Chapter 5: Normal Probability Distributions

Section 1

Introduction to Normal Distributions & the Standard Normal Distribution

Section 3

Normal Distributions: Finding Values

*Transforming a "z" - Score to an "x" - Value* (p. 254)

To transform a standard *z* - score to an *x* - value in a given population, use the formula: *x* = *μ* + (*z* × *σ*).

continuous probability distribution

The probability distribution of a continuous random variable has an infinite number of possible values that can be represented by an interval on a number line. (p. 234)

probability density function (p.d.f.)

A graph used for continuous probability distributions that has two requirements: *(1)* the total area under the curve is equal to 1, and *(2)* the function can never be negative. (p. 234)

sampling distribution

The probability distribution of a sample statistic that is formed when random samples of size *n* are repeatedly taken from a population. If the sample statistic is the sample mean, then the distribution is the *"sampling distribution of sample means"*. Every sample statistic has a sampling distribution. (p. 261)

standard normal distribution

A normal distribution with a mean of 0 and a standard deviation of 1. The total area under its normal curve is 1. (p. 237)

Section 4

Sampling Distributions & the Central Limit Theorem

normal curve

The graph of a normal distribution. (p. 234)

normal distribution (p. 234)

A continuous probability distribution for a random variable *x*. The graph of a normal distribution is called the *"normal curve"*. A normal distribution has these properties. *1.)* The mean, median, and mode are equal. *2.)* The normal curve is bell-shaped and is symmetric about the mean. *3.)* The total area under the normal curve is equal to 1. *4.)* The normal curve approaches, but never touches, the *x* - axis as it extends farther and farther away from the mean. *5.)* Between *μ* - *σ* and *μ* + *σ* (in the center of the curve), the graph curves downward. The graph curves upward to the left of *μ* - *σ* and to the right of *μ* + *σ*. The points at which the curve changes from curving upward to curving downward are called *"inflection points"*.

inflection point

The point at which the curve changes from curving upward to curving downward. (p. 234)

*The Central Limit Theorem* (p. 263)

*1)* If random samples of size *n*, where *n* ≥ 30, are drawn from any population with a mean *μ* and a standard deviation *σ*, then the sampling distribution of sample means approximates a normal distribution. The greater the sample size, the better the approximation. *2.)* If random samples of size *n* are drawn from a population that is normally distributed, then the sampling distribution of sample means is normally distributed for *"any"* sample size *n*. In either case, the sampling distribution of sample means has a mean equal to the population mean. *μ*∨*x̄* = *μ* (*Mean of the sample means*) The sampling distribution of sample means has a variance equal to 1/*n* times the variance of the population and a standard deviation equal to the population standard deviation divided by the square root of *n*. *σ*²∨x̄ = (σ²)/(*n*) (*Variance of the sample means*) *σ*∨x̄ = (σ)/(√*n*) (*Standard deviation of the sample means*)

*Finding Areas Under the Standard Normal Curve* (p. 239)

*1.)* Sketch the standard normal curve and shade the appropriate area under the curve. *2.)* Find the area by following the directions for each case shown. ~ *(a.)* To find the area to the *left* of *"z"*, find the area that corresponds to *"z"* in the Standard Normal Table. ~~ (1): Use the table to find the area for the *z* - score. ~~ (2): The area to the left of *z* = 1.23 is *0.8907*. ~ *(b.)* To find the area to the *right* of *"z"*, use the Standard Normal Table to find the area that corresponds to *z*. Then subtract the area from 1. ~~ (1): Use the table to find the area for the *z* - score. ~~ (2): The area to the left of *z* = 1.23 is *0.8907*. ~~ (3): Subtract to find the area to the right of *z* = 1.23: 1 - *0.8907* = *"0.1093"*. ~ *(c.)* To find the area between two *z* - scores, find the area corresponding to each *z* - score in the Standard Normal Table. Then subtract the smaller area from the larger area. ~~ (1): Use the table to find the area for the *z* - score. ~~ (2): The area to the left of *z* = 1.23 is *0.8907*. ~~ (3): The area to the left of *z* = -0.75 is *0.2266*. ~~ (4): Subtract to find the area of the region between the two *z* - scores: *0.8907* - *0.2266* = *"0.6641"*.

*Properties of the Standard Normal Distribution* (p. 237)

*1.)* The cumulative area is close to 0 for *z* - scores close to *z* = -3.49. *2.)* The cumulative area increases as the *z* - scores increase. *3.)* The cumulative area for *z* = 0 is 0.5000. *4.)* The cumulative area is close to 1 for *z* - scores close to *z* = 3.49.

*Properties of Sampling Distributions of Sample Means* (p. 261)

*1.)* The mean of the sample means *μ*∨*x̄* is equal to the population mean *μ*. *μ*∨*x̄* = *μ* *2.)* The standard deviation of the sample means *σ*∨x̄ is equal to the population standard deviation *σ* divided by the square root of the sample size *n*. *σ*∨*x̄* = (σ)/(√*n*) The standard deviation of the sampling distribution of the sample means is called the *standard error of the mean*.