1Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH

EDITIONEDITION

ELEMENTARY STATISTICSSection 5-4 Normal Distributions: Finding Values

2Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

1. Don’t confuse z scores and areas.

Z scores are distances away from the mean along the horizontal  scale, but areas are regions under the  normal curve.

2. Use invNorm() to find a z-score for a given percentage to the left(percentile).

3. A z score must be negative whenever it is located to the left of the centerline of 0.

Cautions to keep in mind

3Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

0

0.95

1.64

95%

z = invNorm(.95)

Finding z Scores when Given Probabilities

FIGURE 5-11 Finding the 95th Percentile(z score will be positive)

4Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

0

0.80

-.840.20

80%

Finding the top 80%

Bottom 20%

20%

Finding z Scores when Given Probabilities

(z score will be negative)z = invNorm(.20)

5Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Procedure for Finding Values (x)1. Sketch a normal distribution curve, enter the given probability or

percentage in the appropriate region of the graph, and identify the x value(s) being sought.

2. Use invNorm() to find the z score corresponding to the region bounded by x. Cautions:

• You must input the percentile (area below) into invNorm()

• If the area(%) is above you must enter its complement into invNorm()

• If the area(%) is the middle you must enter area(%) in the tail into invNorm(). You will have two opposite z-scores to define the interval.

3. Enter the values for µ, , and the z score found in step 2, then solve for x.

x = µ + (z • ) (z-score formula solved for x)

4. Refer to the sketch of the curve to verify that the solution makes sense in the context of the graph and the context of the problem.

6Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

What weight denotes the 10th percentile of women’s weight? Assume women’s weights are normally distributed with a mean of 143 pounds and standard deviation of 29 pounds.

Example

7Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

143

.10

x = ?

10%

Finding P10 for Weights of Women

FIGURE 5-17

Weight

8Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

143x = ?

0.10

Finding P10 for Weights of Women

0-1.28

Weight

z = invNorm(.10) = -1.28

z

9Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

143x = ?

0.10

Finding P10 for Weights of Women

0-1.28

x = + (z• )

x = 143 + (-1.28 • 29) = 105.88

Weight

10Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

143x = 106

0.10

Finding P10 for Weights of Women

FIGURE 5-17 0-1.28

The weight of 106 lb (rounded) separates the lowest 10% from the highest 90%.

Weight

11Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

143x = 180

0.10

Forgot to make z score negative???

01.28

x = 143 + (1.28 • 29) = 180

Weight

12Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

143x = 180

0.10

Forgot to make z score negative???

01.28

UNREASONABLE ANSWER!

Weight

13Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

x z %normalcdf(zleft, zright)

invNormal(% to Left)x = +z

xz