THE NORMAL CURVE

1
THE NORMAL CURVE
2
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 m s to represent the mean standard
  deviation

3
z score

• Standardized score
• Creates the standard normal density curve
• Has m 0 s 1

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What do these z scores mean?

• -2.3
• 1.8
• 6.1
• -4.3

2.3 s below the mean
1.8 s above the mean
6.1 s above the mean
4.3 s below the mean
5
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
m 45 and s 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
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Chebyshevs Rule
At least what percent of observations is within 2
standard deviations of the mean for any shape
distribution?

• The percentage of observations that are within k
  standard deviations of the mean is at least
• where k gt 1
• can be used with any distribution

75
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Chebyshevs Rule- what to know

• Can be used with any shape distribution
• Gives an At least . . . estimate
• For 2 standard deviations at least 75

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Normal Curve

• Bell-shaped, symmetrical curve
• Transition points between cupping upward
  downward occur at m s and m s
• As the standard deviation increases, the curve
  flattens spreads
• As the standard deviation decreases, the curve
  gets taller thinner

9
Empirical Rule

• Approximately 68 of the observations are within
  1s of m
• Approximately 95 of the observations are within
  2s of m
• Approximately 99.7 of the observations are
  within 3s of m
• See p. 181

Can ONLY be used with normal curves!
10
The height of male students at PWSH 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? b) Taller than 73.5 inches? c)
Between 66 73.5 inches?
About 2.5
About 16
About 81.5
11
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 3.5 0.7
Assembly 21.8 2.4
Tuning 12.3 2.7
2.5