Transcript Standard Normal Calculations

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STANDARD NORMAL
CALCULATIONS
Section 2.2

What should we do if we need to compare two
normal distributions with different means and
standard deviations?
Sometimes we need to compare distributions that
are Normal and have different means and standard
deviations. In order to do this, we need to
standardize the distributions so that they have
the same mean and standard deviation.
x  mean
Remember from Sec. 2.1: z 
standard deviation
STANDARD NORMAL DISTRIBUTION
 If
a variable x has any Normal distribution
N(µ,σ), then the standardized variable
𝑥−𝜇
𝑧=
𝜎
has the standard Normal distribution N(0,1).
THE STANDARD NORMAL TABLE

Table A (front of your book) is a table of areas
under the standard normal curve. The table
entry for each value z is the area under the curve
to the left of z.
STEPS FOR FINDING
NORMAL DISTRIBUTION
Step 1: State the problem in terms of the observed variable x.
Step 2: Standardize x to restate the problem in terms of a standard
normal curve. Draw a picture of the distribution and shade the area
of interest under the curve.
Step 3: Find the required area under the standard normal curve,
using Table A and the fact that the total area under the curve is 1.
Step 4: Write your conclusion in the context of the problem.
The Normal Distribution
Lake Maggiore, St. Petersburg, Florida
Alligators are only native to the United States and China.
The mean length of an American adult male is 11.2 feet
with a standard deviation of 2.06 feet and an approximately
normal distribution.
What proportion of adult males are less than 9 feet?
𝑥<9
x  11.2 9  11.2

2.06
2.06
𝑧 < −1.07
To find in table:
*Locate the ones and
tenths digits on the left.
*Locate the hundredths
digit at the top
*Where they intersect is
your proportion.
Alligators are only native to the United States and China.
The mean length of an American adult male is 11.2 feet
with a standard deviation of 2.06 feet and an approximately
normal distribution.
What proportion of adult males are less than 9 feet?
.1423
𝑧 < −1.07
The proportion of adult
male alligators that are less
than 9 feet long is .1428.
The mean length of an American adult male is 11.2 feet
with a standard deviation of 2.06 feet. The largest alligator
recorded in Florida was 17.4 feet.
What is the probability an adult male alligator is more
than 17 feet?
𝑥 > 17
x  11.2 17  11.2

2.06
2.06
𝑧 > 2.82
.9976
But this is the proportion
for the left side of z.
What should we do?
Subtract from 1!!!!
1 – .9976 = .0024
The mean length of an American adult male is 11.2 feet
with a standard deviation of 2.06 feet. The largest alligator
recorded in Florida was 17.4 feet.
What is the probability an adult male alligator is more
than 17 feet?
.0024
𝑧 > 2.82
The proportion of adult
male alligators that are
more than 17 feet long is
.0024.
The mean length of an American adult male is 11.2 feet
with a standard deviation of 2.06 feet.
What is the probability an adult male alligator is between
12 and 14 feet long?
12 < 𝑥 < 14
12 − 11.2 𝑥 − 11.2 14 − 11.2
<
<
2.06
2.06
2.06
0.39 < 𝑧 < 1.36
For intervals:
*Find the z-score of both
ends of the interval.
*Find the proportion that
corresponds to each z-score.
z = .39  .6517
z = 1.36  .9131
*Subtract the proportions.
.9131 - .6517 = .2614
The mean length of an American adult male is 11.2 feet
with a standard deviation of 2.06 feet.
What is the probability an adult male alligator is between
12 and 14 feet long?
.2614
0.39 < 𝑧 < 1.36
The proportion of adult
male alligators that are
between 12 and 14 feet
long is .2614.
The mean length of an American adult male is 11.2 feet
with a standard deviation of 2.06 feet. What is the cut off
length for the largest 15% of American alligators?
.15
11.2
x
x  11.2
z
2.06
When looking for a value
given a proportion:
• If the proportion given is
the “top-most” part of the
graph, you will have to
subtract from 1.
1 – 0.15 = 0.85
• Look in the table for the
closest proportion.
• Go backwards and find
the z-score.
• We will then work
through the equation
backwards.
The mean length of an American adult male is 11.2 feet
with a standard deviation of 2.06 feet. What is the cut off
length for the largest 15% of American alligators?
.15
11.2
x
x  11.2
z
2.06
x  11.2
1.04 
2.06
x ≈ 13.34 feet
In China, the alligator is extremely endangered
and can only be found along the Yangtze River.
The average length of the Chinese alligator is 5
feet with a standard deviation of 0.67. Chinese
legend says that the largest Chinese alligator was
measured to be 10 feet long.
The largest alligator recorded in the U.S. was an
alligator from Louisiana that measured 19 feet 2
inches. N(11.2, 2.06)
In China, the alligator is extremely endangered and can only be found
along the Yangtze River. The average length of the Chinese alligator is
5 feet with a standard deviation of 0.67. Chinese legend says that the
largest Chinese alligator was measured to be 10 feet long.
The largest alligator recorded in the U.S. was an alligator from
Louisiana that measured 19 feet 2 inches. N(11.2, 2.06)
Which alligator measures to
be in the highest percentile
for their country?
Chinese: 10 feet
N(5, .67)
American: 19 feet 2 inches  19.17 feet
N(11.2, 2.06)
Which alligator measures to be in the
highest percentile for their country?
Chinese: 𝑧 =
10−5
.67
American: 𝑧 =
≈ 7.46
19.17−11.2
2.06
≈3.87
Because the Chinese
alligator has a higher zscore, it would be in a
higher percentile than
the American alligator.
HW:
P.
130 #48,52,54,55 DUE: FRIDAY