Probability and the Normal Distribution

1
Probability and the Normal Distribution

• Module 11

2
The Normal Distribution

• The following are useful tools for working with
  the Normal Distribution
• Suppose X N (m, s2),
• Given a value z, find the corresponding x that it
  came from.
• How many standard deviations is x from m?
• Find Pr (X gt x).
• Find Pr (X lt x1 or X gt x2).
• Find Pr ( x1 lt X lt x2).
• Find the x such that Pr ( X lt x ) a, where a
  is some probability.

3
The Normal Distribution

• Find the x such that Pr ( X gt x ) a, where a
  is some probability.
• Find z such that Pr (-z lt Z lt z) 1 - a,
  where a is some probability.

4
The Normal Distribution

• Given a value z, find the corresponding x that it
  came from.
• Suppose that height of a students desk is
  normally distributed with mean 2.5 feet and
  standard deviation 6 inches.

• If z 1.45, what is the corresponding desk
  height?

5
The Normal Distribution
X N (2.5, 0.52).
If z 1.45, what is the corresponding desk
height?
6
The Normal Distribution

• How many standard deviations is x from m?
• Suppose X N (-2, 22) and x -4.5

7
The Normal Distribution

• How many standard deviations is x from m?
• Suppose X N (1, 102) and x -4.5

8
The Normal Distribution

• Find Pr (X gt x).
• Suppose the scores of an exam are distributed
  Normally with mean 80 and standard deviation 10.
  What is the probability that a student gets
  higher than 95?

9
The Normal Distribution

• Find Pr (X gt x).
• Suppose the speed at which paint dries is
  distributed Normally with mean 1.5 hours and
  standard deviation 15 minutes. What is the
  probability that mine takes more than 1 hour to
  dry?

10
The Normal Distribution

• Find Pr(X lt x1 or X gt x2)
• Suppose the scores of an exam are distributed
  normally with mean 80 and standard deviation 10.
  What is the probability that a student gets lower
  than 70 or higher than 92?

11
The Normal Distribution

• Find Pr(X lt x1 or X gt x2)
• Suppose the average number of pages in a college
  text is distributed normally with mean 550 and
  variance 100. What is the probability that my
  textbook contains more than 1000 or less than 800
  pages?

12
The Normal Distribution

• Find Pr ( x1 lt X lt x2).
• Suppose the scores of an exam are distributed
  normally with mean 80 and standard deviation 10.
  What is the probability that a student gets
  between 84 and 94?

13
The Normal Distribution

• Find Pr ( x1 lt X lt x2).
• Suppose the amount of time spent in a car each
  day is distributed normally with mean 48 minutes
  and standard deviation 20 minutes. What is the
  probability that someone spends between 7 and 48
  minutes on a given day?

14
The Normal Distribution

• Find the x such that Pr ( X lt x ) a, where a
  is some probability.
• Suppose the scores of an exam are distributed
  normally with mean 80 and standard deviation 10.
  Find the score such that 75 of the scores are
  lower.

15
The Normal Distribution

• Find the x such that Pr ( X lt x ) a, where a
  is some probability.
• Suppose the number of hamburgers eaten during the
  year by a person is distributed normally with
  mean 34 and variance 49. Find the number of
  hamburgers eaten such that 38 of the numbers are
  lower.

16
The Normal Distribution

• Find the x such that Pr ( X gt x ) a, where a
  is some probability.
• Suppose the scores of an exam are distributed
  normally with mean 80 and standard deviation 10.
  Find the score such that 10 of the scores are
  higher.

17
The Normal Distribution

• Find the x such that Pr ( X gt x ) a, where a
  is some probability.
• Suppose the number of illnesses during ones
  lifetime is distributed normally with mean 34 and
  standard deviation 12. Find the number of
  illnesses such that 14 of the numbers are
  higher.

18
The Normal Distribution

• Find z such that Pr (-z lt Z lt z) 1 - a,
  where a is some probability.
• Find z for a 0.40
• Find z for a 0.99