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Chapter 7: The Normal Probability Distribution

Section 7.1: Properties of the Normal Distribution

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. Section 1.1:
      • continuous quantitative data
    2. Section 3.1:
      • mean
      • median
    3. Section 3.2:
      • standard deviation
      • variance
      • Empirical Rule (a.k.a., 68-95-99.7 Rule)
    4. Section 3.4:
      • z-score
      • percentiles
    5. Section 5.1:
      • probability
      • outcome
      • random event
      • probability rules
      • unusual event
    6. Section 5.2:
      • addition rule for probability
      • complement of an event, Ec
      • complement rule for probabilities
      • keywords for probability:
        1. or
        2. and [sometimes you will need to identify and even though word is not used]
        3. not
  2. procedural knowledge
    1. Section 5.1:
      • verify probability models
    2. Section 5.2:
      • how to calculate the probability of an event using the addition rule
  3. conditional knowledge
    1. Section 1.1:
      • how to identify the difference between qualitative data and quantitative data
      • how to identify the difference between discrete quantitative data and continuous quantitative data
    2. Section 5.1:
      • know when it is appropriate to use the concept of probability
      • know the importance of the concept of randomness or chance in probability
      • interpret value of probability
      • identify when an event is considered unusual
    3. Section 5.2:
      • how to determine which probability rule to use for a given problem
      • how to determine if two events are disjoint
      • how to identify the complement of an event
      • know that probability, proportion, and percentage are equivalent

Learning Goals

  1. declarative knowledge (definitions)
    1. continuous probability distribution
    2. uniform probability distribution
    3. probability density functions and its properties
    4. normal curve (a.k.a., normal probability distribution or bell-curve)
    5. normal density curve and its properties
    6. standard normal curve
  2. procedural knowledge
    1. calculate probabilities using a uniform distribution
    2. identify μ and σ from the graph of a normal curve
    3. calculate probabilities based on normal curve using the empirical rule
    4. sketch a normal curve and shade appropriate areas using the TI83/84: http://stats.jjw3.com/math1431/ti83normArea.htm
  3. conditional knowledge
    1. determine if a curve is a probability density function
    2. determine if data can be modeled by a normal distribution
    3. how the inflection points of the normal curve relate to the mean and standard deviation
    4. how the standard deviation affects the shape of a normal curve
    5. how the area under the normal probability density function relates to probability, proportion, and percentage
    6. how to identify which inequality ≤ [equivalent to <] or ≥ [equivalent to >] is needed to calculate the probability of a continuous variable

Section 7.2: Applications of the Normal Distribution

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. Section 1.1:
      • continuous quantitative data
    2. Section 3.1:
      • mean
      • median
    3. Section 3.2:
      • standard deviation
      • variance
      • Empirical Rule (a.k.a., 68-95-99.7 Rule)
    4. Section 3.4:
      • z-score
      • percentiles
    5. Section 5.1:
      • probability
      • outcome
      • random event
      • probability rules
      • unusual event
    6. Section 5.2:
      • addition rule for probability
      • complement of an event, Ec
      • complement rule for probabilities
      • keywords for probability:
        1. or
        2. and [sometimes you will need to identify and even though word is not used]
        3. not
    7. Section 7.1:
      • all
  2. procedural knowledge
    1. Section 5.1:
      • verify probability models
    2. Section 5.2:
      • how to calculate the probability of an event using the addition rule
    3. Section 7.1:
      • all
  3. conditional knowledge
    1. Section 1.1:
      • how to identify the difference between qualitative data and quantitative data
      • how to identify the difference between discrete quantitative data and continuous quantitative data
    2. Section 5.1:
      • know when it is appropriate to use the concept of probability
      • know the importance of the concept of randomness or chance in probability
      • interpret value of probability
      • identify when an event is considered unusual
    3. Section 5.2:
      • how to determine which probability rule to use for a given problem
      • how to determine if two events are disjoint
      • how to identify the complement of an event
      • know that probability, proportion, and percentage are equivalent
    4. Section 7.1:
      • all

Learning Goals

  1. declarative knowledge (definitions)
    1. standard normal curve and its properties
    2. mean and standard deviation of z
    3. α
    4. zα
    5. percentile
  2. procedural knowledge
    1. how to calculate the area under the normal curve using the TI83/84: http://stats.jjw3.com/math1431/ti83norm.htm
    2. how to calculate the area under the standard normal curve using the TI83/84: http://stats.jjw3.com/math1431/ti83norm.htm
    3. how to calculate the percentile rank of a value of a normal distribution using the TI83/84: http://stats.jjw3.com/math1431/ti83norm.htm
    4. sketch a normal curve and shade appropriate areas using the TI83/84: http://stats.jjw3.com/math1431/ti83normArea.htm
    5. given the area under the normal curve find the z-score using the TI83/84: http://stats.jjw3.com/math1431/ti83invNorm.htm
    6. given the percentile (i.e., area to left) under the normal curve find the z-score using the TI83/84: http://stats.jjw3.com/math1431/ti83invNorm.htm
    7. sketch a standard normal curve and shade appropriate areas using the TI83/84: http://stats.jjw3.com/math1431/ti83normArea.htm
    8. calculate zα using the TI83/84
    9. find z-scores that separate the middle p% of normal distribution from the area in the tails
  3. conditional knowledge
    1. purpose of the standard normal curve
    2. relationship between α and percentile
  4. important notes
    1. you are not expected to use the z distribution table, you are expected to use the TI83/84

Section 7.3: Assessing Normality

Knowledge Prerequisites

  1. declarative knowledge (definitions)
    1. Section 7.1:
      • all
    2. Section 7.2:
      • all
  2. procedural knowledge
    1. Section 7.1:
      • all
    2. Section 7.2:
      • all
  3. conditional knowledge
    1. Section 7.1:
      • all
    2. Section 7.2:
      • all

Learning Goals

  1. declarative knowledge (definitions)
    1. normal probability plot (a.k.a., Q-Q plot)
  2. procedural knowledge
    1. how to construct a normal probability plot using TI83/84
  3. conditional knowledge
    1. how to use the normal probability plot to determine if a distribution is normal

Chapter 7: Required Formulas – Need to Know for Tests

  1. Population z-Score: Population z-Score