Sample Mean, A Formula

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Sample Mean, A Formula
The Sample Mean. A Formula page 6
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Normal Distributions, Standard Normal
Normal Distributions. Standard Normal
1. STANDARD NORMAL(Mean 0 Standard deviation
1) In The Sample Mean we derived the probability
density function for the standard normal random
variable Z. We can use integration and fZ to
compute probabilities for Z. Example 1. Compute
P(?0.74 ? Z ? 1.29). As we saw in
Integration, To evaluate the integral open
Integrating.xls and enter the function
as (1/SQRT(2PI()))EXP(-0.5x2). Recall that x
is the only variable that can be used in
Integrating.xls.
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Normal, General Normal
Normal Distributions. General Normal
2. GENERAL NORMAL(Any Mean Any Standard
deviation/But its standardization is a standard
normal distribution) The adjective standard,
used in standard normal distributions, implies
that there are non-standard normal
distributions. This is indeed the case. A
random variable, X, is called normal if its
standardization, has a standard normal
distribution. It can be shown that the
probability density function for a normal random
variable, X, with mean ?X and standard deviation
?X has the following form.
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Normal Distributions-

• Standard Normal Random Variable (Z)
• p.d.f.
• Can use Integrating.xls to find probabilities

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Normal Distributions-

• Ex1. Find
• Soln show ex1 excel file

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Normal Distributions

• Ex. Find a number so that .
• Solnshow ex2 excel file

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Normal Distributions

• The previous example tells us that 97.5 of all
  data for a standard normal random variable lies
  in the interval .
• This means that 2.5 of the data lies above z
  1.96
• Graphically, we have the following

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Normal Distributions

• The shaded region corresponds to 97.5 of all
  possible area (note 2.5 is not shaded)
• 1.96

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Normal Distributions

• Due to symmetry, we get 95 of the area shaded
  with 5 not shaded (2.5 on each side)
• -1.96 1.96

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Normal Distributions

• This means that a 95 confidence interval for the
  standard normal random variable Z is (-1.96,
  1.96)
• -1.96 1.96

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Normal Distributions

• A 95 confidence interval tells you how well a
  particular value compares to known data or sample
  data
• The interval that is constructed tells you that
  there is a 95 probability that the interval will
  contain the mean of X.
• Another interpretation is that 95 of all values
  found in a sample should lie within this 95
  confidence interval.

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Normal Distributions

• Possible formulas
• Z Standard Normal random variable

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Normal Distributions

• Possible formulas

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Important

• Possible formulas

If is unknown The sample standard
deviation, , will be a very good
approximation for
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Normal Distributions

• Remember, that -1.96 and 1.96 were special values
  that apply to a 95 confidence interval
• You need to find different values for other types
  of confidence intervals.
• Ex. Find a 99 confidence interval for Z. Find a
  90 confidence interval for Z.

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Normal Distributions

• Soln
• Ex. What is the confidence interval for Z with 1
  standard deviation? 2 standard deviations? 3
  standard deviations?

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Normal Distributions

• Soln

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Normal Distributions

• Soln

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Normal Distributions

• Since Z is a standard normal random variable, Z
  would have standardized some variable X.
• So,

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Normal Distributions

• Ex. Suppose X is a normal random variable with
  and . Find a 95 confidence
  interval for X if the 95 confidence interval for
  Z is (-1.96, 1.96).

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Normal Distributions

• Soln
• So, the 95 confidence interval for X is (90.6,
  149.4). We are 95 confident that this interval
  contains the true mean

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Normal Distributions

• Ex. Suppose X is a normal random variable. If a
  sample of size 34 was taken with
  and , find a 95 confidence
  interval for the sample mean-remember this is
• if the 95 confidence interval for Z is (-1.96,
  1.96).

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Normal Distributions

• Soln

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Normal Distributions

• Soln
• So, the 95 confidence interval for is
  (114.9579, 125.0421).
• We are 95 confident that this interval contains
  the true mean

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Normal Distributions

• General Normal Random Variable
• p.d.f.
• Probabilities are done similarly to Standard NRV

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Normal Distributions

• Ex. If X is a normal random variable representing
  exam 1 scores with mean 75 and standard deviation
  10, find .
• Soln

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Normal Distributions

• NORMDIST function in Excel
• Can calculate p.d.f. and c.d.f. values for a
  normal random variable
• Ex. If X is a normal random variable representing
  exam 1 scores with mean 75 and standard deviation
  10, find .

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Normal Distributions(show excel)

• Soln

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Normal Distributions

• Specific values for the p.d.f. can also be
  calculated using NORMDIST
• Ex. Find height of p.d.f. of a normal random
  variable X at X 90 that has a normal
  distribution with and .

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Normal Distributions

• Soln Two ways to solve0.009
• (1) Evaluate
• (2) Evaluate NORMDIST(90, 71, 12, FALSE) using
  Excel

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Normal Distributions

• How does the mean and standard deviation affect
  the shape of the Normal Random Variable graph?
• Ex. Graph the p.d.f. of a normal random variable
  with the following characteristics
• (1) and
• (2) and
• (3) and
• (4) and

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Normal Distributions
Why? Recall-General Normal random variable p.d.f

• Soln (1) and
• Max Height 0.40
• The y-values of the graph
• around x -3 and x 3
• are very small

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Normal Distributions

• General Normal Random Variable
• p.d.f.

e(0)1
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Normal Distributions-sec1-4/13
Why?

• Soln (2) and
• Max Height 0.08
• (This is 0.40 std. dev.)
• Y values very small
• Around
• x -15 and
• x 15
• (This is 3 standard
• deviations from the mean- )

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Normal Distributions

• General Normal Random Variable
• p.d.f.

e(0)1
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Normal Distributions

• Soln (3) and
• Max Height 0.40
• (At x 4)
• Y values very small
• around x 1 and x 7
• (This is 3 standard
• deviations from
• the mean)

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Normal Distributions

• Soln (4) and
• Max Height 0.08
• (This is 0.40 std. dev.)
• Y values very small
• around x -11 and x 19
• (This is 3 standard
• deviations from the
• mean)

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Normal Distributions

• Ex. Find the mean and standard deviation for the
  following normal random variables graphed.
• (A)

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Normal Distributions

• Mean is 6 and standard deviation is 3

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Normal Distributions

• (B)

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Normal Distributions

• Mean is -7 and standard deviation is 2

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Normal Distributions

• (C)

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Normal Distributions

• Mean is 300 and standard deviation is 50

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Normal Distributions

• Relationship between p.d.f. and c.d.f.
• So,