Y2, C3 - Normal Distribution

Question 1

Is the normal distribution discrete or continuous

Answer 1

Continuous

Question 2

What is σ^2

Answer 2

Variance

Question 3

What does a large variance do to the bell shaped curve

Answer 3

Widens it

Question 4

What is μ

Answer 4

Mean

Question 5

What is the area under a probability graph

Answer 5

1

Question 6

What is the y axis of a probability graph

Answer 6

The density of the probability

Question 7

Where are the points of inflection on a normal distribution curve

Answer 7

At μ ± σ

Question 8

What 2 things are equal to the mean of a normal distribution

Answer 8

Mean = mode = median

Question 9

What % of data is within one standard deviation of the mean

Answer 9

68%

Question 10

What % of data is within two standard deviations of the mean

Answer 10

95%

Question 11

What % of data is within three standard deviations of the mean

Answer 11

99.7%

Question 12

What is σ

Answer 12

One standard deviation

Question 13

X~N(20, 3^2), how would you find P(16 < X < a) = 0.3

Answer 13

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

Question 14

What is a Z value

Answer 14

The number of standard deviations above the mean

Question 15

Using IQ distributed as X~N(100, 15^2), what would the z value be for 130 and 85

Answer 15

130 –> Z = 2
85 –> Z = -1

Question 16

What is the distribution formula for the standard normal distribution

Answer 16

Z~N(0, 1^2)
Mean 0, standard deviation, 1

Question 17

What is the coding formula to turn X into Z

Answer 17

Z = (X - μ) / σ

Question 18

In a Z table, what does Ф(a) mean

Answer 18

P(Z < a)
It is the probability Z is less than a constant, a

Question 19

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)

Answer 19

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)

Question 20

Determine a such that P(-a < Z < a) = 0.6

Answer 20

Symmetrical therefore P(Z < a) = 0.8
Plug into inverse normal calc
a = 0.8416

Question 21

X~N(μ, 3^2). Given that P(X > 20), how would you find μ

Answer 21

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

Question 22

Given that P(Y < 160) = 0.99 and P(Y > 152) = 0.90, how would you find μ and σ

Answer 22

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

Question 23

Why do we approximate a binomial distribution using a normal distribution

Answer 23

It takes a lot of computational power to use factorials in high order binomial calculations (over 50)

Question 24

What two things need to happen to be able to accurately approximate binomial distributions as normal distributions (n and p)

Answer 24

n must be large
p must be close to 0.5

Question 25

X~B(n, p) what are n and p and X

Answer 25

n = number of trials
p = probability of successful trials
X = number of successful trials

Question 26

When converting between binomial and normal, how do you calculate μ

Answer 26

μ = n * p

Question 27

When converting between binomial and normal, how do you calculate σ

Answer 27

σ = root(np(1-p))

Question 28

X~B(6, 0.3), how does this convert to a normal distribution Y~N(μ, σ^2)

Answer 28

Y~N(1.8, 1.26) (σ would equal root(1.26))

Question 29

Apply a continuity correction on ‘6 hours to the nearest hour’ to convert it from discrete to continuous

Answer 29

5.5 <= Y < 6.5

Question 30

Apply a continuity correction on X < 9 to convert it from discrete to continuous

Answer 30

Y < 8.5

Question 31

What are the steps for applying a continuity correction

Answer 31

1) If > or <, convert to >=, <= first
2) Enlarge the range by 0.5

Question 32

Convert P(3 < X <= 6) from discrete to continuous

Answer 32

P(4 <= X <= 6)
P(3.5 < Y < 6.5)

Question 33

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

Answer 33

p = 0.5
When p = 0.5, the binomial distribution is symmetrical like the normal distribution

Question 34

For a random sample of size n taken from a random variable X, what is the sample mean (X-bar) distributed with (formula)

Answer 34

X-bar ~ N(μ, (σ^2) / n)

Question 35

What happens to the variance if more sample means are taken

Answer 35

It decreases

Question 36

What does X-bar mean

Answer 36

The sample means (the mean of the samples taken)

Question 37

P(X < μ - 15) = 0.35
Find P((X > μ + 15) l (X > μ - 15))

Answer 37

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