Module 5 The Normal Distribution

Question 1

A __________ variable is a variable that can assume any value on a continuum (can assume an uncountable number of values):
*time required to complete a task
*height, in inches.

Answer 1

continuous variable

Question 2

A ___________ density function is a mathematical expression that defines the distribution of the values for a continuous variable.

Answer 2

probability density function

Question 3

examples of reasons to know population density

Answer 3

-My test score is 80 and how many are above and below it?

-Our family’s annual income is $90,000 and how many other family’s are above and below?

-My height is 6 feet, how many people my age are taller than me?

Question 4

*Bell Shaped.
*Symmetrical.
*Mean, Median, and Mode are Equal.

Location is determined by the mean, μ.

Spread is determined by the standard deviation, σ.

The random variable has an infinite theoretical range:
-∞ to +∞

Answer 4

The Normal Distribution

Question 5

Where f(X) is the probability that X occurs
e= the mathematical constant approximated by 2.71828
π = the mathematical constant approximated by 3.14159
μ = the population mean
σ= the population standard deviation
X= any value of our interest (e.g., test score of 80)

Answer 5

The Normal Distribution
-Density Function

Question 6

The Normal Distribution bell shape is put on a _ and () Y graph

Answer 6

X and f(X)

Question 7

*Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution (Z).
*To compute normal probabilities need to transform X units into Z units.
The standardized normal distribution (Z) has a mean of 0 and a standard deviation of 1.

Answer 7

The Standardized Normal

Question 8

Why should we transform the normal distribution into the standardized normal distribution?

Answer 8

There could be an infinite number of combinations of mean and standard deviation. it’s impossible to provide the corresponding infinite number of each normal distribution table to calculate the probabilities that the event of X occurs.

Question 9

Why should we transform the normal distribution into the standardized normal distribution? (Cont.)

Answer 9

If we transform and use the standardized normal distribution, we need only one table of the cumulative standardized normal distribution because its mean is 0 and standard deviation is 1.

Question 10

Translate from X to the standardized normal (the “Z” distribution) by subtracting the mean of X and dividing by its standard deviation:

The Z distribution always has a mean = 0 and standard deviation = 1

Answer 10

Translation to the Standardized Normal Distribution

Question 11

Z=X-μ /σ

The Z distribution always has a mean = 0 and standard deviation = 1

Answer 11

Translation to the Standardized Normal Distribution

Question 12

*Also known as the “Z” distribution
*Mean is 0.
*Standard Deviation is 1.

Answer 12

The Standard Normal Distribution

Question 13

The total area under is 1 because the sum of all the probabilities that the event of Z could occur is 1, so ___%

Answer 13

100%

Question 14

What is P(Z<0)?

Answer 14

The area under the curve to the left of the mean is 0.5, so 50%. That’s because the total area is 1 and the standard normal distribution is symmetric. This can be expressed as P(Z<0).

Question 15

What is P(Z>0)?

Answer 15

A way to solve:
P(Z>0) = 1 - P(Z<0) = 1 -0.5 = 0.5

Question 17

Finding Normal Probabilities

Answer 17

Probability is measured by the area under the curve.
ex: P(a≤ X ≤ b)
=P (a < X < b)