Transcript 2.1.2 - GEOCITIES.ws

Normal Distributions
Section 2.1.2
Starter
•
A density curve starts at the origin and
follows the line y = 2x. At some point on
the line where x = p, the curve drops
vertically to return to the x axis.
1. Draw the curve.
2. What is the value of p?
3. What is the median of the curve?
– Show it as a vertical line on the curve.
Answer
If x = p, then y = 2p
Since area = 1 in a density curve:
1
1  ( p )(2 p )
2
1  p2
p 1
If M is the x value of the median, then 2M is the height of the triangle.
Since one half the area is to the left of the median, then triangle area is one half.
1
.5  ( M )(2 M )
2
.5  M 2
M  .5 
1
2

 .707
2
2
Today’s Objectives
• Draw a normal curve and show μ, µ±σ,
µ±2σ, µ±3σ on the graph
• Use the Empirical Rule (a.k.a. 68-95 Rule)
to answer questions about percents and
percentiles
Normal Curves
• Draw a bell-shaped curve above an x axis
• Draw the vertical line of symmetry
– Label the x axis “μ” at this point
• Show the two “inflection points” on the curve
• The left inflection point is where the curve stops getting more
steep and starts getting less steep
• The other is symmetric to it about the mean line
– Label the x axis “µ+σ” and “µ-σ” below the points
• Using the same scale, label the x axis with µ
plus and minus 2σ and 3σ
Normal Curves
• Normal curves are special case density
curves
– The area under the curve is 1
• This is true of ALL density curves
– The curve is symmetric and “bell-shaped”
• So mean = median
• We normally speak of the mean rather than median
– The inflection points of the curve are one
standard deviation (σ) above and below the
mean
The Empirical Rule
or: 68-95 Rule
• In a normal distribution with mean μ and
standard deviation σ:
– This is called the N(μ, σ) distribution
– About 68% of the observations fall within σ of μ
– About 95% fall within 2σ of μ
– About 99.7% fall within 3σ of μ
Example
• Suppose the heights of American men are
known to be N(69 in, 2.5 in)
– Draw the normal curve and label the axis
– What percent of men are between 69 inches
and 71.5 inches tall?
• Since 68% are between 66.5 and 71.5, and the
graph is symmetric, there are 34% between 69 and
71.5 inches tall.
Example Continued
• What percent of men are taller than 74 in?
– Since 95% of observations fall within ±2σ of μ,
then 5% fall outside those borders.
– By symmetry, 2.5% fall more than 2σ above
the mean.
• In this case, that is 69 + 2 x 2.5 = 74 in
• So 2.5% of men are taller than 74 in
Example Concluded
• In what percentile is a man who is 71.5 in
tall?
• Recall that “percentile” means the percent of
observations equal to or less than the specified
value
– By definition, 50% fall below 69 inches
– 71.5 inches is one σ above the mean, so 34%
must fall between 69 and 71.5
– Thus 71.5 inches is the 84th percentile
• 50 + 34 = 84
Exploring Normal Data
• 50 Fathoms Demo 3
– What Do Normal Data Look Like?
Today’s Objectives
• Draw a normal curve and show μ, µ±σ,
µ±2σ, µ±3σ on the graph
• Use the Empirical Rule (a.k.a. 68-95 Rule)
to answer questions about percents and
percentiles
Homework
• Read pages 73 – 77
• Do problems 6 – 9