IAL Edexcel S1 Normal Distribution

IAL Edexcel S1 Normal Distribution

-Understand the Normal Distribution Curve

-Use Normal Distribution tables

Multiple Choice

The height of Miss Davidson

• discrete data

• continuous data

Multiple Choice

The number of pets per family in a city.

• discrete data

• continuous data

Continuous Data

Continuous random variable can take any value in a given range.

A continuous random variable has a continuous probability distribution - this can be shown as a curve on a graph.

The total area under the curve is 1

Continuous Data

A Normal Distribution is when the distribution is symmetrical about the mean.

The total area under the curve is 1

The probability is found by finding the area below the curve.

Normal Distribution

A Normal Distribution is when the distribution is symmetrical about the mean.

Mean = Median = Mode

(no skew)

50% of the values are less than the mean and 50% are greater than the mean

Normal Distribution

The mean is easy to spot!

A value is;

Likely to be within 1 standard deviation

Very likely to be within 2 standard deviations

Almost certainly within 3 standard deviations

Fill in the Blank

What is the mean of this Normal Distribution?

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What is the standard deviation of this Normal Distribution?

Fill in the Blank

What is the mean of this Normal Distribution?

Fill in the Blank

What is the standard deviation of this Normal Distribution?

Multiple Choice

What percentage of values lie within 1 standard deviation of the mean?

• 50%

• 68%

• 95%

• 99.7%

Multiple Choice

What percentage of values lie within 2 standard deviations of the mean?

• 50%

• 68%

• 95%

• 99.7%

Multiple Choice

What percentage of values lie less than the mean?

• 50%

• 68%

• 95%

• 99.7%

Normal Distribution

This curve shows the normal distribution of heights of a group of people

Fill in the Blank

What is the mean height of a person in the group?

Fill in the Blank

What is the standard deviation of the group?

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What is the probability that someone has a height between 1.1 and 1.7

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What is the probability that someone has a height between 1.25 and 1.55

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What is the probability that a persons height is exactly 1.1

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What is the probability that a person is taller than 1.7

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What is the probability that a person is taller than 1.85

Normal Distribution

 $X\sim N\left(\mu,\ \sigma^2\right)$  

 $\mu\ =\$ Mean
 $\sigma\ =$  Standard deviation

Multiple Choice

 $X\sim N\left(2,\ 9\right)$  

What is the mean?

• 2

• 3

• 4.5

• 9

Multiple Choice

 $X\sim N\left(2,\ 9\right)$  

What is the variance?

• 2

• 3

• 4.5

• 9

Multiple Choice

 $X\sim N\left(2,\ 9\right)$  

What is the standard deviation?

• 2

• 3

• 4.5

• 9

Multiple Choice

 $X\sim N\left(2,\ 9\right)$  

What is  $P\left(X>2\right)$  

• 0.5

• 0.68

• 0.95

• 0.05

Multiple Choice

 $X\sim N\left(2,\ 9\right)$  

What is  $P\left(X\ge2\right)$  

• 0.5

• 0.68

• 0.95

• 0.55

Multiple Choice

 $X\sim N\left(2,\ 9\right)$  

What is  $P\left(-1<X<5\right)$  

• 0.5

• 0.68

• 0.95

• 0.55

Multiple Choice

 $X\sim N\left(2,\ 9\right)$  

What is  $P\left(-4<X<8\right)$  

• 0.5

• 0.68

• 0.95

• 0.55

Standard Normal Distribution

There are an infinite number of possibilities of means and standard deviations.

Therefore, there is a table, which makes use of the standardised normal distribution curve.

