MP1.12_SECTION 2.2: DENSITY CURVES AND NORMAL DISTRIBUTIONS
THE STANDARD NORMAL DISTRIBUTION
The standard normal distribution has a  = _______ and a  = ______
In order to properly compare one normal distribution to another where UNITS don’t matter, only the , we
have to STANDARDIZE our value. We do this with the z-score.
If we want to find probabilities of an event with CLEAN  (-1 and 1, -2 and 2, -3 and 3) then the
68-95-99.7 rule will work.
If we want to find probabilities of an event with  that are NOT whole numbers, then we have to use a
z-table (aka Table A)
What is the standard normal distribution notation: __________________________
Draw what standardizing a distribution means (from slide):
USING TABLE A
1) All table entries for z are the ____________ under the standard normal curve to the ____________ of z.
2) The area to the left is also called the “__________________” where p stands for _____________________ or
____________________________
3) Suppose we wanted to find P(Z < 0.81), find 0.8 in the LEFT-HAND COLUMN, then find .01 in the
TOP ROW, then find the INTERSECTION.
The number is ____________. This is the DESIRED PROBABILITY.
In other words: the AREA in a standard normal distribution
To the left of the point z = 0.81 is 0.7910.0
Using Table A, fill in the missing values:
Z-Score
P-value
-2.23
1.65
.52
.79
.23
PRACTICE: In the 2008 Wimbledon tennis tournament, Rafael Nadal averaged 115 miles per hour (mph) on
his first serves. Assume that the distribution of his first serve speeds is normal with a mean of 115 mph and a
standard deviation of 6.2 mph.
About what proportion of his first serves would you expect to be less than 120 mph? Greater than?
Steps For Finding Areas In a NORMAL Distribution
STEP 1:
STEP 2: a)
b)
STEP 3:
PRACTICE: When Tiger Woods hits his driver, the distance the ball travels can be described by N(304, 8).
What percent of Tiger’s drives travel between 305 and 325 yards?
STEP 1:
STEP 2(a):
STEP 2(b):
STEP 3:
PRACTICE: The continuous random variable X = the ITBS grade-equivalent vocabulary score for a randomly
selected seventh-grade student in Gary, Indiana. The distribution is approximately normal with mean µ = 6.84
and standard deviation σ = 1.55.
What is the probability that a randomly selected seventh-grader scores below the fourth-grade level on the ITBS
vocabulary test—aka P(X < 4)?
WORKING BACKWARD
I know the percentile, but I don’t know the corresponding z-score.
Let’s find the 90th percentile of the standard normal distribution. We’re looking for the z-score that has 90% of
the area to its left
KEY: Find the AREA in the table, then figure out how you got there
Z=
How to Find Values (X) From Areas
STEP 1:
STEP 2: a)
b)
STEP 3:
PRACTICE: The continuous random variable X = the ITBS grade-equivalent vocabulary score for a randomly
selected seventh-grade student in Gary, Indiana. The distribution is approximately normal with mean µ = 6.84
and standard deviation σ = 1.55.
What test score would place a Gary seventh-grader at the 90th percentile of the distribution?
STEP 1:
STEP 2(a):
STEP 2(b):
STEP 3:
PRACTICE: Suzy bombed her recent AP Stats exam; she scored at the 25th percentile. The class average was
a 170 with a standard deviation of 30. Assuming the scores are normally distributed, what score did Suzy earn
on the exam? (Use a calculator)
HW (page 130): 49, 51, 53, 55, 69, 71, 73, 75