Download Chapter 3: The Normal Distributions Density curves: A density curve

Chapter 3: The Normal Distributions
Density curves:
A density curve describes the overall pattern of a distribution.
The area under the curve and above any range of values on the horizontal axis is
the proportion of all observations that fall in that range.
A density curve is a curve that is always on or above the horizontal axis.
A density curve is a curve that has an area of exactly 1 underneath it.
Normal Distribution
Many continuous variables have distributions that are bell-shaped and are called
approximately normally distributed variables.
Symmetric Bell-shaped
Mathematical equation for the normal distribution:
x−µ 2
e− 2 ( σ )
√
y=
σ 2π
1
e = 2.71828
π = 3.14159
Any specific Normal curve is described by giving its mean µ and standard deviation σ.
The properties of the Theoretical Normal Distribution :
1. The normal distribution curve is bell-shaped.
2. The mean, median, and mode are equal and located at the center of the distribution.
3. The curve is symmetrical about the mean.
4. The curve is continuous - i.e., there are no gaps or holes.
5. The curve never touches the x axis.
6. The total area under the normal distribution curve is equal to 1.00, or 100%.
7. Empirical Rule (The 68-95-99.7 rule)
The area under the normal curve that lies
within one standard deviation of the mean is approximately 68%;
within two standard deviation of the mean is approximately 95%;
within three standard deviation of the mean is approximately 99.7%.
Standardizing and z-Score
z-Scores: The z-Score measures how far each data value is from the mean of a data set using
Standard Deviation as a yardstick.
The z-Score can be used to compare values from different data sets, or to compare
values within the same data set.
z-Score z =
x−µ
: Population
σ
z-Score z =
x−x
: Sample
s
How to use z-score?
Ordinary values: −2 ≤ z-Score ≤ 2
Unusual values:
z-Score < −2
or
z-Score > 2
If the z-Score is positive, the score is above the mean.
If the z-Score is 0, the score is the same as the mean.
If the z-Score is negative, the score is below the mean.
Round-off Rule for z-Scores: Round z-Score to two decimal place.
Standard Normal Distribution
A normal probability distribution with µ = 0, and σ = 1. The total area under its density
curve is equal to 1.
If a variable x has any Normal distribution N (µ, σ) with mean µ and standard deviation σ,
then the standardized variable
z=
x−µ
σ
(Standardisation Formula)
has the standard Normal distribution, N (0, 1).
Using the standard Normal Table A
Finding Normal proportions (probabilities)
Finding Z Scores from known Areas or Normal proportions (probabilities)
Step 1.
Draw a Density Curve.
Step 2.
Draw a straight line below the density curve as a scale for z.
Step 3.
To find a proportion (probability), plot the value of z on the straight line.
Step 4.
To find a value of z, shade the desired region.
Finding Probability. (You need to find z-scores at first.)
Ex. 1: If the scores for the test have a mean of 100 and standard deviation of 15, find
the percentage of scores that will fall below 112.
Ex. 2: Each month, an American household generates and average of 28 pounds of newspaper
for garbage or recycling. Assume the standard deviation is 2 pounds. If a household is
selected at random, find the probability of its generating
a. Between 27 and 31 pounds per month.
b. More than 30.2 pounds per month.
Assume the variable is approximately normally distributed.
Finding Values from Known Probability. (You need to change % to z-scores.)
Ex. 3: An exclusive college desires to accept only the top 10% of all graduating seniors based on
the results of a national placement test. This test has a mean of 500 and a standard
deviation of 100. Find the cutoff score for the exam.
Assume the variable is normally distributed.
Ex. 4: A special enrichment program in English is to be offered to the top 15% of students
in a school district. A standardized English achievement test given to all students
has a mean of 72.8 and a standard deviation of 18.
Find the cutoff score. Assume the variable is normally distributed.