Ch 7.2 Applications of the Normal Distribution

Modular 12 Ch 7.2 Part II to 7.3

Ch 7.2 Part II Applications of the Normal Distribution Objective B : Finding the Z-score for a given probability Objective C : Probability under a Normal Distribution Objective D : Finding the Value of a Normal Random Variable

Ch 7.3 Assessing Normality Ch 7.4 The Normal Approximation to the Binomial Probability Distribution (Skip) Objective A : Continuity Correction

Objective B : A Normal Approximation to the Binomial

Ch 7.2 Applications of the Normal Distribution Objective B : Finding the Z-score for a given probability

Area  0 . 5

Area  0 . 5

Area  0 . 5

Example 1 : Find the Z-score such that the area under the standard normal curve to its left is 0.0418.  1 . 73 )  0.0418 P ( Z  _____

From Table V 0 . 03

0 . 0418  1 .7 Z ?

0

0 . 0418

Example 2 : Find the Z-score such that the area under the standard normal curve to its right is 0.18. From Table V 0 . 92 P ( Z  ______ )  0.18 0 . 02

1  0 . 18  0 . 82

0 . 18 0 .9

0 . 8186

0 . 8212

(Closer to 0.82)

Z ?

Example 3 : Find the Z-score such that separates the middle 70%.  1 . 04  Z  _____ 1 . 04 )  0.70 P ( _____

70 % Z ?

From Table V 0 . 04

Z ?

15 %

15 %

or 0 . 15

or 0 . 15

Z   1 . 04

 1 .0

Z  1 . 04 ( Due to symmetry )

0 . 1515

0 .1 4 9 2

(Closer to 0.15)

Ch 7.2 Part II Applications of the Normal Distribution Objective B : Finding the Z-score for a given probability

Objective C : Probability under a Normal Distribution Objective D : Finding the Value of a Normal Random Variable

Ch 7.3 Assessing Normality Ch 7.4 The Normal Approximation to the Binomial Probability Distribution Objective A : Continuity Correction

Objective B : A Normal Approximation to the Binomial

Ch 7.2 Applications of the Normal Distribution Objective C : Probability under a Normal Distribution Step 1 : Draw a normal curve and shade the desired area.

Step 2 : Convert the values of

X

to

Z

– scores using

Z 

X 



.

Step 3 : Use Table V to find the area to the left of each Z – score found in Step 2. Step 4 : Adjust the area from Step 3 to answer the question if necessary.

Example 1 : Assume that the random variable X is normally distributed with mean   50 and a standard deviation   7 . (Note: This is not a standard normal curve because   0 and   1 .)  (a) P ( X  58 ) Z   

X 

 58  50

50

7



8

 1 . 14

58

X

0 .8 7 2 9

7

P ( Z  1 . 14 )

From Table V  0 .8 7 2 9

0 1 . 14

Z

(b) P ( 45  X  63 ) X  63

X  45 Z  

X 



X 

Z 

45  50

0 . 2389

63  50



7

5

7 Z   0 . 71

0 . 9686



7 





13

7 Z  1 . 86

P (  0 . 71  Z  1 . 86 )

From Table V  0 . 71  0 . 2389 1 . 86  0 . 9686  0 . 9686  0 . 2389  0 . 7297 ( the whole blue area )

 0 . 71

0

1 . 86

Z

Example 2 : GE manufactures a decorative Crystal Clear 60-watt light bulb that it advertises will last 1,500 hours. Suppose that the lifetimes of the light bulbs are approximately normal distributed, with a mean of 1,550 hours and a standard deviation of 57 hours, what proportion of the light bulbs will last more than 1650 hours? 

P ( X  1650 ) X  1650 ,   1550 ,   57 Z 



0 . 0401

X 



0

1650  1550 57



0 . 9599

100 57

Z  1 . 75

P ( Z  1 . 75 )

From Table V 1 . 75  0 . 9599

 1  0 . 9599  0 . 0401

1 . 75

Z

Ch 7.2 Part II Applications of the Normal Distribution Objective B : Finding the Z-score for a given probability Objective C : Probability under a Normal Distribution Objective D : Finding the Value of a Normal Random Variable Objective E : Applications

Ch 7.3 Assessing Normality Ch 7.4 The Normal Approximation to the Binomial Probability Distribution Objective A : Continuity Correction

Objective B : A Normal Approximation to the Binomial

Ch 7.2 Applications of the Normal Distribution Objective D : Finding the Value of a Normal Random Variable Step 1 : Draw a normal curve and shade the desired area. Step 2 : Find the corresponding area to the left of the cutoff score if necessary. Step 3 : Use Table V to find the Z – score that corresponds to the area to the left of the cutoff score. Step 4 : Obtain x from

Z

by the formula

Z 

X 



or

x    z

.

Ch 7.2 Applications of the Normal Distribution Objective D : Find the Value of a Normal Distribution Example 1 : The reading speed of 6th grade students is approximately normal (bell-shaped) with a mean speed of 125 words per minute and a standard deviation of 24 words per minute. (a) What is the reading speed of a 6th grader whose reading speed is at the 90% percentile?   125 ,   24 , Z  1 . 28 90 % or 0 . 9

Solve for X Z  1 . 28

From Table V 0 . 08

Z  1 . 28 

X 

 X  125 24

1 . 28 ( 24 )  X  125 1 .2

0 . 8997

(Closer to 0.9)

0 . 9015

X  1 . 28 ( 24 )  125 X  155 . 72

words per minute

(b) Determine the reading rates of the middle 95% percentile.   125 ,   24 , solve for X

95 % or 0 . 95

Z   1.96

Z ?

Z ?

2 .5 %

2 .5 %

or 0 . 025

or 0 . 025

Z   1.96

Z  1.96 due to sym m etry

Z 

 1.96 

X 

 X  125 24

 1.96 (24)  X  125

From Table V 0 .0 6

X   1.96 (24)  125 X  7 7 .9 6

 1 .9

0 . 025

words per minute

Z  1 .9 6 Z 

1.96 

X 

 X  125 24

1.96 (24)  X  125

X  1.96 (24)  125 X  1 7 2 .0 4

words per minute

Ch 7.2 Part II Applications of the Normal Distribution Objective B : Finding the Z-score for a given probability Objective C : Probability under a Normal Distribution Objective D : Finding the Value of a Normal Random Variable

Ch 7.3 Assessing Normality Ch 7.4 The Normal Approximation to the Binomial Probability Distribution Objective A : Continuity Correction

Objective B : A Normal Approximation to the Binomial

Ch 7.3 Assessing Normality Ch 7.3 Normality Plot Recall: A set of raw data is given, how would we know the data has a normal distribution? Use histogram or stem leaf plot. Histogram is designed for a large set of data.

For a very small set of data it is not feasible to use histogram to determine whether the data has a bell-shaped curve or not. We will use the normal probability plot to determine whether the data were obtained from a normal distribution or not. If the data were obtained from a normal distribution, the data distribution shape is guaranteed to be approximately bell-shaped for n is less than 30.

Perfect normal curve. The curve is aligned with the dots.

Almost a normal curve. The dots are within the boundaries.

Not a normal curve. Data is outside the boundaries.

Example 1: Determine whether the normal probability plot indicates that the sample data could have come from a population that is normally distributed.

(a)

Not a normal curve. The sample data do not come from a normally distributed population.

(b)

A normal curve. The sample data comes from a normally distribute population.