what are inferential procedures typically built around?
- probability
- Specifically, the relationships between samples and populations are usually defined in terms of probability.
- Probability is used to predict the type of samples that are likely to be obtained from a population.
Probability
For a situation in which several different outcomes are possible, the probability for any specific outcome is defined as a fraction or a proportion of all the possible outcomes. If the possible outcomes are identified as A, B, C, D, and so on, then
- probability is defined as a proportion, or a part of the whole
Probability equation
probability of A = number of outcomes classified as A / total number of possible outcomes
Random Sample
A random sample requires that each individual in the population has an equal chance of being selected. A sample obtained by this process is called a simple random sample.
Independent random sample
An independent random sample requires that each individual has an equal chance of being selected and that the probability of being selected stays constant from one selection to the next if more than one individual is selected.
- For a population with N individuals, each individual must have the same probability, P = 1/N, of being selected
Sampling with replacement
A sampling technique that returns the current selection to the population before the next selection is made. A required part of random sampling.
probability and frequency distribution
a particular portion of the graph corresponds to a particular probability in the population. Thus, whenever a population is presented in a frequency distribution graph, it will be possible to represent probabilities as proportions of the graph.
The normal distribution. The exact shape of the normal distribution is specified by an equation relating each X value (score) with each Y value (frequency). The equation is
Y (frequency) = 1/square root of 2pivariance * e^ - (X-mu)^2 - 2*variance
z-score and percentages in a normal distrubtion
in 0 - 1 z-score, percentage = 34.13%
in 1-2 z-score, percentage = 13.59%
in +2 z-score, percentage = 2.28%
unit normal table
A table listing proportions corresponding to each z-score location in a normal distribution.
- The first column (A) lists z-score values corresponding to different positions in a normal distribution.
- Columns B and C in the table identify the proportion of the distribution in each of the two sections. Column B presents the proportion in the body (the larger portion), and column C presents the proportion in the tail.
- Finally, we have added a fourth column, column D, that identifies the proportion of the distribution that is located between the mean and the z-score.
Facts about unit normal table
- The body always corresponds to the larger part of the distribution whether it is on the right-hand side or the left-hand side. Similarly, the tail is always the smaller section whether it is on the right or the left.
- Because the normal distribution is symmetrical, the proportions on the right-hand side are exactly the same as the corresponding proportions on the left-hand side.
- Although the z-score values change signs (+ and −) from one side to the other, the proportions are always positive. Thus, column C in the table always lists the proportion in the tail whether it is the right-hand tail or the left-hand tail.
to answer probability questions about scores (X values) from a normal distribution, you must use the following two-step procedure:
- Transform the X values into z-scores.
- Use the unit normal table to look up the proportions corresponding to the z-score values.
Finding value from the probability or proportion
we begin with a specific proportion, use the unit normal table to look up the corresponding z-score, and then transform the z-score into an X value. The following example demonstrates this process.
binomial
When a variable is measured on a scale consisting of exactly two categories, the resulting data are called binomial
- Binomial data can occur when a variable naturally exists with only two categories. For example, people can be classified as male or female
binomial distribution
Using the notation presented here, the binomial distribution shows the probability associated with each value of X from X=0 to X=n.
binomial values are discrete
binomial distribution is continuous
- it is the area under the distribution that is used to find probabilities.
the binomial distribution will approximate a normal distribution with the following parameters:
Mean (mu) = pn
Standard deviation (o) = square root of npq
z = X-mu/o = X - pn / square root of npq
normal approximation
provides an extremely accurate model for computing binomial probabilities in many situations.
shows the difference between the true binomial distribution, the discrete histogram, and the normal curve that approximates the binomial distribution
To gain maximum accuracy when using the normal approximation, you must remember that each X value in the binomial distribution actually corresponds to a bar in the histogram
if our sample is located in the tail beyond one of the ±1.96 boundaries, then we can conclude:
- The sample is an extreme value, nearly 2 standard deviations away from average, and therefore is noticeably different from most individuals in the original population.
- The sample is a very unlikely value with a very low probability if the treatment has no effect.
