Transcript Document

Lesson 7 - QR
Quiz Review
Objectives
• Review for the chapter 7 quiz on sections 7-1
through 7-3
Vocabulary
• Continuous random variable – has infinitely many
values
• Uniform probability distribution – probability
distribution where the probability of occurrence is
equally likely for any equal length intervals of the
random variable X
• Normal curve – bell shaped curve
• Normal distributed random variable – has a PDF or
relative frequency histogram shaped like a normal
curve
• Standard normal – normal PDF with mean of 0 and
standard deviation of 1 (a z statistic!!)
Continuous Uniform PDF
1
0.75
P(x=1) = 0
P(x ≤ 1) = 0.33
P(x ≤ 2) = 0.66
P(x ≤ 3) = 1.00
0.5
0.25
0
0
1
2
3
Since the area under curve must equal one. The height
or P(x) will always be equal to 1/(b-a), where b is the
upper limit and a the lower limit. Probabilities are just
the area of the appropriate rectangle.
Properties of the Normal Density Curve
• It is symmetric about its mean, μ
• Because mean = median = mode, the highest point
occurs at x = μ
• It has inflection points at μ – σ and μ + σ
• Area under the curve = 1
• Area under the curve to the right of μ equals the area
under the curve to the left of μ, which equals ½
• As x increases or decreases without bound (gets
farther away from μ), the graph approaches, but
never reaches the horizontal axis (like approaching
an asymptote)
• The Empirical Rule applies
Empirical Rule
μ ± 3σ
μ ± 2σ
μ±σ
99.7%
95%
68%
2.35%
34%
0.15%
μ - 3σ
13.5%
μ - 2σ
μ-σ
2.35%
34%
13.5%
μ
μ+σ
μ + 2σ
0.15%
μ + 3σ
Normal Curves
• Two normal curves with different means (but
the same standard deviation) [on left]
– The curves are shifted left and right
• Two normal curves with different standard
deviations (but the same mean) [on right]
– The curves are shifted up and down
Area under a Normal Curve
The area under the normal curve for any interval of
values of the random variable X represents either
• The proportion of the population with the
characteristic described by the interval of values or
• The probability that a randomly selected individual
from the population will have the characteristic
described by the interval of values
[the area under the curve is either a proportion or the
probability]
Standardizing a Normal Random
Variable
our Z statistic from before
X-μ
Z = ----------σ
where μ is the mean and σ is the standard deviation of
the random variable X
Z is normally distributed with mean of 0 and standard
deviation of 1
Z measures the number of standard deviations away
from the mean a value of X is
Normal Distributions on TI-83
• normalcdf cdf = Cumulative Distribution Function
This function returns the cumulative probability from zero
up to some input value of the random variable x.
Technically, it returns the percentage of area under a
continuous distribution curve from negative infinity to the
x. You can, however, set a different lower bound.
• Syntax: normalcdf (lower bound, upper bound,
mean, standard deviation)
(note: we use -E99 for negative infinity and E99 for
positive infinity)
Normal Distributions on TI-83
• invNorm
inv = Inverse Normal PDF
This function returns the x-value given the probability
region to the left of the x-value. (0 < area < 1 must be
true.) The inverse normal probability distribution
function will find the precise value at a given percent based
upon the mean and standard deviation.
• Syntax: invNorm (probability, mean, standard
deviation)
Obtaining Area under Standard Normal Curve
Approach
Graphically
Solution
Shade the area to the left of za
Use Table IV to find the row and
column that correspond to za. The
area is the value where the row
and column intersect.
Find the area to the
left of za
P(Z < a)
Normcdf(-E99,a,0,1)
a
Shade the area to the right of za
Find the area to the
right of za
Use Table IV to find the area to the
left of za. The area to the right of za
is 1 – area to the left of za.
Normcdf(a,E99,0,1) or
1 – Normcdf(-E99,a,0,1)
P(Z > a) or
1 – P(Z < a)
a
Shade the area between za and zb
Find the area
between za and zb
Use Table IV to find the area to the
left of za and to the left of za. The
area between is areazb – areaza.
Normcdf(a,b,0,1)
P(a < Z < b)
a
b
Problems
• Standard Normal Random Variable
P(Z < 1.96) = 0.975
normalcdf(-E99,1.96)
P(Z > 0.57) = 0.284
normalcdf(0.57,E99)
P(-2.71 < Z < 1.09) = 0.859
normalcdf(-2.71,1.09)
• Regular Normal Random Variable
P(x<4) = 0.965
normalcdf(-E99,4,2,1.1)
with μ=2 σ=1.1
P(x>16) = 0.965
normalcdf(16,E99,10,3.84)
with μ=10 σ=3.84
Problems
• Standard Normal Random Variable
What is the Z value associate with 91st percentile?
Z = invNorm(0.91) =
• Regular Normal Random Variable
What is the X value associated with 57% to the right with
μ = 11 and σ = 3?
X = invNorm(1-0.57,11,3) = 10.47
invNorm uses area to the left!