Stats Ch. 6 - Probability, Normal Distribution, adn z scores

empirical rule
(68% of scores lie in 1 SD, 95% of scores lie in 2 SD, 99.7% of scores lie in 3 SD)

When data in the population are normally distributed, we use the _____ ____ to determine the probability of an outcome in a research study.

1. ND is mathematically define (by formula)
2. ND is theoretical (bc behavioral data rarely conforms excatly to ND)
3. Mean, median, mode all located at 50th percentile (half data above/below)
4. ND is symmetrical
5. Mean can equal any value (any # from positive to neg infinity)
6. SD can equal any positive value greater than 0
7. Total area under curve of ND is equal to 1.00 (50% above & below so 1.00)
8. Tails of a ND are asymptotic (tails always approaching x-axis as traveling away from mean but never touch it, allowing for the possibility of outliers in a ND)

8 characteristics of a normal distribution:

-95% of scores in a ND will fall within 2 SD of the mean
-the 5% that fall outside of 2 SD are consider "not normal" bc they fall far from the mean

what is considered the statistical norm?

standard normal distribution

the likelihood of obtaining a score in any normal distribution; all other normal distributions can be converted to this using z transformation

standard normal distribution

Converting a score in any normal distribution to a z score will give you the position of that score in the _____ _____ _____

z score

a standard deviation, tells you the # of SDs that a score deviates from the mean in that distribution

population z-score formula

z = (x - μ)/σ

μ is population mean
σ is population SD

sample z-score formula

z = (x-x̄)/SD

x̄ is sample mean

standard normal distribution/z-scores

mean of 0, Sd of 1

z-score

a value on the x-axis of the z distribution that specifies the distance from the mean in SDs

z-score

We can find the probability of obtaining a score when we know its _____ in the standard normal distribution

unit normal table

The proportion of area or probability of z-scores in the standard normal distribution is listed in the ___ _____ ____

columns A, B, and C

to read the z table, we use what 3 columns?

column A

list only positive z scores

column B

lists the area b/w a z score and the mean

identical

a proportion associated with a z score above the mean is _____ to that below the mean bc z distribution is symmetrical

increases

As a z score is farther from the mean, the area under the curve b/w that z score and the mean _____

column C

Lists the area from a z score toward the tail

decreases

As a z score is farther from the mean, the area under the curve b/w that z score and the tail _____

1. transform a raw score into a z score
2. locate the corresponding proportion for the z score in the unit normal table

what 2 steps do we follow to locate the proportion, or probability, of a score in any normal distribution?

1. locate a z score associated with a given proportion in the unit normal table
2. transform the z score into a raw score

what two steps do we follow to locate the raw score in a normal distribution based on the probability of that score occurring?

real limits

The upper and lower values within which the probability of obtaining a binomial outcome is contained. The real limits for a binomial outcome, x, are x ± 0.5

binomial outcome/distribution

a type of distribution that has two possible outcomes.
For example, a coin toss has only two possible outcomes: heads or tails and taking a test could have two possible outcomes: pass or fail.

A Binomial Distribution

shows either (S)uccess or (F)ailure.

a smaller probability of occurence

larger z scores are associated with what?

to determine the distance that a score deviates from the mean and the probability of its occurrence

we can use z scores for what?

the z distribution is symmetrical so any proportion in the positive tail of the distribution will be mirrored in the negative tail
-if the value is negative, the proportion is on left or bottom half of distribution

only positive values are listed in the unit normal table why?