Y2, C3 - Normal Distribution Flashcards
Is the normal distribution discrete or continuous
Continuous
What is σ^2
Variance
What does a large variance do to the bell shaped curve
Widens it
What is μ
Mean
What is the area under a probability graph
1
What is the y axis of a probability graph
The density of the probability
Where are the points of inflection on a normal distribution curve
At μ ± σ
What 2 things are equal to the mean of a normal distribution
Mean = mode = median
What % of data is within one standard deviation of the mean
68%
What % of data is within two standard deviations of the mean
95%
What % of data is within three standard deviations of the mean
99.7%
What is σ
One standard deviation
X~N(20, 3^2), how would you find P(16 < X < a) = 0.3
Find P(X < 16) = 0.0912 (normal CD)
0.0912 + 0.3 is the area to the left of a
Inverse normal:
area = 0.3912
σ = 3
μ = 20
Answer of a = 19.17
What is a Z value
The number of standard deviations above the mean
Using IQ distributed as X~N(100, 15^2), what would the z value be for 130 and 85
130 –> Z = 2
85 –> Z = -1
What is the distribution formula for the standard normal distribution
Z~N(0, 1^2)
Mean 0, standard deviation, 1
What is the coding formula to turn X into Z
Z = (X - μ) / σ
In a Z table, what does Ф(a) mean
P(Z < a)
It is the probability Z is less than a constant, a
The random variable X~N(50, 4^2), write in terms of Ф(z) for some value of z.
a) P(X < 53)
b) P(X >= 55)
a) P(X < 53) = P(Z < (53-50) / 4)
P(Z < 0.75)
ans = Ф(0.75)
b) P(Z > (55-50) / 4)
P(Z > 1.25)
1 - Ф(1.25)
Determine a such that P(-a < Z < a) = 0.6
Symmetrical therefore P(Z < a) = 0.8
Plug into inverse normal calc
a = 0.8416
X~N(μ, 3^2). Given that P(X > 20), how would you find μ
P(X > 20) = 0.2
P(Z > a) = 0.2
inverse normal calc, a = 0.8416
Use coding formula:
(20 - μ) / 3 = 0.8416
Therefore μ = 17.5
Given that P(Y < 160) = 0.99 and P(Y > 152) = 0.90, how would you find μ and σ
Put into 2 forms of P(Z < a) = …
Sub values into coding formula
Solve simultaneous equations to find μ and σ
μ = 158.8
σ = 2.2
Why do we approximate a binomial distribution using a normal distribution
It takes a lot of computational power to use factorials in high order binomial calculations (over 50)
What two things need to happen to be able to accurately approximate binomial distributions as normal distributions (n and p)
n must be large
p must be close to 0.5
X~B(n, p) what are n and p and X
n = number of trials
p = probability of successful trials
X = number of successful trials
When converting between binomial and normal, how do you calculate μ
μ = n * p
When converting between binomial and normal, how do you calculate σ
σ = root(np(1-p))
X~B(6, 0.3), how does this convert to a normal distribution Y~N(μ, σ^2)
Y~N(1.8, 1.26) (σ would equal root(1.26))
Apply a continuity correction on ‘6 hours to the nearest hour’ to convert it from discrete to continuous
5.5 <= Y < 6.5
Apply a continuity correction on X < 9 to convert it from discrete to continuous
Y < 8.5
What are the steps for applying a continuity correction
1) If > or <, convert to >=, <= first
2) Enlarge the range by 0.5
Convert P(3 < X <= 6) from discrete to continuous
P(4 <= X <= 6)
P(3.5 < Y < 6.5)
X~B(n, p)
What is the most accurate estimate when approximating the binomial distribution by a normal distribution.
Give a reason to support the value
p = 0.5
When p = 0.5, the binomial distribution is symmetrical like the normal distribution
For a random sample of size n taken from a random variable X, what is the sample mean (X-bar) distributed with (formula)
X-bar ~ N(μ, (σ^2) / n)
What happens to the variance if more sample means are taken
It decreases
What does X-bar mean
The sample means (the mean of the samples taken)
P(X < μ - 15) = 0.35
Find P((X > μ + 15) l (X > μ - 15))
P(A l B) = P(A n B) / P(B)
P(X > μ + 15 X > μ - 15) / P(X > μ - 15)
= P(X > μ + 15) / 0.65
= 0.35 / 0.65
ans = 7/13
