Chapter 8 - Probability and Why It Counts

hypothesis testing & z-scores

- Any event can have a probability associated with it - Probability values help determine how "unlikely" the event is - Researchers usually use a probability of .05 (sometimes .01) - If there is less than a 5% (or 1%) chance of the event occurring, you have a "statistically significant" result

finding probabilities above or below certain z scores

- how to use the normal curve table - what is the probability of getting a z-score of +2.00 or greater? steps: 1. sketch a distribution 2. look up z-score of +2.00 in the table 3. determine probability

what z-scores represent

- the areas of the curve that are covered by different z-scores also represent the probability of a certain score occurring for example: 1. what is the probability of getting a score of 110 or above? 2. what is the probability of getting a score of 90 or below?

normal curve (aka Bell Curve)

- visual representation of a distribution of scores three characteristics: - mean, median, mode are EQUAL - perfectly symmetrical about the mean - tails are asymptotic - allows for all possible scores - in general, many events occur in the middle of a distribution with few on each end

Comparing Different Distributions

- z scores allow us to compare distributions with different means and SDs - z scores across different distributions are comparable because scores have been converted so that M = 0 and SD = 1

probability examples

1. tossing a coin: chance of getting tails = 1/2 = 50% = .50 2. selecting a card from a deck: drawing an ace = 4/52 = 7.92% = .0792

Converting raw scores to z scores

Example: Mean = 10, SD = 2, 1. 10 = 10-10/2 = 0/2 = 0 2. 13 = 13-10/2 = 3/2 = 1.5 3. 8 = 8-10/2 = -1 4. 6.5 = 6.5-10/2 = -3.5/2 = -1.75

examples of comparing different distributions

Two people take different IQ tests: - Emily got a score of 113 on a test with M = 100 and SD = 10. - Gabriel got a score of 54 on a test with M = 50 and SD = 6. Which person has the highest IQ? 113-100/10 = 3/10 = .333 54-50/6 = 4/6 = 1.333

z score

a standard score that is the result of dividing the amount that a raw score differs from the mean of distribution by the standard deviation z = (x - x̄) / s x = raw score (individual score) x̄ = sample mean of the distribution s = standard deviation of distribution

Central Limit Theorem

in a world of somewhat random events (meaning somewhat random values), this theory explains the occurrence of somewhat normally distributed sample values - two basic tenant: First, the value (such as the sum or the mean) associated with a large number of independent observations will be distributed approximately in a normal fashion. - Second, this "normality" gets more and more normal as the number of observations or samples increases.

normal curve

is the basis for understanding probability of a possible outcome - allows us to infer results from a sample to a population - basis for determining the degree of confidence that an outcome is "real" what is the probability of obtaining a certain study result by chance?

asymptotic

means that they come closer and closer to the horizontal axis, but never touch the axis.

more about z-scores

scores below the mean are negative. those above are positive - A z-score represents the number of SD's away from the mean for a particular raw score

variability

uncertainty (sampling error) - probability lets us access that uncertainty

what exactly is probability?

when several outcomes are possible, the probability for any particular outcome is a fraction probability of an outcome = A number of outcomes classified as A / total number of possible outcomes - can be expressed as fractions, percentages, or proportions

finding probabilities with raw scores

with a distribution where M = 75 and SD = 5, What is the probability of getting a score of 87 or above? - Sketch the distribution - Convert the raw score into a z score - Look up z -score in the table - Determine the probability