Normal Distribution FIS

S1 Normal Distribution

-Understand the Normal Distribution Curve

-Use Normal Distribution tables

Multiple Choice

The height of high school studnets

• 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

The Normal Curve

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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

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

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

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

Normal Distribution

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

Standard Normal Distribution

Standard Normal Distribution

Normal Distribution

The Normal Distribution is SYMMETRICAL

Multiple Choice

Select an equivalent probability to

$$P\left(Z>-0.5\right)$$  

• $$P\left(Z<-0.5\right)$$  

• $$P\left(Z<0.5\right)$$  

• $$P\left(Z>0.5\right)$$  

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

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

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

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

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