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Computing Probabilities From the Standard Normal Distribution

Table B.1

p. 687

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Table B.1A

Closer Look

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dx

deP

x

x

X 2

1

22 2/)(

22

1

The Normal Distribution: why use a table?

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From x or z to PTo determine a percentage orprobability for a normally distributed variable

Step 1 Sketch the normal curve associated with the variable

Step 2 Shade the region of interest and mark the delimiting x-values

Step 3 Compute the z-scores for the delimiting x-values found in Step 2

Step 4 Use Table B.1 to obtain the area under the standard normal curve delimited by the z-scores found in Step 3

Use Geometry and remember that the total area under the curve is always 1.00.

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From x or z to PFinding percentages for a normally distributed variable from areas under the standard normal curve

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Finding percentages for a normally distributed variable from areas under the standard normal curve

1. , are given.

2. a and b are any two values of the variable x.

3. Compute z-scores for a and b.

4. Consult table B-1

5. Use geometry to find desired area.

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Given that a quiz has a mean score of 14 and an s.d. of 3, what proportion of the class will score between 9 & 16?

1. = 14 and = 3.

2. a = 9 and b = 16.

3. za = -5/3 = -1.67, zb = 2/5 = 0.4.

4. In table B.1, we see that the area to the left of a is 0.0475 and that the area to the right of b is 0.3446.

5. The area between a and b is therefore 1 – (0.0475 + 0.3446) = 0.6079 or 60.79%

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Finding the area under the standard normal curve to the left of z = 1.23

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What if you start with x instead of z?

z = 1.50: Use Column C; P = 0.0668

What is the probability of selecting a random student who scored above 650 on the SAT?

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Finding the area under the standard normal curve to the right of z = 0.76

The easiest way would be to use Column C, but lets use Column B instead

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Finding the area under the standard normal curve that lies between z = –0.68 and z = 1.82

One Strategy: Start with the area to the left of 1.82, then subtract the area to the right of -0.68.

P = 1 – 0.0344 – 0.2483 = 0.7173

Second Strategy: Start with 1.00 and subtract off the two tails

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Determination of the percentage of people having IQs between 115 and 140

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From x or z to PReview of Table B.1 thus far

Using Table B.1 to find the area under the standard normal curve that lies

(a) to the left of a specified z-score,

(b) to the right of a specified z-score,

(c) between two specified z-scores

Then if x is asked for, convert from z to x

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From P to z or x Now the other way around

To determine the observations corresponding to a specified percentage or probability for a normally distributed variable

Step 1 Sketch the normal curve associated the the variable

Step 2 Shade the region of interest (given as a probability or area

Step 3 Use Table B.1 to obtain the z-scores delimiting the region in Step 2

Step 4 Obtain the x-values having the z-scores found in Step 3

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From P to z or xFinding z- or x-scores corresponding to a given region.

Finding the z-score having area 0.04 to its left

Use Column C: The z corresponding to 0.04 in the left tail is -1.75

x = σ × z + μ

If μ is 242 σ is 100, thenx = 100 × -1.75 + 242

x = 67

zx

xz

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The z Notation

The symbol zα is used to denote the z-score having area α (alpha) to its right under the standard normal curve. We read “zα” as “z sub α” or more simply as “z α.”

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The z notation : P(X>x) = α

This is the z-score that demarks an area under the curve with P(X>x)= α

P(X>x)= α

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The z notation : P(X<x) = α

This is the z-score that demarks an area under the curve with P(X<x)= α

P(X<x)= α

Z

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The z notation : P(|X|>|x|) = α

This is the z-score that demarks an area under the curve with P(|X|>|x|)= α

P(|X|>|x|)= α

α/2 α/21- α

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Finding z 0.025

Use Column C: The z corresponding to 0.025 in the right tail is 1.96

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Finding z 0.05

Use Column C: The z corresponding to 0.05 in the right tail is 1.64

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Finding the two z-scores dividing the area under the standard normal curve into a middle 0.95 area and two outside 0.025 areas

Use Column C: The z corresponding to 0.025 in both tails is ±1.96

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Finding the 90th percentile for IQs

z0.10 = 1.28

z = (x-μ)/σ

1.28 = (x – 100)/16

120.48 = x

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What you should be able to do

1. Start with z-or x-scores and compute regions

2. Start with regions and compute z- or x-scores

zx

xz