STA 291 Fall 2009
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STA 291 Fall 2009 Lecture 10 Dustin Lueker
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STA 291 Fall 2009. Lecture 10 Dustin Lueker. Normal Distribution. Perfectly symmetric and bell-shaped Characterized by two parameters Mean = μ Standard Deviation = σ Standard Normal μ = 0 σ = 1 Solid Line. Examples. For a normally distributed random variable, find the following - PowerPoint PPT Presentation
Transcript of STA 291 Fall 2009
STA 291Fall 2009
Lecture 10Dustin Lueker
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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
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Examples For a normally distributed random variable,
find the following◦ P(Z>.82) =
◦ P(-.2<Z<2.18) =
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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 z=1.28
The 90th percentile of a normal distribution is 1.28 standard deviations above the mean
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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 =
STA 291 Fall 2009 Lecture 10
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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
xz
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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
xz
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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
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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?
650 5001.5
10025 18
1.176
SAT
ACT
xz
xz
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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?
