Introduction to statistical concepts Part III

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Introduction to statistical concepts(Part III)

• Dr. Yan Liu
• Department of Biomedical, Industrial and Human
  Factors Engineering
• Wright State University

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Normal Distribution

• The normal distribution (also known as Gaussian
  distribution) with parameters µ and s, N(µ,s), is
  the continuous probability distribution with the
  following probability density function
• Normal distribution is the cornerstone of the
  field of statistical inference
• Many distributions can be well approximated by
  the normal distribution
• e.g. weight, height, bolt diameter, resistance
  of wire, construction error, etc.

(Eq. 1)
Where p 3.14159, e 2.71828
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Normal Distribution

• A bell curve, symmetric about X µ
• Shape of the curve is determined by s
• the larger s is, the flatter the curve is

s2 gt s1
Graph of the Probability Dense Function of Normal
Distribution
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Standard Normal Distribution

• The standard normal distribution is the normal
  distribution with parameters µ0 and s1, N(0,1),
  and has the following probability density
  function
• For ZN(0,1), P(Zltz) is equal to the area of the
  region to the left of the ordinate Z z that is
  bounded by the graph of its density function and
  the horizontal axis
• Why N(0,1)?
• We can use the table of standard normal
  distribution to calculate probabilities (can be
  downloaded from the course website)

(Eq. 2)
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What is the total area under curve?
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P(Zlt-z) P(Zgtz), for zgt0
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Z Score

• The z score of a random variable X N(µ,s) is
  computed as

(Eq. 3)
X N(µ20, s2), P(X gt23)?
ZX (X-20)/2
P(X gt23) P(ZX gt (23-20)/2) P(ZX gt1.5) 1-
P(ZX lt 1.5) 1- 0.9332
0.0668 0.07
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Suppose the resistance of a carbon composition
resistor is normally distributed with mean µ
1,000 and variance s2 900. What is the
probability that the resistor has measurement in
excess of 1,060 ohms?
Define X
X the resistance of a carbon composition
X N(µ1000, s30), P(X gt1060)?
ZX (X-1000)/30
P(X gt1060) P(ZX gt (1060-1000)/30) P(ZX gt2)
1- P(ZX lt 2) 1- 0.9772
0.0228 0.02
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Students t-Distribution

• Students t-distribution or t-distribution is
  used to replace normal distribution when the
  standard deviation s is not known and thus has to
  be estimated using sample standard deviation s

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Students t-Distribution

• Suppose a random sample of size n is drawn from a
  normal distribution with mean µ and standard
  deviation s (unknown), then

(Eq. 4)
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Confidence Interval

• A confidence interval gives an estimated range of
  values which is likely to include an unknown
  population parameter it is calculated from a
  given set of sample data
• Confidence Level
• How sure the confidence interval includes the
  true value of the population parameter to be
  estimated

We are 95 confident that between 60 and 70
people will agree with this proposal.
95 is our confidence level (60, 70) is our
confidence interval
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Confidence Interval for Mean

• Suppose a random sample of size n is drawn from a
  normal distribution with mean µ and standard
  deviation s (unknown), then the confidence
  interval for µ is

(Eq. 6)
where a is significance level and equal to (1
confidence level)

• t-distribution table can be downloaded from the
  course website

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Suppose these grades are sampled from a normal
distribution with mean µ and standard deviation s
(unknown), find the confidence interval of the
mean of the grades with 95 confidence level
a 0.05, n10
ta/2 (10-1) ta/2 (9) 2.262
The confidence interval for µ is (86.2-2.2625.75
/v10, 86.22.2625.75/v10) (82.09, 91.31)
X grades of student
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Exponential Distribution

• Often used to model the time interval between
  independent events that happen at a constant
  average rate
• The probability density function of an
  exponential function has the form

(Eq. 7)
? gt 0 is a parameter of the distribution, often
called the rate parameter

• The cumulative distribution is given by

(Eq. 8)

• Mean and Variance

(Eq. 10)
(Eq. 9)
var(X) 1/?2
E(X) 1/?
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fX(x)
X
Probability Dense Function of Exponential
Distribution
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Suppose the life in hours of a certain type of
tube is a random variable that has an exponential
distribution with a mean of 1,000 hours. What is
its probability density function? What is the
probability that such a tube will last at least
1,250 hours?
X life in hours of the tube
E(X) 1000 1/? ? ? 1/1000
f(x) 1/1000 e(-1/1000)x
P(X 1250) 1- P(X lt1250) 1- F(X1250) 1- (1-
e(-1/1000)1250 ) e(-1/1000)1250
0.287