Normal Distribution

Question 1

Characteristics of the normal distribution

PHOTO 1

Answer 1

Bell-shaped

Symmetrical

Mean, median and mode are equal

Central location is determined by the mean

Spread is determined by the standard deviation (IT IS THE POPULATION STANDARD DEVIATION)

The random variable x has an infinite theoretical range

Question 2

Many different normal distributions

Answer 2

PHOTO 2 SLIDE 4

Identical mean on yellow and blue

Larger mean on pink

yellow smallest sd
Blue medium sd
Pink largest sd

Question 3

What is the height of the curve a measure of

Answer 3

Probability

Question 4

What must the area under the curve be

Answer 4

1

Question 5

Shape of the normal distribution

Answer 5

Photo 3

Question 6

Normal distribution always refers to

Answer 6

Population because Greek letters

Question 7

Translation to the standardised normal distribution

Answer 7

Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardised normal distribution (Z).

Translate any X to the 
Standardised Normal (the Z
distribution) by subtracting
the population mean from
any particular X value
and dividing by the 
population standard deviation

Question 8

Normal distribution pdfs

Answer 8

PHOTOS 4-5

Question 9

The standardised normal distribution

PHOTO 7 SLIDE 9

Answer 9

Also known as the Z distribution
Mean is 0
Standard deviation is 1
Values above the mean have positive Z-values. Values below the mean have negative Z-values.

Question 10

Standardised normal distribution example

Answer 10

PHOTO 6 Slide 10

Question 11

General procedure for finding probabilities

Answer 11

To find P(a < X < b) when X is distributed normally:

Draw the normal curve for the problem in terms of X.

Translate X-values to Z-values and put Z values on your diagram.

Use the Standardised Normal Table.

Question 12

Photo 8

Answer 12

Transformation of scales

Question 13

Finding the x value for a known probability

Answer 13

Photos 9-12

Question 14

Methods of evaluating normality

Answer 14

Compare data set characteristics with properties of normal distribution.

Constructing charts and observing their appearance.

Calculate descriptive numerical measures.

Evaluate how data are distributed.

Construct normal probability plot.

Question 15

Constructing charts and observing their appearance.

Answer 15

• For small- or moderate-sized data sets, do stem-and-leaf display and box-and-whisker plots look symmetric?
  For large data sets, does the histogram or polygon appear bell-shaped?

Question 16

Calculate descriptive numerical measures.

Answer 16

Do the mean and median have similar values? (Remember there may be no unique mode or there may be multiple modes.)
Is the interquartile range approximately 1.33 times the standard deviation?
Is the range approximately 6 times the standard deviation?

Question 17

Evaluate how data are distributed.

Answer 17

Do approximately 2/3 of the observations lie within mean 1 standard deviation?
Do approximately 80% of the observations lie within mean 1.28 standard deviations?
Do approximately 95% of the observations lie within mean 2 standard deviations?

Question 18

Normal probability plot

Answer 18

Arrange data into ordered array.
Find corresponding standardised normal quantile values.
Plot the pairs of points with observed data values on the vertical axis and the standardised normal quantile values on the horizontal axis.
Evaluate the plot for evidence of linearity.

Question 19

Normal probability plot example

Answer 19

Photos 13-14

Question 20

Continuous probability density function

Answer 20

Mathematical expression that defines the distribution of the values for a continuous random variable.