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Normal Distribution

- Perfectly symmetric and bell-shaped
- Characterized by two parameters
- Mean = μ
- Standard Deviation = σ

- Standard Normal
- μ = 0
- σ = 1
- Solid Line

STA 291 Fall 2009 Lecture 10

Examples

- For a normally distributed random variable, find the following
- P(Z>.82) =
- P(-.2<Z<2.18) =

STA 291 Fall 2009 Lecture 10

Finding z-Values for Percentiles

- For a normal distribution, how many standard deviations from the mean is the 90th percentile?
- What is the value of z such that 0.90 probability is less than z?
- P(Z<z) = .90

- If 0.9 probability is less than z, then there is 0.4 probability between 0 and z
- Because there is 0.5 probability less than 0
- This is because the entire curve has an area under it of 1, thus the area under half the curve is 0.5

- Because there is 0.5 probability less than 0
- z=1.28
- The 90th percentile of a normal distribution is 1.28 standard deviations above the mean

- What is the value of z such that 0.90 probability is less than z?

STA 291 Fall 2009 Lecture 10

Working backwards

- We can also use the table to find z-values for given probabilities
- Find the following
- P(Z>a) = .7224
- a =

- P(Z<b) = .2090
- b =

- P(Z>a) = .7224

STA 291 Fall 2009 Lecture 10

Standard Normal Distribution

- When values from an arbitrary normal distribution are converted to z-scores, then they have a standard normal distribution
- The conversion is done by subtracting the mean μ, and then dividing by the standard deviation σ

STA 291 Fall 2009 Lecture 10

z-Scores

- The z-score for a value x of a random variable is the number of standard deviations that x is above μ
- If x is below μ, then the z-score is negative

- The z-score is used to compare values from different normal distributions
- Calculating
- Need to know
- x
- μ
- σ

- Need to know

STA 291 Fall 2009 Lecture 10

Example

- SAT Scores
- μ=500
- σ=100
- SAT score 700 has a z-score of z=2
- Probability that a score is above 700 is the tail probability of z=2
- Table 3 provides a probability of 0.4772 between mean=500 and 700
- z=2

- Right-tail probability for a score of 700 equals 0.5-0.4772=0.0228
- 2.28% of the SAT scores are above 700

- Now find the probability of having a score below 450

STA 291 Fall 2009 Lecture 10

z-Scores

- The z-score is used to compare values from different normal distributions
- SAT
- μ=500
- σ=100

- ACT
- μ=18
- σ=6

- What is better, 650 on the SAT or 25 on the ACT?
- Corresponding tail probabilities?
- How many percent have worse SAT or ACT scores?
- In other words, 650 and 25 correspond to what percentiles?

- How many percent have worse SAT or ACT scores?

- Corresponding tail probabilities?

- SAT

STA 291 Fall 2009 Lecture 10

Example

- The scores on the Psychomotor Development Index (PDI) are approximately normally distributed with mean 100 and standard deviation 15. An infant is selected at random.
- Find the probability that the infant’s PDI score is at least 100
- P(X>100)

- Find the probability that PDI is between 97 and 103
- P(97<X<103)

- Find the z-score for a PDI value of 90
- Would you be surprised to observe a value of 90?

- Find the probability that the infant’s PDI score is at least 100

STA 291 Fall 2009 Lecture 10