Applications of Normal Distributions

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Section 6-3

• Applications of Normal Distributions

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• To find probabilities with normal distributions
• (xs), we must first standardize the values
• (convert them to z-scores).
• Then, find the probability as outlined in section
  5-2.
• Recall,

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Converting to Standard Normal Distribution

P
P
?
x
z
0
(a)
(b)
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Example 1

• General Electric manufactures a decorative
  Crystal Clear 60-watt light bulb that it
  advertises will last 1500 hours. Suppose the
  lifetimes of the light bulbs are approximately
  normally distributed with a mean of 1500 hours
  and a standard deviation of 57 hours.
• a. What proportion of the light bulbs will last
  less than the advertised time?

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• b. What is the probability that a randomly
  selected GE Crystal Clear 60-watt light bulb will
  last between 1625 and 1725 hours?
• c. If there are 500 bulbs, how many of them will
  last more than 1650 hours?

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Example 2

• The lengths of pregnancies are
• normally distributed with a mean
• of 268 days and a standard deviation of 15 days.
• One classic use of the normal distribution is
  inspired by a letter to Dear Abby in which a
  wife claimed to have given birth 308 days after a
  brief visit from her husband, who was serving in
  the Navy. Given this information, find the
  probability of a pregnancy lasting 308 days or
  longer.

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• Next, well see how to find a specific score,
• given some preliminary information the
• score separating the top 5 from the
• bottom 95, etc.

95
5
5 or 0.05
0.50
0.45
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• Cautions to keep in mind
• 1. Dont confuse z-scores and areas
• ?areas are regions under the curve
• between specified boundaries
• ?z-scores represent the DISTANCE
• the score is from the mean (its the
• number of standard deviations a
• score is from the mean).

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• 2. Choose the correct side of the graph
  carefully draw a sketch to help you see the
  problem better!
• 3. Although you can have negative z-scores
  (located LEFT of center), areas are always
  positive amounts.

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• To find x, once you have z, think about the
  formula for z
• Rewrite this formula in terms of x
• ?x - ? z?
• ?x ? z?

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Example 3

• Assume that adults have IQ scores that are
  normally distributed with a mean of 100 and a
  standard deviation of 15.
• Find which IQ score separates the bottom 80 from
  the top 20.

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Example 4

• Engineers must consider the breadths of male
  heads when designing motorcycle helmets. Men
  have head breadths that are normally distributed
  with a mean of 6 inches and a standard deviation
  of 1 inch. Due to financial constraints, the
  helmets will be designed to fit all men except
  those with head breadths that are in the smallest
  2.5 or the largest 2.5. Find the minimum and
  maximum head breadths that will fit men.