Standard Normal Calculations

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STANDARD NORMAL CALCULATIONSSection 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.

Remember from Sec. 2.1: mean

standard deviationxz

STANDARD NORMAL DISTRIBUTION If a variable x has any Normal distribution

N(µ,σ), then the standardized variable

has the standard Normal distribution N(0,1).

𝑧=𝑥−𝜇𝜎

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 DISTRIBUTIONStep 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?

11.2 9 11.22.06 2.06x

𝑥<9

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 The proportion of adult

male alligators that are less than 9 feet long is .1428.

𝑧<−1 .07

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?

11.2 17 11.22.06 2.06x

𝑥>17

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?

.0024The proportion of adult male alligators that are more than 17 feet long is .0024.

𝑧>2.82

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 .22 .06 <

𝑥−11 .22 .06 <

14−11 .22 .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<1.36The 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?

11.22.06xz

11.2

.15

x

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? 11.2

2.06xz

x ≈ 13.34 feet

11.2

.15 11.21.042.06x

x

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: American: 3.87

Because the Chinese alligator has a higher z-score, it would be in a higher percentile than the American alligator.

HW: P. 130 #48,52,54,55 DUE: FRIDAY