QBM117Business Statistics

Probability and Probability Distributions

The Normal Distribution

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Objectives

• Learn how to transform a normal random variable to a standard normal random variable.

• Calculate probabilities for any normal random variable.

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Calculating Probabilities for Any Normal Distribution

• Probabilities for all normal distributions are calculated using the standard normal distribution.

• To calculate probabilities for any normal distribution we transform the normal random variable to a standard normal random variable and use the standard normal probability tables (Table 3).

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Standardizing the Normal Distribution

• A normal random variable X with mean and standard deviation is transformed to a standard normal random variable Z with mean 0 and standard deviation 1 using the following formula

• Hence if then the transformation

results in .

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XZ

2~ ( , )X N /)( XZ )1,0(~ NZ

Steps for Calculating Probabilities for Any Normal Distribution

• Sketch the normal curve and mark on the mean • Shade the area corresponding to the probability that

you want to find.• Calculate the z-scores for the boundaries of the

shaded area.• Sketch the standard normal curve and mark on the

mean at 0.• Shade the area corresponding to the area on the

normal curve.• Use table 3 to find the area of the shaded area and

hence the probability that you are looking for.

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Example 1

Suppose that a random variable X is normally distributed with a mean of 10 and a standard deviation of 2.

a. What is the probability that X is less than 9?

b. What is the probability that X is between 10 and 14?

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2~ (10,2 )X N

X10

• What is the probability that X is greater than 12?

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X10 12

)12( XP

9

)1(

21012

)12(

ZP

ZPXP

Z0 1

)1( ZP

1587.0

)1()12(

Therefore

1587.0

3413.05.0)1(

XPXP

ZP

b. What is the probability that X is between 6 and 14?

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X6 14

(6 14)P X

10

12

Z2 2

( 2 2)P Z

0

6 10 14 10(6 14)

2 2

( 2 2)

P X P Z

P Z

( 2 2) 2 (0 2)

2 0.4772

0.9544

Therefore

(6 14) ( 2 2)

0.9544

P Z P Z

P Z P X

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Example 2

A consultant was investigating the time it took factory workers in a car factory to assemble a particular part after the workers had been trained to perform the task using an individual learning approach. The consultant determined that the time in seconds to assemble the part for workers trained with this method was normally distributed with a mean of 75 seconds and a standard deviation of 6 seconds.

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a. What is the probability that a randomly selected factory worker can assemble the part in under 60 seconds?

b. What is the probability that a randomly selected worker can assemble the part in under 80 seconds?

c. What is the probability that a randomly selected worker can assemble the part in 65 to 75 seconds?

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Let X = the time in seconds to assemble the part

2~ (75,6 )X N

X75

a. What is the probability that a randomly selected factory worker can assemble the part in under 60 seconds?

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X60 75

Z2.5 0

( 60)

60 75

6

( 2.5)

0.5 0.4938

0.0062

P X

P Z

P Z

b. What is the probability that a randomly selected worker can assemble the part in under 80 seconds?

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( 80)

80 75

6

( 0.83)

0.5 0.2967

0.7967

P X

P Z

P Z

X

Z

75

0

80

0.83

c. What is the probability that a randomly selected worker can assemble the part in 65 to 75 seconds?

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(65 75)

65 75 75 75

6 6

( 1.67 0)

0.4525

P X

P Z

P Z

X

Z

7565

1.67 0

Exercise 1

The attendance of football games at a certain stadium is normally distributed with a mean of 25000 and a standard deviation of 3000.

a. What percentage of the time will attendance be between 24000 and 28000?

b. What is the probability of the attendance exceeding 30000?

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Exercise 2

Mensa is the international high-IQ society. To be a Mensa member, a person must have an IQ of 132 or higher. If IQ scores are normally distributed with a mean of 100 and a standard deviation of 15, what percentage of the population qualifies for membership in Mensa?

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Exercise 3

Battery manufacturers compete on the basis of the amount of time their product lasts in cameras and toys. A manufacturer of alkaline batteries has observed that its batteries last for an average of 26 hours when used in a toy racing car. The amount of time is normally distributed with a standard deviation of 2.5 hours.

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a. What is the probability that a battery lasts between 24 hours and 28 hours?

b. What is the probability that a battery lasts longer than 24 hours?

c. What is the probability that a battery lasts less than 20 hours?

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Exercise 4

The waiting time at a certain bank is normally distributed with a mean of 3.7 minutes and a standard deviation pf 1.4 minutes.

a. What is the probability that a customer has to wait no more than 2 minutes?

b. What is the probability that a customer has to wait between 4 and 5 minutes?

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Exercise 5

The amount spent by students on textbooks in a semester is normally distributed with a mean of $235 and a standard deviation of $15.

a. What is the probability that a student spends between $220 and $250 in any semester?

b. What percentage of students spend more than $270 on textbooks in any semester?

c. What percentage of students spend less than $225 in a semester?

Exercises

• 5.55• 5.59

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