Introduction to Normal Distributions and the Standard Distribution § 5.1 Introduction to Normal Distributions and the Standard Distribution

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 6 3 9 15 12 18 24 21 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.

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.

Density Function Consider the function f(x) below defined on the interval [-1, 1]. Which of the following statements is true? 1 -1 1 f(x) could not be a continuous probability density function P(−1≤𝑥≤.5)= 2 3 c) P(−1≤𝑥≤.5)= 7 8 d) P(−1≤𝑥≤.5)= 11 12 e) P(−1≤𝑥≤.5)= 3 4

Uniform Distribution Let 𝑋~𝑈𝑛𝑖𝑓𝑜𝑟𝑚 0,25 , state the piecewise function that defines this uniform distribution. 𝑓 𝑥 = 1 25 0<𝑥<25 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 a. Using your uniform distribution, sketch a graph of the probability function.

Complete 5.1 Handout Uniform Density Function Practice

Properties of Normal Distributions The most important probability distribution in statistics is the normal distribution. x Normal curve 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 𝑵~(𝝁,𝝈)

Means and Standard Deviations A normal distribution can have any mean and any positive standard deviation. Inflection points The mean gives the location of the line of symmetry. Inflection points 3 6 1 5 4 2 x 3 6 1 5 4 2 9 7 11 10 8 x Mean: μ = 3.5 Standard deviation: σ  1.3 Mean: μ = 6 Standard deviation: σ  1.9 The standard deviation describes the spread of the data.

Means and Standard Deviations Example: Which curve has the greater mean? Which curve has the greater standard deviation? 3 1 5 9 7 11 13 A B x 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.

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 inflection points are one standard deviation away from the mean. 6 8 7 9 10 Height (in feet) x μ = 8 σ  0.7 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.

Complete 5.1 Handout Normal Distribution Practice