Standard Normal Distribution

 $Z\sim N\left(0,1^2\right)$  

 $P\left(Z<0\right)=0.5$  
 $P\left(-1<Z<1\right)=0.68$  
 $P\left(-2<Z<2\right)=0.95$  

Standard Normal Distribution

 $Z\sim N\left(0,1^2\right)$  

All values are cumulative

  $P\left(Z<z\right)=P\left(Z\le z\right)$  

Standard Normal Distribution

 $Z\sim N\left(0,1^2\right)$  

 $P\left(Z<1.5\right)=\phi\left(1.5\right)=0.9332$  

 $P\left(Z<2.2\right)=\phi\left(2.2\right)=0.9861$  

Fill in the Blank

 $P\left(Z<1.01\right)$  

Fill in the Blank

 $P\left(Z<1.68\right)$  

Fill in the Blank

 $\phi\left(2.18\right)$  

Fill in the Blank

 $\phi\left(0.94\right)$  

Fill in the Blank

 $P\left(Z>0.94\right)$  

Fill in the Blank

 $\phi\left(z\right)=0.7190$  

What is the value of  $z$  

Fill in the Blank

 $P\left(Z>z\right)=0.1335$  

NORMAL TABLE always tells us  $P\left(Z<z\right)$  

What is the value of  $z$  

Normal Distribution

The Normal Distribution is SYMMETRICAL

Multiple Choice

Select an equivalent probability to

 $P\left(Z>-0.5\right)$  

•  $P\left(Z<-0.5\right)$  

•  $P\left(Z<0.5\right)$  

•  $P\left(Z>0.5\right)$  

Fill in the Blank

Find

 $P\left(Z>-0.5\right)$  

Multiple Choice

Select an equivalent probability to

 $P\left(Z<-1.2\right)$  

•  $P\left(Z>-1.2\right)$  

•  $P\left(Z<1.2\right)$  

•  $P\left(Z>1.2\right)$  

Multiple Choice

Select 

 $P\left(Z<-1.2\right)$  

• 0.8849

• -0.8849

• 0.1151

• -0.1151

Multiple Choice

Select an equivalent probability to

 $\phi\left(-0.6\right)$  

•  $\phi\left(0.6\right)$  

•  $1-\phi\left(0.6\right)$  

•  $1-\phi\left(-0.6\right)$  

Standard Normal Distribution

 $\phi\left(z\right)=P\left(Z<z\right)$  

 $\phi\left(-z\right)=1-\phi\left(z\right)$  

Fill in the Blank

What is

 $P\left(1<Z<2\right)$  

Fill in the Blank

What is

 $P\left(-0.5<Z<1.08\right)$  

How can we use the Standard Normal Distribution for all Normal Distributions

 $X\sim N\left(1010,\ 20^2\right)$  

 $z=\frac{x-\mu}{\sigma}$  

 $z=\frac{1010-1010}{20}=0$  

How can we use the Standard Normal Distribution for all Normal Distributions

 $X\sim N\left(1010,\ 20^2\right)$  

 $z=\frac{x-\mu}{\sigma}$  

 $z=\frac{1030-1010}{20}=1$  

How can we use the Standard Normal Distribution for all Normal Distributions

 $X\sim N\left(1010,\ 20^2\right)$  

 $z=\frac{x-\mu}{\sigma}$  

 $Z\sim N\left(0,1^2\right)$  

Multiple Choice

 $X\sim N\left(1010,\ 20^2\right)$  

Calculate the z-value when x=1050

• 1

• 2

• 3

• 4

Multiple Choice

 $X\sim N\left(1010,\ 20^2\right)$  

Calculate the z-value when x=1040

• 1

• 3

• 1.5

• 2

Multiple Choice

 $X\sim N\left(1010,\ 20^2\right)$  

Select the diagram that represents  $P\left(X<1040\right)$  

Multiple Choice

 $X\sim N\left(1010,\ 20^2\right)$  

Calculate  $P\left(X<1040\right)$  

• 0.9332

• 0.8531

• 0.6915

Multiple Choice

 $X\sim N\left(1010,\ 20^2\right)$  

Select the diagram that represents  $P\left(X>1040\right)$  

Multiple Choice

 $X\sim N\left(1010,\ 20^2\right)$  

Calculate  $P\left(X>1040\right)$  

• 0.1332

• 0.8531

• 0.0668

• 0.6668