QBM117 Business Statistics

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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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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.
• What is the probability that X is less than 9?
• What is the probability that X is between 10 and
  14?

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

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• What is the probability that X is between 6 and
  14?

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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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• What is the probability that a randomly selected
  factory worker can assemble the part in under 60
  seconds?
• What is the probability that a randomly selected
  worker can assemble the part in under 80 seconds?
• 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

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

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

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

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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.
• What percentage of the time will attendance be
  between 24000 and 28000?
• 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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1. What is the probability that a battery lasts
   between 24 hours and 28 hours?
2. What is the probability that a battery lasts
   longer than 24 hours?
3. 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.
• What is the probability that a customer has to
  wait no more than 2 minutes?
• 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.
• What is the probability that a student spends
  between 220 and 250 in any semester?
• What percentage of students spend more than 270
  on textbooks in any semester?
• What percentage of students spend less than 225
  in a semester?

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• Exercises
• 5.55
• 5.59

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