Chapter 7: The Normal Probability Distribution

Normal curve

A ____________ is used to describe continuous random variables that are said to be normally distributed.

Uniform Probability Distribution

If any two intervals of equal length are equally likely, a random variable, X, is said to follow _________________.

Normal approximation to the binomial distribution

If np(1-p)>=10, the binomial random variable, X, is approximately normally distributed.

Standard normal distribution

Is a special normal distribution. It has a mean μ = 0 and standard deviation σ = 1. A standard normal variable is always denoted by the variable Z so it's often referred to as the "Z distribution." Z is only use to denote a standard normal variable. Z sub alpha denotes the Z-score having area, α specifically to its right under the standard normal curve. (α is usually a very small amount of probability).

Standard normal random variable, Z

__________________ normal random variable whose mean is 0 and standard deviation is 1.

Example Interpreting the Area Under a Normal Curve

The probability that X is less than 2100 (the purple area to the left of the curve) is .3085

Use Normal Probability Plots to Assess Normality

The idea behind finding the expected z-score is that, if the data comes from normally distributed population, we could predict the area to the left of each of the data value. The value of fi represents the expected area left of the ith observation when the data come from a population that is normally distributed. For example, f1 is the expected area to the left of the smallest data value, f2 is the expected area to the left of the second smallest data value, and so on. If sample data is taken from a population that is normally distributed, a normal probability plot of the actual values versus the expected Z-scores will be approximately linear.

Correction for continuity

Add or subtract 0.5 from a normal random variable because we are using a continuous density function to approximate a discrete probability.

Drawing a Normal Probability Plot

Step 1 & 2: see image: where i is the index (the position of the data value in the ordered list) and n is the number of observations. The expected proportion of observations less than or equal to the ith data value is f. Step 3 Find the z-score corresponding to fi from Table V. Step 4 Plot the observed values on the horizontal axis and the corresponding expected z-scores on the vertical axis.

Area under a Normal Curve

Suppose that a random variable X is normally distributed with mean μ and standard deviation σ. 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.

Normal score

The expected z-score of the data value, assuming that the distribution of the random variable is normal. A normal probability plot is a graph that plots observed data versus normal scores. A _______________ is the expected​ z-score of a data​ value, assuming that the distribution of the random variable is normal.

Properties of the Normal Curve

1. The normal curve is symmetric about its mean, μ. 2. Because the mean = median = mode, the normal curve has a single peak and the highest point occurs at x = μ. 3. The normal curve has inflection points at μ - σ & μ + σ. 4. The area of the normal curve is 1. 5. The area under the normal curve to the right of μ equals the area under the left of μ, which equal ½. 6. As x increases without bound (gets larger and larger), the graph approaches, but never reaches, the horizontal axis. As x decreases without bound (gets more and more negative), the graph approaches but never reaches, the horizontal axis. 7. The Empirical Rule: Approximately 68% of the area under the normal curve is between x = μ − σ and x = μ + σ; approximately 95% of the area is between x = μ − 2σ and x = μ + 2σ; approximately 99.7% of the area is between x = μ − 3σ and x = μ + 3σ.

Normal probability plot

A __________________________ is a graph that plots observed data versus normal scores. Used to assess whether a random variable may come from a population that is normally distributed. A method for assessing normality of a random sample X in a small data set. A ____________________ is a graph that plots observed data versus normal scores. Versus a normal score which is the expected​ z-score of the data​ value, assuming that the distribution of the random variable is normal. The expected​ z-score of an observed value depends on the number of observations in the data set. Random variable X is normally distributed provided the histogram of the data is symmetric and bell shaped. This works for large data sets. A histogram drawn from a small sample of observations does not always accurately represent the shape of the population.

Normally distributed, normal probability distribution

A continuous random variable is __________________, or has a _______________________ if its relative frequency histogram has the shape of a normal curve.

Normal probability distribution

A continuous random variable whose relative frequency histogram has the shape of a normal curve.

Inflection points

Points where the curvature of the graph changes.

Model

In mathematics, a ___________ is an equation, table, or graph used to describe reality.

Probability Density Function (pdf)

_______________________is an equation used to compute probabilities of continuous random variables. It must satisfy the following two properties: 1. The total area under the graph of the equation over all possible values of the random variable must equal 1. 2. The height of the graph of the equation must be greater than or equal to 0 for all possible values of the random variable.