Download stat226_2-2-16 - Iowa State University

Session packet
Statistics 226
Supplemental Instruction
Iowa State University
Leader:
Course:
Instructor:
Date:
Luyun
Stat 226
Anna Peterson
2/2/16
RECAP:
The standard normal distribution
 Is a “special” normal distribution.
 Has a mean ________ and a variance ________.
 Is donated by ________________.
 Nearly all the area is between ____ and _____.
Having knowledge of the mean μ and the standard deviation σ of a normal distribution allows us
to determine
 What proportion of individuals we can expect to fall within a specific range.
 What percentile a given individual falls at if you know their data value.
 What data value corresponding to a given percentile. (backward calculation)
For the standard normal distribution, the proportion of observations falling into a specified range
is tabulated.
 This is the only normal distribution for which we have readily available tabulated values.
 We therefore need to transform any normal distribution with mean μ and variance to a
standard distribution, i.e. the values from any _________ are transformed to the
corresponding values from a _________.
 This is called s____________.
Standardizing, z-score
If x is an observation from a normally distributed random variable X, i.e. X follows a normal
distribution with mean μ and standard deviation σ, then the standardized value Z is given by
Z=
A standardized value is often called a z-score.
 A z-score tells us how many standard deviations the original observation is off the mean
and in which direction.
 Observations _______ than the mean are positive when standardized, and
observations________ than the mean are negative when standardized.
1. Let x denote an observation from a normal distribution with a mean of 50 and a standard
deviation of 10. For any given x-values, find the corresponding z-score. For any given zscore, find the corresponding x-value from the distribution. Interpret each z-score/xvalue.
a. What is the normal distribution notation for this distribution?
b. X=33
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c. X=8
d. X=49
e. X=63
f. X=72
g. Z=3.4
h. Z=1.2
i. Z=0
j. Z=-2.7
k. Z=-0.4
2. Former ISU basketball player Kelvin Cato is 83 inches tall. Assuming that heights follow
approximately a normal distribution with mean 70 and standard deviation σ = 3，
a. What is his corresponding z-score?
b. What proportion of men are taller than him?
3. For each problem below draw a picture of the normal curve and shade the area you have
to find. Let Z represent a variable following a standard normal distribution.
a. Find the proportion that is less than z=2.00
b. Find the proportion that is between z = -0.13 and z = 1.75
c. Find the proportion that is greater than z = 1.86
d. Find the z-score for the 64th percentile.
e. Find the z-score for the 24th percentile.