Objectives for Unit Five
The Normal Distribution and Probability
1. Know the characteristics of
empirical and theoretical distributions.
Empirical distribution are composed of actual cases in which a sample or population size can be determined and an actual plot can be made of the distribution. The shape of an empirical distribution rarely conforms to a precise shape such as normal, rectangular or symmetrical.
A theoretical distribution is one that is generated by a formula or a description and does not have actual cases. The shape of the distribution reflects the definition of the distribution.
2. Know the characteristics of
a normal distribution.
The normal distribution is a theoretical distribution that never occurs with real data, only approximated. It is symmetrical, unimodal, the mean, median and mode are equal, bell-shaped, has inflection points at z = ±1.00, and is asymptotic to the X axis.
3. Know the proportions of a
normal distribution within one, two, and three standard deviations
of the mean.
The percentages of a normal distribution within one, two, and three standard deviation of the mean are 68%, 95%, and 99%.
4. Know the information needed
to plot a normal curve.
All that is needed to plot a normal curve is the mean and standard deviation of the distribution. A standard normal distribution has a mean of 0 and a standard deviation of 1.
5. Know situations when a distribution
close to normal is and is not likely to occur.
Distribution approximately normally distributed occur when many random events combine to result in one outcome. For example flipping a coin many times to result in one number indicating the number of heads in the series will be approximately normally distributed if it is repeated a large number of times. Many human characteristics (physical, behavioral, and psychological) approximately normally distributed because they are the result of many causes, many of which are random. Scores from moderately difficult tests with large numbers of questions are frequently approximately normally distributed.
Non-normal distributions occur whenever there are limitations for how high or low the scores may be, if there are relatively few causes for the event or characteristic, if the sampling of scores or subjects is by a specific process (not random or by chance), if two or more homogeneous subgroups are combined to form a larger group, or if a small number of cases are used. Examples include eye color (a small number of genes), test scores on very easy or very hard tests (limits on a difficult test of 0% if a subjective test or chance percent if an objective test or 100% percent on an easy test), a distribution of height for men and women combined (a bi-modal distribution), or if only high-scoring students are selected.
6. Know how to use a normal curve
table to compute for a normal distribution the percentages of cases (or
probability of cases being) above and below a positive or negative z score,
between two z scores (both positive, both negative, or one positive and
one negative), outside of two z scores (one positive and one negative),
to figure the z scores within which or outside of which a certain percentage
of cases fall and to figure z scores corresponding to percentile ranks.
Examples of questions would be:
percent above a z score of +0.50
percent above a z score of -0.50
percent below a z score of +0.50
percent below a z score of +0.50
percent between z scores of +1.03 and +1.36
percent between z scores of -0.55 and -2.14
percent between z scores of -1.52 and +0.48
percent more extreme than +1.00 and -1.00
z scores for the middle 50%
z scores for the most extreme 10%
z score corresponding to a percentile rank of 83
7. Know the meaning of mutually
Mutually exclusive categories are those that cannot be true for a case at the same time. Political party is a variable composed of mutually exclusive categories. A person cannot be a Democrat and a Republican at the same time.
8. Know how to figure probabilities
of individual events.
The probability of an individual event is determined by dividing either the number of ways the specific event or outcome indicated can occur by the number of possible events or outcomes of any kind that could occur or by dividing the number of elements with the indicated characteristic by the number of elements of any kind. To find the probability of drawing a heart from a deck of cards you would take the number of hearts in a deck (13) and divide by the total number cards in a deck (52).
9. Know how to figure the probability
of one of two or more mutually exclusive outcomes occurring.
The probability of one or the other of two or more mutually exclusive events occurring is computed by taking the sum of the individual probabilities.
10. Know the meaning of independent
Independent events are those where one event is not dependent on the other.
11. Know the meaning of sampling
with and without replacement.
Sampling with replacement means that with repeated sampling, it would be possible to select the same case each time. Without replacement means that once a case has been selected it is not eligible to be selected in succeeding selections.
12. Know how to figure the probability
of two or more independent events (e.g., with replacement) or non-independent
events (e.g., without replacement).
The probability of two or more independent events all occurring is computed by taking the product of the individual probabilities. For non-independent events, the conditional probabilities of each event (the probability given that the other events have occurred previously) are multiplied together.
The probability of randomly selecting
the same person twice in a row from a class of 20 students would be 1/20
times 1/20. The probability of selecting from a class of 10 boys and 10
girls a team of two students where both students were boys would be 10/20
times 9/19 (there would be 9 boys left and 19 students left in the class
from which to randomly select the second person on the team).