Probability and the Normal Distribution

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Probability and the Normal Distribution
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Why is probability important?

• Allows us to make inferences about populations.

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Random Sampling

• Two Requirements of Random Sampling
• Each individual in a population has an equal
  chance of being selected
• We use Sampling with Replacement each sample is
  replaced before the next selection is made.

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Probability and Frequency Distributions
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Probability and Frequency Tables

• Whats the probability of scoring an 8 on the
  quiz?
• p f/N 3/30 .10

• Whats the probability of scoring an 8 or higher
  on the quiz?
• p f/N 6/30 .20

• Whats the probability of scoring at least a 5 on
  the quiz?
• p f/N 20/30 .67

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The Standard Normal Curve and Probability
Assessments
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Uses for the Standard Normal Curve
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Assumptions of Standard Normal Curve Problems

• 1. The variable is normally distributed
• 2. The mean and standard deviation for the
  population is known.
• The SNC does not apply if these two assumptions
  are not met.

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Finding Probabilities Associated with Z-scores

• What proportion of the distribution falls above z
  1.5?

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Finding Probabilities Associated with Z-scores

• What percentage of the distribution falls below z
  -.50?

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Finding Probabilities Associated with Z-scores

• What percentage of the distribution falls between
  the mean and z -.50?

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Finding Probabilities Associated with Z-scores

• What percentage of the distribution falls between
  z 1.00 and z 1.50?

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Finding Probabilities Associated with Z-scores

• What percentage of the distribution falls between
  z 1.00 and z 1.50?

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Finding Probabilities Associated with Z-scores

• What percentage of students score below 70 on the
  test?

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Finding Probabilities Associated with Raw Scores

• What percentage of students score below 70 on the
  test?

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Finding Probabilities Associated with Raw Scores

• What proportion of students score between 60 and
  80 on the test (µ 60 and s 10)?

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Finding Probabilities Associated with Raw Scores

• What z-score separates the upper 10 from the
  lower 90 of a distribution?

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Finding Probabilities Associated with Raw Scores

• What z-score separates the lower 25 from the
  upper 75 of a distribution?

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Finding Probabilities Associated with Raw Scores

• What z-score separates the lower 25 from the
  upper 75 of a distribution?

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Determining Raw Scores Given a Probability

• In a normal distribution with µ 50 and s 10,
  what raw score separates the lowest 25 from
  upper 75 of the distribution?
• X µ z s
• X 50 (-.67)(10) 43.30

Z-score value found in Unit Normal Table (see
previous problem)
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Determining Raw Scores Given a Percentile

• Students in the 95th percentile or higher in your
  bio class get an A. Whats the lowest score that
  you could earn to get an A if µ 60 and s 10
  in this class?

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Determining Raw Scores Given a Percentile

• For the 95th percentile z 1.64
• X µ z s
• X 60 (1.64)(10) 76.40
• You need to earn at least 76.40 to get an A in
  the course.