Probability and Statistics – Mrs. Leahy
Unit 7: Normal Curves and Sampling Distributions
Unit 7: Section 1 --- Graphs of Normal Probability Distributions
What is a Normal Distribution and a Normal Curve?
Example 1:
Two Normal Curves, A & B
Do these two distributions have the same mean?
If so, what is it?
One curve has a standard deviation σ=1 and the other has
a standard deviation σ=3. Which is which?
Area under a Normal Curve
Example 2: What percentage of the area under
the normal curve lies:
a) to the left of µ ?
b) between µ - σ and µ + σ ?
c) between µ - 3σ and µ + 3σ ?
Example 3: The scores of a recent SAT test are
normally distributed with a mean of µ=1100 and a
standard deviation σ=200.
a) What is the probability that a score selected at random will be between 1100 and 1500?
b) Is the probability that the score is between 700 and 1100 the same as the probability that the score is between
1100 and 1500?
c) What is the probability that the score is between 1100 and 1300?
d) What is the probabilty that the score is between 1300 and 1500?
Example 4:
Unit 7: Section 2 --- Standard Units and Areas Under the Standard Normal
Distribution
What is a Z-score and a Raw Score?
Suppose Tina and jack are in two different sections of the same course. Each
section is quite large and the scores on the midterm exams of each section
follow a normal distribution. In Tina’s section, the average (mean) was 64 and
her score was 74. In Jack’s section, the mean was 72 and his score was 82. Tina and
Jack were pleased that their scores were each 10 points above the mean of their
sections.
Looking at the normal curves at the right, who did better with respect to the other
students in their section?
One way to express this numerically is with a “z value” or “z score”
Example 1:
Example 2:
Sometimes it is also useful to be able to convert a z score into a raw score.
Example 3:
Example 4:
Standard Normal Distribution:
Areas under the Standard normal
Curve
Example 5: Let z be a random variable with a standard normal distribution. Find the indicated probability and
shade the corresponding area under the normal curve.
a) P( z  1.00)
b) P( z  2.14)
c) P( z  1.98)
d) P( z  0.42)
e) P( z  1.33)
f) P( z  0.45)
g) P( 0.25  z  1.35)
h) P( 1.68  z  2.59)
.
Unit 7: Section 3 --- Areas under any Normal Curve A
Normal Curve vs. Standard Normal Curve
Normal curves:
Show mean and standard deviation values
Generally not a table of areas for EVERY
value of x to reference and find probabilities.
Standard Normal curves:
Show z-scores (#standard deviations from the mean)
Has a table of areas for EVERY z value (to 2 dec)
to reference to find probabilities.
In other words:
P(8.2  x  10.5) =
P( 1.35  z  1.80) =
To find the probability that x will fall into an interval from a to b on a normal curve, we must _______________
the original measurements for x, a, and b into z values.
Example 1: Suppose x has a normal
distribution with µ=2 and σ=3.
Find P( x  7 )
Example 2: Suppose x has a normal distribution with µ=2 and σ=3.
Find P( x  4)
Example 3: Assume that x has a normal distribution with the specified mean and standard deviation. Find the
indicated probabilities.
a)
Inverse Normal Distribution
How to find x, given a desired probability.
1. Use your table (from yesterday) to locate
the probability. If you can’t find it exactly, use
the closest answer to estimate z
2. Translate your z answer into x using the
equation: x  z  
Case 1: Left Tail
Find z by looking up probability “A” in table.
Example 4.
Example 5:
Case 2: Right Tail
Find z by looking up the probability “1 – A” in table.
Example 6:
Example 7:
Case 3: Symmetrical Center Tail
Example 8:
Find z by looking up the probability “(1 – A)/2” in table.
Unit 7: Section 4 --- The Central Limit Theorem A
Probability of a MEAN rather than a SINGLE SCORE
Example 1: Children between the ages of 2 and 5 watch an average of 25 hours of TV per week with a standard
deviation of 3 hours. If a sample of 20 children are selected, find the probability that the mean number of hours
watched per week will be greater than 26.3?
Variables in this problem:
Central Limit Theorem:
µ = mean = ______
n = sample size = ______
σ = standard deviation = _______
𝑥̅ = mean of sample = _______
Standard Error: The standard deviation of a
sampling distribution.
standard error:  x  
Conclusion we can draw:
z
x

n
Our example: P( x  26.3)
Step 1: Convert to z-score
Step 2: Find Probability from Standard Normal Distribution Table
n
Example 3:
Example 4: