Chapter 5 and 6 Probability Distributions and Normal Probability Distributions

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Chapter 5 and 6 Probability Distributions and
Normal Probability Distributions
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Chapter Goals
Combine the ideas of frequency distributions and
probability to form probability distributions

• Learn about the normal, bell-shaped, or Gaussian
  distribution

• How probabilities are found

• How probabilities are represented

• How normal distributions are used in the real
  world

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5.2 Random Variables

• Random Variable A variable that assumes a unique
  numerical value for each of the outcomes in the
  sample space of a probability experiment

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Examples of Random Variable

• 1. Let the number of computers sold per day by a
  local merchant be a random variable. Integer
  values ranging from zero to about 50 are possible
  values.

2. Let the number of pages in a mystery novel at
a bookstore be a random variable. The smallest
number of pages is 125 while the largest number
of pages is 547.
3. Let the time it takes an employee to get to
work be a random variable. Possible values are
15 minutes to over 2 hours.
4. Let the volume of water used by a household
during a month be a random variable. Amounts
range up to several thousand gallons.
5. Let the number of defective components in a
shipment of 1000 be a random variable. Values
range from 0 to 1000.
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Discrete Continuous Random Variables

• Discrete Random Variable A quantitative random
  variable that can assume a countable number of
  values

• Intuitively, a discrete random variable can
  assume values corresponding to isolated points
  along a line interval. That is, there is a gap
  between any two values.

Note Usually associated with counting
Continuous Random Variable A quantitative random
variable that can assume an uncountable number of
values

• Intuitively, a continuous random variable can
  assume any value along a line interval, including
  every possible value between any two values

Note Usually associated with a measurement
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Example

• Example Determine whether the following random
  variables are discrete or continuous

1. The barometric pressure at 1200 PM 2. The
length of time it takes to complete a statistics
exam 3. The number of items in the shopping cart
of the person in front of you at the checkout
line 4. The weight of a home grown
zucchini 5. The number of tickets issued by the
PA State Police during a 24 hour period 6. The
number of cans of soda pop dispensed by a machine
placed in the Mathematics building on
campus 7. The number of cavities the dentist
discovers during your next visit
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Probability Distribution Function

• Probability Distribution A distribution of the
  probabilities associated with each of the values
  of a random variable. The probability
  distribution is a theoretical distribution it is
  used to represent populations.

• Notes
• The probability distribution tells you everything
  you need to know about the random variable.
• The probability distribution may be presented in
  the form of a table, chart, function, etc.

Probability Function A rule that assigns
probabilities to the values of the random variable
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6.2 Normal Probability Distributions

• The normal probability distribution is the most
  important distribution in all of statistics

• Many continuous random variables have normal or
  approximately normal distributions

• Need to learn how to describe a normal
  probability distribution

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Normal Probability Distribution

• 1. A continuous random variable

3. Recall the probability that x lies in some
interval is the area under the curve
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The Normal Probability Distribution
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Notation

• If x is a normal random variable with mean m and
  standard deviation s, this is often denoted x
  N(m, s2)

• Example Suppose x is a normal random variable
  with m 35 and s 6. A convenient notation to
  identify this random variable is x N(35, 62).

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Probabilities for a Normal Distribution
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Percentage, Proportion Probability

• Percentage (30) is usually used when talking
  about a proportion (3/10) of a population

• Probability is usually used when talking about
  the chance that the next individual item will
  possess a certain property

• Area is the graphic representation of all three
  when we draw a picture to illustrate the situation

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6.3 Standard Normal Distribution

• Properties
• The total area under the normal curve is equal to
  1
• The distribution is mounded and symmetric it
  extends indefinitely in both directions,
  approaching but never touching the horizontal
  axis
• The distribution has a mean of 0 and a standard
  deviation of 1
• The mean divides the area in half, 0.50 on each
  side
• Nearly all the area is between z -3.00 and z
  3.00

• Notes
• Table 3, Appendix B lists the probabilities
  associated with the intervals from the mean (0)
  to a specific value of z
• Probabilities of other intervals are found using
  the table entries, addition, subtraction, and
  the properties above

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Table 3, Appendix B Entries

• The table contains the area under the standard
  normal curve between 0 and a specific value of z

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Example

• Example Find the area under the standard normal
  curve between z 0 and z 1.45

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Example

• Example Find the area under the normal curve to
  the right of z 1.45 P(z gt 1.45)

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Example

• Example Find the area to the left of z 1.45
  P(z lt 1.45)

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Notes

• The symmetry of the normal distribution is a key
  factor in determining probabilities associated
  with values below (to the left of) the mean. For
  example the area between the mean and z -1.37
  is exactly the same as the area between the mean
  and z 1.37.

• When finding normal distribution probabilities, a
  sketch is always helpful

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Example

• Example Find the area between the mean (z 0)
  and z -1.26

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Example

• Example Find the area to the left of -0.98 P(z
  lt -0.98)

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Example

• Example Find the area between z -2.30 and z
  1.80

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Example

• Example Find the area between z -1.40 and z
  -0.50

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Normal Distribution Note

• The normal distribution table may also be used to
  determine a z-score if we are given the area
  (working backwards)

• Example What is the z-score associated with the
  85th percentile?

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Solution

• In Table 3 Appendix B, find the area entry that
  is closest to 0.3500

. . .
. . .

• The area entry closest to 0.3500 is 0.3508
• The z-score that corresponds to this area is 1.04
• The 85th percentile in a standard normal
  distribution is 1.04

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Example

• Example What z-scores bound the middle 90 of a
  standard normal distribution?

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Solution

• The 90 is split into two equal parts by the
  mean. Find the area in Table 3 closest to 0.4500

• 0.4500 is exactly half way between 0.4495 and
  0.4505
• Therefore, z 1.645
• z -1.645 and z 1.645 bound the middle 90 of
  a normal distribution

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6.4 Applications of Normal Distributions

• Apply the techniques learned for the z
  distribution to all normal distributions

• Start with a probability question in terms
  ofx-values

• Convert, or transform, the question into an
  equivalent probability statement
  involvingz-values

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Standardization

• Suppose x is a normal random variable with mean m
  and standard deviation s

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Example

• Example A bottling machine is adjusted to fill
  bottles with a mean of 32.0 oz of soda and
  standard deviation of 0.02. Assume the amount
  of fill is normally distributed and a bottle is
  selected at random

1) Find the probability the bottle contains
between 32.00 oz and 32.025 oz 2) Find the
probability the bottle contains more than 31.97 oz
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Solution Continued
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Example, Part 2
2)
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Notes

• The normal table may be used to answer many kinds
  of questions involving a normal distribution

• Often we need to find a cutoff point a value of
  x such that there is a certain probability in a
  specified interval defined by x

• Example The waiting time x at a certain bank is
  approximately normally distributed with a mean
  of 3.7 minutes and a standard deviation of 1.4
  minutes. The bank would like to claim that 95
  of all customers are waited on by a teller
  within c minutes. Find the value of c that
  makes this statement true.

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Solution
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Example

• Example A radar unit is used to measure the
  speed of automobiles on an expressway during
  rush-hour traffic. The speeds of individual
  automobiles are normally distributed with a mean
  of 62 mph. Find the standard deviation of all
  speeds if 3 of the automobiles travel faster
  than 72 mph.

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Solution
-
m
x


z


s
.

1
88
10
s