Normal distribution

Question 1

Introduction

Answer 1

Normal distribution is continuous, the probabilities are found between 2 values of the variable. The probability is represented by the area under the curve, which can be found by integration or the table of values. The distribution is often shaped like a bell, regular bell shaped distribution is called normal distribution. Normal distribution and normal curve are symmetrical about the mode, median and mean.

Note: To compare normal distributions the mean and standard deviation must be known.

Question 2

Standard normal distribution

Answer 2

As the shape of normal distribution varies according to the mean and standard deviation, each distribution is changed to a standard normal curve

X ~ N (µ, σ²) –> Z ~ N (0, 1)

A standard normal curve has the mean zero and standard deviation 1.

If the area wanted to be found is negative as the curve is symmetrical subtract its positive value from 1

To read the table the first column gives the first decimal place, the second columns give the second decimal place and the third column gives the third decimal place that is added in the 3rd decimal place to the values given from the table.

Note: values are added in the third decimal place given values such as 4 and 3 are considered 04 and 03 eg 0.0004 not 0.004

Question 3

Inverse standard normal distribution

Answer 3

Draw a bell shaped curve and use context and the table to work out which probability aligns with the given value

Question 4

Standardise normal distribution

Answer 4

If the normal distribution does not have a mean of 0 and standard deviation of 1 we must standardise the values from a distribution using the formula..

X ~ N (µ, σ²) where Z = [ x - µ ] / σ

Z is the number of standard deviations from the mean
µ is the mean
X is the value being considered
σ is the standard deviation

P(Z ≤ z) = Φ(z)
z = [ x - µ ] / σ

Question 5

Inverse normal distribution

Answer 5

with the given values sub them into the formula to convert P(X ≤ [ ] ) to P(Z ≤ [ ]) and equate it to what is given

Φ( the value of P(Z)) = value specified
use the table to find the Φ value
equate the incomplete equation for z to the located Φ value and solve for the variable [ ]

Note: for a real life problem the values have to be standardised and converted to z scores, sketching the normal distribution may help.

Question 6

Normal approximation to the binomial distribution,

Continuity correction

Answer 6

Only applied when converting discrete distributions eg (binomial) to continuous distributions, so not for an already given continuous distribution.

The normal distribution is the probability distribution of a continuous random variable, sometimes a normal distribution is used to model a discrete random variable to continuous random variable.

Scores in an exam or rounded measurements, when this happens continuity correction is used.

If a measurement is given to the nearest unit it could actually be half of the unit on either side of the given measurement.

When calculation draw out an ig number line diagram, the sign remains constant regardless of the numerical logic and then the sign is followed to ensure the probability fits the original equation.

Question 7

Distributions with inequalities

Answer 7

Calculate the phi value for each values for the inequality then sketch a graph with the probabilities and values. Based on the graphical representation and desired area add or subtract the values obtained from the table to solve for the correct probability.

Question 8

Normal approximation to binomial distribution

Answer 8

If X ~ B (n, p) where np > 5 and nq > 5, then this distribution of X can be reasonably approximated by a normal distribution,
If X ~ B (n, p) –> X ~ N (µ, σ²)

Steps:
State the parameters (n and p) of the binomial distribution
Check the conditions
Convert the binomial to normal distribution
Apply continuity correction
Standardise the normal distribution using Z = [ x - µ ] / σ
Use the table to find the probability