Chapter 5: Probability and the Normal Distribution

What is the statistical definition of probability?

The relative likelihood of occurrence of any given outcome or event.

Which probability rule is used to determine the probability that an event will NOT occur?

The Converse Rule.

According to the Converse Rule, how is the probability of an event not occurring calculated?

By subtracting the probability that the event will occur from 1.

The Addition Rule is used to determine __ probability by adding the probabilities of separate outcomes.

Cumulative

What condition must be met for the simple Addition Rule P(A or B) = P(A) + P(B) to apply?

The outcomes must be mutually exclusive.

What is the formula for the Addition Rule if the outcomes are not mutually exclusive?

P(A or B) = P(A) + P(B) - P(A and B)

Which rule is used to find the probability of successive, independent outcomes occurring together?

The Multiplication Rule.

What is the formula for the Multiplication Rule when outcomes A and B are independent?

P(A and B) = P(A) × P(B)

If the assumption of independence is dropped, what does P(B|A) represent in the Multiplication Rule formula?

Conditional probability.

What is the term for a function that describes the probability of a random variable taking certain values?

Probability distribution.

To what descriptive statistical tool are probability distributions most closely analogous?

Frequency distributions in percentage terms.

What is the sum of the probabilities of all possible events in a probability distribution?

1

In a flip of two coins, what is the probability of obtaining exactly one head?

0.50

In a flip of two coins, what is the probability of obtaining zero heads?

0.25

When rolling two dice, what is the probability of the combined number being 7?

6/36 (or approximately .167)

When rolling two dice, which combined score has the lowest probability of occurrence?

2 or 12 (each at 1/36)

Definition: A variable whose values result from a random process or are associated with a probability distribution.

Random Variable

What is the specific term used for the mean of a random variable?

Expected Value (E)

How is the expected value of a random variable mathematically defined?

As the weighted average of all possible values the random variable can take.

Which Greek letter is commonly used to symbolize the mean of a random variable?

μ

Which Greek letter is used to represent the standard deviation of a random variable?

σ

Which symbol represents the variance of a random variable in a theoretical distribution?

σ²

In notation, what is the difference between the symbol s² and the symbol σ²?

s² is used for observed sample data, while σ² is used for theoretical distributions or populations.

What determines whether a probability distribution is discrete or continuous?

The values that the associated random variable can take.

What is the graphic presentation of a continuous probability distribution called?

A curve.

On the graph of a continuous probability distribution, what does the value on the y-axis represent?

The probability that a random variable takes a certain value.

For a continuous variable, the total area under the probability curve is equal to __.

1

Why is it necessary to understand probability distributions in inferential statistics?

To make reasonably accurate estimations about a population using sample statistics.

Which probability distribution is considered the most theoretically useful for statistical estimation?

The normal distribution.

List two alternative names for the normal distribution.

The Gauss distribution or the bell curve.

How is the symmetry of the normal curve described?

It is perfectly symmetrical.

What is the modality of a normal curve?

Unimodal.

Where do the mean, median, and mode lie in a normal distribution?

They coincide at the exact center of the distribution.

What happens to the tails of the normal curve as they move away from the mean?

They continue infinitely closer to the baseline without ever touching it.

According to the normal curve, what percentage of cases fall within one standard deviation of the mean ([μ - σ, μ + σ])?

68.26%

What percentage of cases in a normal distribution fall within two standard deviations of the mean?

95.44%

What percentage of cases in a normal distribution fall within three standard deviations of the mean?

99.74%

What is the specific area under the normal curve between the mean (μ) and +1σ?

34.13%

What is the term for a random variable that follows a normal distribution with a mean of 0 and a standard deviation of 1?

Z-score (or standard score).

What is the name of a normal distribution consisting of z-scores?

Standard normal distribution.

Formula: z = (X - μ) / σ

Definition: The formula for calculating a z-score by subtracting the mean from a raw score and dividing by the standard deviation.

In the z-score formula, what does the symbol X represent?

A raw score.

What two things does a z-score indicate about a raw score's relation to the mean?

Direction and degree of deviation in sigma units.

What is the first step in finding the probability under the normal curve for a specific variable value?

Convert the value of the variable into a z-score.

After converting a value to a z-score, what resource is used to decide the final probability?

The Appendix table of probabilities provided by statisticians.

Why is the z-score process referred to as a 'standardization process'?

It allows the use of universal statistical tables to determine areas regardless of the original scale's mean or standard deviation.

Concept: Expected Value of X (denoted E(X))

Definition: The long-run average value of a random variable, calculated as the sum of each possible value multiplied by its probability.

The standard deviation of a z-score distribution is always equal to __.

1

The mean of a standard normal distribution is always equal to __.

0

What portion of cases in a normal distribution fall above the mean?

50%

In Image [SOURCE_IMAGE_4], what percentage of the area lies between +1σ and +2σ?

13.6%

In Image [SOURCE_IMAGE_4], what percentage of the area lies beyond 3σ from the mean in a single tail?

0.1%

What is the point on the normal curve where the 'reversed S-shaped portion' shifts direction?

Exactly one standard deviation from the mean (1σ).