Normal Distribution - Calculations (CIE A Level Maths: Probability & Statistics 1)

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Normal Distribution - Calculations

Throughout this section we will use the random variable X tilde straight N left parenthesis mu comma sigma squared right parenthesis . For normal, X can take any real number. Therefore any values mentioned in this section will be assumed to be any real number.

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Calculating Normal Probabilities

How do I find probabilities using a normal distribution?

  • The area under a normal curve between the points x equals a and x equals b is equal to the probability P(a < X < b )
    • Remember for a normal distribution P left parenthesis a less or equal than X less or equal than b right parenthesis equals P left parenthesis a less than X less than b right parenthesis so you do not need to worry about whether the inequality is strict (< or >) or weak (≤ or ≥)
  • The equation of a normal distribution curve is complicated so the area must be calculated numerically
  • You will be expected to standardise all normal distributions to z and use the table of the normal distribution to find the probabilities
    • It is likely that your calculator has a function that can find normal probabilities, if so it is a good idea to learn to use it so that you can check your probabilities
    • However you must show your calculations to get the z values and use the tables to get all the marks

How do I calculate the probability for a normal distribution?

  • A random variable X tilde straight N left parenthesis mu comma sigma squared right parenthesis  can be coded to model the standard normal distribution Z tilde straight N left parenthesis 0 comma 1 squared right parenthesis using the formula

Z equals fraction numerator X minus mu over denominator sigma end fraction

  • You can calculate a probability straight P left parenthesis X less than x right parenthesis using the relationship straight P left parenthesis X less than x right parenthesis equals straight P open parentheses Z less than fraction numerator x minus mu over denominator sigma end fraction close parentheses
  • Always sketch a quick diagram to visualise which area you are looking for
  • Once you have determined the z value use the table of the normal distribution to find the probability
    • Refer to your sketch to decide if you need to subtract the probability from one

The probability of a single value is always zero for a normal distribution

    • You can picture this as the area of a single line is zero
    • bold P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis bold equals bold 0
  • straight P left parenthesis X less than mu right parenthesis equals straight P left parenthesis X greater than mu right parenthesis equals 0.5
    • You can look at which side of the mean x is on and the direction of the inequality to decide if your answer should be greater or less than 0.5
  • As straight P left parenthesis X equals a right parenthesis equals 0 you can use:
    • straight P left parenthesis X less than a right parenthesis plus straight P left parenthesis X greater than a right parenthesis equals 1
    • straight P left parenthesis X greater than a right parenthesis equals 1 minus straight P left parenthesis X less than a right parenthesis equals 1 minus straight capital phi open parentheses fraction numerator a minus mu over denominator sigma end fraction close parentheses
    • straight P left parenthesis a less than X less than b right parenthesis equals straight P left parenthesis X less than b right parenthesis minus straight P left parenthesis X less than a right parenthesis equals straight capital phi open parentheses fraction numerator b minus mu over denominator sigma end fraction close parentheses minus straight capital phi open parentheses fraction numerator a minus mu over denominator sigma end fraction close parentheses

Worked example

The random variable X tilde straight N left parenthesis 20 space comma space 5 squared right parenthesis. Calculate:

(a)
P left parenthesis X space less or equal than space 22 right parenthesis,

 

(b)
straight P left parenthesis 18 less or equal than X less than 27 right parenthesis
(a)
P left parenthesis X space less or equal than space 22 right parenthesis,

 3-3-3-calculating-normal-probabilities-we-solution-1_a

(b)
straight P left parenthesis 18 less or equal than X less than 27 right parenthesis

3-3-3-calculating-normal-probabilities-we-solution-1_b

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Inverse Normal Distribution

Given the value of P(X < a)  or P(X > a)  how do I find the value of a?

  • Given a probability you will have to look through the table of the normal distribution to locate the z-value that corresponds with that probability
  • Look at whether your probability is greater or less than 0.5 and the direction of the inequality to determine whether your z-value will be positive or negative
    • If straight P left parenthesis X less than a right parenthesis is more than 0.5 or straight P left parenthesis X greater than a right parenthesis is less than 0.5 then a should be bigger than the mean
      • z will be positive
    • If straight P left parenthesis X less than a right parenthesis is less than 0.5 or straight P left parenthesis X greater than a right parenthesis is more than 0.5 then a  should be smaller than the mean
      • z will be negative
  • You do not need to remember these, a sketch will help you see it
    • Always sketch a diagram

3-3-3-inverse-normal-diagram-1-

  • If your probability is less than 0.5 you will need to subtract it from one to find the corresponding z value
    • Remember that the position of the z-value will not change, only the direction of the inequality
  • Once you have the correct z value substitute it into the formula z equals fraction numerator a minus mu over denominator sigma end fraction   and solve to find the value of a
  • Always check that your answer makes sense by considering where a is in relation to the mean

Given the value of P(µ- a < X < µ + a) I find the value of a  ?

  • A sketch making use of the symmetry of the graph is essential
  • If you are given P left parenthesis mu minus a less than X less than mu plus a right parenthesis equals alpha percent sign  then straight P left parenthesis X less than mu plus a right parenthesis will be open parentheses fraction numerator 100 plus alpha over denominator 2 end fraction close parentheses percent sign 
    • This is easier to see from a sketch than to remember
    • You can then look through the tables for the corresponding z-value and substitute into the formula  z equals fraction numerator left parenthesis mu plus a right parenthesis minus mu over denominator sigma end fraction equals a over sigma

3-3-3-inverse-normal-diagram-2

Worked example

The random variable W tilde straight N left parenthesis 50 comma 36 right parenthesis  

Find the value of w such that  P left parenthesis W greater than w right parenthesis equals 0.7676

3-3-3-inverse-normal-we-solution-2

Examiner Tip

  • The most common mistake students make when finding values from given probabilities is forgetting to check whether the z-value should be negative or not.  Avoid this by checking early on using a sketch whether z is positive or negative and writing a note to yourself before starting the other calculations.

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.