Chapter 6

6.3 Checking the Normality Assumption (pg.

Almost every inferential statistics procedure requires certain assumptions, for example, that observations are from a normal distribution. ~Therefore, it seems reasonable to be able to perform some kind of check for normality, to make sure there is no evidence to refuse this assumption. -If the population parameters are not given you can use the sample parameters. 1.) Graphs 2.) Backward Empirical Rule 3.) IQR/s 4.) Normal Probability Plot

-Standardization Rule (pg.

If X is a normal random variable with mean and variance, then a standard normal random variable is given by Equation (6.8). -The process of converting from X to Z is called *standardization.* Z is a *standardized* random variable.

-Interpolation

Is a method of approximation. It is often used to estimate a value at a position between given values in table. Linear interpolation assumes that the two known values lie on a straight line.

-Standard Normal Random Variable (pg.

The standard normal distribution is not common, but it is used extensively as a *reference* distribution. ~Any probability statement involving any normal random variable can be transformed into an equivalent expression (with same probability) involving a Z random variable. We will often refer to a s standard normal distribution as a "Z world."

-Cumulative Distribution Function (pg.

This function starts at 0 and is always increasing, until it reaches a maximum value of 1. -The mean, and the variance , for a continuous random variable are computed using calculus. Although we will not consider any of these calculations, we will interrupt and use these values as usual. ~Mean is a measure of the *center* of the distribution ~Variance is a measure of the spread, or *variability,* of the distribution

-Probability Distribution for a Continuous Random Variable (pg.

Is given by a smooth curve called a *density curve,* or *probability density function* (pdf). -The curve is defined so the the probability that X takes on a value between *a* and *b*(a<b) is the area under the cave between *a* and *b.* -The density curve, or *probability density function,* is usually denoted by *f.* It is a *function,* defined for *all* real numbers. ~f(x) is *not* the probability that the random variable X equals the specific value x. Rather, the function *f* leads to, or conveys, probability through area. -The shape of the graph of a density function can vary considerably. However, a density function must satisfy the following two properties: a.) f must be defined so that the total area under the curve is 1. The total probability associated with any random variable must be 1. f(x), a specific value of the density function, *may* be greater than 1(while the total area under the curve is still exactly 1). b.) f(x)>= 0 for all x. Therefore, the entire graph lies on or above the x axis. -If X is a continuous random variable with density function f, the probability that X equal *any* one specific value is 0. That is, P(X=a)=0 for any a. The reason: There is no area under a single point. ~This seems like a contradiction. Certainly we can observe specific values of X, yet the probability of observing any single value is 0. Recall: Probability is a *limiting relative frequency.* There are an *uncountably) infinite number of value for any continuous random arable. Therefore, the limiting relative frequency of occurrence of any single value is 0. ~The only reasonable probability questions concerning continuous random variable involve intervals. And we can almost always sketch a graph to visualize these probabilities, or regions.

-Normal Probability Distribution (pg.

Is very common and is the most important distribution in all of statistics. This *bell-shaped* density curve can be used to model many natural phenomena, the normal distortion is used extensively in statistical inference. -Equation 6.6 means that x can be any real number (the density curve continues forever in both directions), the mean can be any real number (positive or negative), and the variance can be any positive real number. -For ANY mean and variance, the density curve is symmetric about the mean, unimodal, and bell-shaped as shown in Figure 6.21. ~The mean is equal to the median because the normal distributions symmetric. ~As the variance increases, the total area under the probability density function (1) is rearranged. The graph is compressed down and pushed out (on the tails).

-(Continuous) Uniform Distribution (pg

Provides a good opportunity to illustrate the connection between area under the curve and probability. -For this random variable, the total probability, 1, is distributed evenly, or uniformly, between two points. Computing probabilities associated with this random variable reduces to finding the area of a rectangle. -a and b can be any real numbers, as long as *a* is less then b(a<b). -ALl of the probability (action) is between *a* and *b.* The probability density function is the constant 1(b-a) between *a* and *b,* and zero outside of this interval. Hence, there is no area and no probability outside the interval [a,b].