Applied Statistics: Probability and the Standard Normal Distribution

Vocabulary practice flashcards covering probability concepts, terminology, formulas, standard normal distributions, Z-score transformations, and percentile ranks based on the lecture notes.

Inferential Statistics

A branch of statistics that uses sample data to answer questions about an entire population, defining relationships between samples and populations in terms of probability.

Probability

Reflects the likelihood of an event occurring,

Probability (con’t)

It’s calculated in its simplest form as:

• The number of outcomes of interest divided by The total number of possible outcomes.

Probability’s purpose

To estimate attributes of the population given information that you know about the sample

Sample Space

A set that contains all possible outcomes of a given experiment or random process.

e.g., A die has six possible outcomes; therefore, the sample space contains those six outcomes.

Event

A particular outcome or set of outcomes in the sample space, forming a subset of it

e.g. Flipping a coin once – an event might be “heads”

Probability Notation $p(A)$

The notation for the probability of event $A$ occurring, with values ranging between $0$ and $1$ ($0 < p(A) < 1$)

e.g. If you flip a fair coin, what is the probability of getting a head? p(head) = 1/2 = .50

Probability Notation p(A) [con’t]

Is expressed as a proportion, fraction, or ratio.

Probability for Continuous Variables

The probability calculated for continuous variables, defined as $p = \text{area under curve for a range of scores}$.

Standard normal distribution

When the distribution of scores (X) is normal

Standard normal distribution (con’t)

We can convert X scores to Z scores and use the standard normal distribution

Standard normal distribution (III)

Allows us to determine the proportion of area within specific areas of the normal distribution

How many standard deviations are -1/+1 away from the mean

68.26%

How many standard deviations are -2/+2 away from the mean

95.44%

How many standard deviations are -3/+3 away from the mean

99.74%

Column B (Standard Normal Table)

The column in Appendix A representing the proportion of area in the body (the total area before $Z$).

Column C (Standard Normal Table)

The column in Appendix A representing the proportion of area in the tail (the area beyond $Z$).

Column D (Standard Normal Table)

The column in Appendix A representing the proportion of area located between the mean ($Z = 0$) and a specific Z-score.

Percentile Rank

An indicator showing that a specific score is greater than or equal to a given percentage of scores within a distribution.

Extreme $5\text{\%}$ Cutoff Z-Scores

The Z-scores ($Z = +1.96$ and $Z = -1.96$) associated with the most extreme $5\text{\%}$ of scores in a normal distribution, dividing the extreme region into $.0250$ in each tail.

Negative Z scores

Percentile rank of a score below the mean

How can negative Z scores be found?

• Column C (area beyond Z score) x 100

• [.50 – Column D (area between mean and Z score)] x 100

Positive Z scores

Percentile rank of a score above the mean

How can positive Z scores be found?

• [1 – Column C (area beyond Z score)] x 100

• [.50 + Column D (area between mean and Z score)] x 100