Download Statistics – Chapter 6 – Normal Distribution – Notes

Statistics – Chapter 6 – Normal Distribution – Notes
Distributions – Continuous Random Variables
I. Normal Distribution (Normal Curve) (6.1 Normal Probability Distribution)
1.
The curve is bell-shaped with the highest point over the mean  .
2.
3.
4.
5.
6.
It is symmetrical about a vertical line through  .
The curve approaches the horizontal axis but never touches or crosses it.
The transition points between cupping upward and downward occur above    and    .
The parameter  controls the spread of the curve.
The area under the curve is probability. The sum of the area under the curve equals 1.
A. Empirical Rule
For a distribution that is symmetrical and bell-shaped (in particular, for a normal distribution):
 Approximately 68% of the data values will lie within    .
 Approximately 95% of the data values will lie within   2 .
 Approximately 99.7% of the data values will lie within   3
B. Control Charts
1. Out-of-Control Signal I: One point falls beyond the 3 level.
2. Out-of-Control Signal II: A run of nine consecutive points on one side of the center line (the line at
target value  ).
3. Out-of-Control Signal III: At least two of three consecutive points lie beyond the 2 level on the same
side of the center line.
II. Standard Normal Distribution (6.2 Standard Units and Areas Under the Standard Normal Dist.)
If the original distribution of x values is normal, then the corresponding z values have a normal distribution
as well. The standard normal distribution is a normal distribution with mean   0 and standard
deviation   1 . Any normal distribution of x values can be converted to the standard normal distribution by
converting all x values to their corresponding z values.
A. Z-Score
The z value of z score tells us the number of standard deviations the original measurement is from the
mean. The z value is in standard units. A positive z score represents a number, x, above the mean. Likewise,
a negative z score corresponds to an x below the mean.
z
x

x  z  
B. Areas Under the Standard Normal Curve
The area under the curve is the same as probability.
C. Using left-tail style standard normal distribution table
1. For areas to the left of a specified z value, use the table entry directly.
2. For areas to the right of a specified z value, look up the table entry for z and subtract the area from 1.
Note: Another way to find the same area is to use the symmetry of the normal curve and look up the
table entry for - z.
3. For areas between two z values, z1 and z2 (where z2  z1 ), subtract the table area for z1 from the table
area for z2 .