Normal distribution

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Normal distribution

1. Learn about the properties of a normal
   distribution
2. Solve problems using tables of the normal
   distribution
3. Meet some other examples of continuous
   probability distributions

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Types of variables Discrete Continuous

• Describe what types of data could be described as
  continuous random variable Xx
• Arm lengths
• Eye heights

3
REMEMBER your box plot
Middle 50
LQ
UQ
range
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The Normal Distribution (Bell Curve)
Average contents 50 Mean µ 50 Standard
deviation s 5
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The normal distribution is a theoretical
probabilitythe area under the curve adds up to
one
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The normal distribution is a theoretical
probabilitythe area under the curve adds up to
one
A Normal distribution is a theoretical model of
the whole population. It is perfectly
symmetrical about the central value the mean µ
represented by zero.
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As well as the mean the standard deviation (s)
must also be known.
The X axis is divided up into deviations from the
mean. Below the shaded area is one deviation from
the mean.
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Two standard deviations from the mean
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Three standard deviations from the mean
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A handy estimate known as the Imperial Rule for
a set of normal data68 of data will fall
within 1s of the µ
P( -1 lt z lt 1 ) 0.683 68.3
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95 of data fits within 2s of the µ
P( -2 lt z lt 2 ) 0.954 95.4
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99.7 of data fits within 3s of the µ
P( -3 lt z lt 3 ) 0.997 99.7
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Simple problems solved using the imperial rule -
firstly, make a table out of the rule
lt-3 -3 to -2 -2 to -1 -1 to 0 0 to 1 1 to 2 2 to 3 gt3
0 2 14 34 34 14 2 0
The heights of students at a college were found
to follow a bell-shaped distribution with µ of
165cm and s of 8 cm.
What proportion of students are smaller than 157
cm
16
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Simple problems solved using the imperial rule -
firstly, make a table out of the rule
lt-3 -3 to -2 -2 to -1 -1 to 0 0 to 1 1 to 2 2 to 3 gt3
0 2 14 34 34 14 2 0
The heights of students at a college were found
to follow a bell-shaped distribution with µ of
165cm and s of 8 cm.
Above roughly what height are the tallest 2 of
the students?
165 2 x 8 181 cm
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Task class 10 minutes finish for homework

• Exercise A Page 76

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The Bell shape curve happens so when recording
continuous random variables that an equation is
used to model the shape exactly.

• Put it into your calculator and use the graph
  function.

Sometimes you will see it using phi .
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Luckily you dont have to use the equation each
time and you dont have to integrate it every
time you need to work out the area under the
curve the normal distribution probability

• There are normal distribution tables

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How to read the Normal distribution table
F(z) means the area under the curve on the left
of z
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How to read the Normal distribution table
F(0.24) means the area under the curve on the
left of 0.24 and is this value here
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Values of F(z)

• F(-1.5)1- F(1.5)

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Values of F(z)

• F(0.8)0.78814 (this is for the left)
• Area 1-0.78814 0.21186

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Values of F(z)

• F(1.5)0.93319
• F(-1.00)
• 1- F(1.00)
• 1-0.84134
• 0.15866
• Shaded area
• F(1.5)- F(-1.00)
• 0.93319 - 0.15866
• 0.77453

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Task

• Exercise B page 79

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Solving Problems using the tables

• NORMAL DISTRIBUTION
• The area under the curve is the probability of
  getting less than the z score. The total area is
  1.
• The tables give the probability for z-scores in
  the distribution XN(0,1), that is mean 0, s.d.
  1.
• ALWAYS SKETCH A DIAGRAM
• Read the question carefully and shade the area
  you want to find. If the shaded area is more than
  half then you can read the probability directly
  from the table, if it is less than half, then you
  need to subtract it from 1.
• NB If your z-score is negative then you would
  look up the positive from the table. The rule for
  the shaded area is the same as above more than
  half read from the table, less than half
  subtract the reading from 1.

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You will have to standardise if the mean is not
zero and the standard deviation is not one
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Task

• Exercise C page 168

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Normal distribution problems in reverse

• Percentage points table on page 155
• Work through examples on page 84 and do questions
  Exercise D on page 85

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Key chapter points

• The probability distribution of a continuous
  random variable is represented by a curve. The
  area under the curve in a given interval gives
  the probability of the value lying in that
  interval.
• If a variable X follows a normal probability
  distribution, with mean µ and standard deviation
  s, we write X ? N (µ, s2)
• The variable Z is called the
  standard normal variable corresponding to X

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Key chapter points cont.

• If Z is a continuous random variable such that Z
  ? N (0, 1) then F(z)P(Zltz)
• The percentage points table shows, for
  probability p, the value of z such that P(Zltz)p