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Density Curves
•
•
•
•
Can be created by smoothing histograms
ALWAYS on or above the horizontal axis
Has an area of exactly one underneath it
Describes the proportion of observations that fall
within a range of values
• Is often a description of the overall distribution
• Uses μ & σ to represent the mean & standard
deviation
z score
• Standardized score
• Creates the standard normal density curve
• Has μ = 0 & σ = 1
z
x

What do these z scores mean?
-2.3
1.8
6.1
-4.3
2.3 σ below the mean
1.8 σ above the mean
6.1 σ above the mean
4.3 σ below the mean
Jonathan wants to work at Utopia
Landfill. He must take a test to see if
he is qualified for the job. The test
has a normal distribution with μ = 45
and σ = 3.6. In order to qualify for the
job, a person can not score lower than
2.5 standard deviations below the
mean. Jonathan scores 35 on this test.
Does he get the job?
No, he scored 2.78 SD below the mean
At least what percent of
Chebyshev’s
Rule
observations
is within 2
standard deviations of the
meanpercentage
for any shape of observations
• The
distribution? 75%
that are within k standard
deviations of the mean is at least

100 1  1

%
k
2
where k > 1
• can be used with any distribution
Chebyshev’s Rule- what to know
• Can be used with any shape
distribution
• Gives an “At least . . .” estimate
• For 2 standard deviations – at least
75%
Normal Curve
• Bell-shaped, symmetrical curve
• Transition points between cupping
upward & downward occur at μ + σ
and μ – σ
• As the standard deviation increases,
the curve flattens & spreads
• As the standard deviation decreases,
the curve gets taller & thinner
Empirical Rule
• Approximately 68% of the
Can ONLY be used
observations are within 1σ of μ
with normal curves!
• Approximately 95% of the
observations are within 2σ of μ
• Approximately 99.7% of the
observations are within 3σ of μ
The height of male students at WHS
is approximately normally
distributed with a mean of 71 inches
and standard deviation of 2.5 inches.
a) What percent of the male students
are shorter than 66 inches? About 2.5%
b) Taller than 73.5 inches? About 16%
c) Between 66 & 73.5 inches?About 81.5%
Remember the bicycle problem? Assume that the
phases are independent and are normal
distributions. What percent of the total setup
times will be more than 44.96 minutes?
First, find the mean
& standard
deviation for the
total setup time.
Phase
Mean
SD
Unpacking
Assembly
Tuning
3.5
21.8
12.3
0.7
2.4
2.7
2.5%