§ 5.1 introduction to normal distributions and the standard distribution
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§ 5.1Introduction to Normal Distributions
and the Standard Distribution
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Properties of Normal Distributions
A continuous random variable has an infinite number of possible values that can be represented by an interval on the number line.
Hours spent studying in a day
0 63 9 1512 18 2421
The time spent studying can be any number between 0 and 24.
The probability distribution of a continuous random variable is called a continuous probability distribution.
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Density Functions•A Density Function is a function that describes the relative likelihood for this random variable to have a given value. For a given x-value, the probability of x, P(x), is called the Density, or Probability Density. •Density Functions can take any shape. •The Sum of all Probabilities, P(x), of a Probability Density Function, that is, Densities, is always 1.•The Sum of probabilities in an interval is the area under the Density Function of the interval.
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Density Function•Consider the function f(x) below defined on the interval [-1, 1]. Which of the following statements is true?
a) f(x) could not be a continuous probability density functionb) P( c) P(d) P( e) P(
-1 1
1
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Uniform Distribution• Let that defines this uniform distribution.
𝑓 (𝑥 )={ 125 0<𝑥<25
0 h𝑜𝑡 𝑒𝑟𝑤𝑖𝑠𝑒}a. Using your uniform distribution, sketch a graph of the probability function.
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Complete 5.1 HandoutUniform Density Function
Practice
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Properties of Normal Distributions
The most important probability distribution in statistics is the normal distribution.
A normal distribution is a continuous probability distribution for a random variable, x. The graph of a normal distribution is called the normal curve. Notation Used is
Normal curve
x
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Means and Standard Deviations
A normal distribution can have any mean and any positive standard deviation.
Mean: μ = 3.5Standard deviation: σ 1.3
Mean: μ = 6Standard deviation: σ 1.9
The mean gives the location of the line of symmetry.
The standard deviation describes the spread of the data.
Inflection points
Inflection points
3 61 542x
3 61 542 97 11108x
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Means and Standard Deviations
Example:1. Which curve has the greater mean?2. Which curve has the greater standard
deviation?
The line of symmetry of curve A occurs at x = 5. The line of symmetry of curve B occurs at x = 9. Curve B has the greater mean.
Curve B is more spread out than curve A, so curve B has the greater standard deviation.
31 5 97 11 13
AB
x
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Interpreting Graphs
Example:The heights of fully grown magnolia bushes are normally distributed. The curve represents the distribution. What is the mean height of a fully grown magnolia bush? Estimate the standard deviation.
The heights of the magnolia bushes are normally distributed with a mean height of about 8 feet and a standard deviation of about 0.7 feet.
μ = 8
The inflection points are one standard deviation away from the mean.
σ 0.7
6 87 9 10Height (in feet)
x
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Complete 5.1 HandoutNormal Distribution
Practice