + μ
Can use simultaneous equations if both are unknown

Question 11

Normal Hypothesis test

Answer 11

Population parameter, μ = value modelled as
H0: μ = population parameter
H1: μ >/!= population parameter (1/2 tail)
Significance level as a decimal
Code for z using (sqrt(n)(meanx - μ))/σ
Find the probability Z > z and compare to α

Question 12

Normal two-tail hypothesis test

Answer 12

Carry out a one-tail with the side of the mean it’s on, compare to α/2

Question 13

P (X = 7) binomial normal approximation

Answer 13

P(6.5 < Y < 7.5)

Question 14

P (X <= 7) binomial normal approximation

Answer 14

P(Y < 7.5)

Question 15

P (X < 7) binomial normal approximation

Answer 15

P(Y < 6.5)

Question 16

Normal binomial approximation conditions

Answer 16

p ≈ 0.5
n is large

If p isn’t close to 0.5 it works if np and nq are both bigger than 5

Where q = 1-p (probability of failure)

Question 17

Binomial to normal μ and σ^2

Answer 17

μ = np
σ^2 = npq