Exam 2 key terms and practice Flashcards
Probability
the relative frequency of an event in the long run
probability= events/#number of outcomes
as a relative frequency
p(x) = # of heads/ # of outcomes = 1/2= 50 %
can find probability by finding z score, then looking at smaller portion to find probability in a-score table
simple experiment
well-defined process that leads to a single well-defined outcome
ex. coin toss
elementary event
the outcome of a simple experiment
ex. heads or tails
event
set of elementary events that share predefined characteristics
ex. dice roll (odd 1,3,5 or even 2,4,6)
mutually exclusive events
cannot co-occur
ex. heads or tails are mutually exclusive, the occurrence of heads precludes the occurrence of tails
exhaustive events
representing all possible outcomes
ex. heads & tails
sample space
set of all possible elementary events that may occur in a sample experiment
ex. dice roll (1-6), six sided
random sampling
takes a small, random portion of the entire population to represent the entire data set, where each member has an equal probability of being chosen
addition rule
probability that one event or another will occur and events are mutually exclusive, can add properties
p(head or tail)= p(head) + p(tails) = 1/2 + 1/2 = 1
Multiplication Rule
if two events are independent, can calculate probability both occur together by multiplying probabilities
p(head & head)= 1/2 x 1/2 = 1/4 = 25%
Probability from Frequency Distribution
p (x) = number of elementary of events (freq) / sample size
Probability of scores vs samples
Sampling Error
a statistical error that occurs when an analyst does not select a sample that represents the entire population of data
Sampling Distribution
a probability distribution of a statistic that is obtained through repeated sampling of a specific population
Sampling Distribution of the Mean
This method shows a normal distribution where the middle is the mean of the sampling distribution
Standard error of the mean
measures how much discrepancy is likely in a sample’s mean compared with the population mean
Central Limit Theorem
regardless of distribution of the original random variable, the sampling distribution of the mean associated with that random variable will be approximately normally distributed when based on a large # of classes
- rule of thumb is n>30 will yield approximately normal sampling distribution of the mean
Hypothesis testing
a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables
- if it’s a normal distribution, we can calculate probabilities from a known table because we want to make accurate inferences about the population mean
formal steps for statistical test (4 steps)
- state hypotheses
Ho & HA
(given standard dev.) - state statistical decision criteria
alpha, type of test, cutoff values , draw picture - calculate test statistic and compare
- state the statistical conclusion
- reject the null, or fail to reject (we reject if in critical region)
- interpret alternative (population mean is or is not equal to x)
- explain how it answers question (ex. students are more extroverted)
One tailed vs. Two tailed Tests
one tailed test
- we want to know if population mean is above or below a certain (single) hypothesized mean value
- we have a directional hypothesis and we conduct a one tailed test
two tailed test
- we want to know if the population mean is greater or less than the hypothesized value (on both ends)
- 2 critical values: one above and below hypothesized value
- in z test 2 critical values with be +/- 1.96 (we look at table to find z score)
Type 1 error
- the 5% we have left in 95% or 0.05 significance
When the Ho is true, and we reject it (false alarm)
the p(reject Ho|Ho is True)= α
incorrect decision
Type 2 error
when Ho is false, and we accept it (miss)
the p(retain Ho|Ho is false) = β
incorrect decision
Power & factors that affect it
1-β
correct decision
reject Ho|Ho is false = 1-β
we want to limit α and maximize β
factors that affect power:
- the probability of a type 1 error (alpha), the priori level of significance, and the criterion for rejecting Ho
- the sample size and variance (as sample size increases, the variance of the sampling distribution decreases. If the distribution is more narrow, then there will be less overlap between the two sampling distributions resulting in fewer type II (false negative) errors and a greater statistical power)
how to increase:
- Switch from a 2-tailed test to a 1-tailed test
- increase mean difference
- Use z distribution instead of t distribution
- decrease std dev.
- increase sample size (most practical way)
z-statistic
relative location of a sample mean with in a sampling distribution of the mean
one sample t-test
- we often will have a hypothesized mean value but not known population variance (sometimes using s^2)
- the t distribution is single peaked and symmetric like the standard normal distribution
- somewhat flatter and relatively heavier tails for a small sample
- as N gets larger, the t distribution approaches the standard normal destitution (i.e the t-statistic coverage)
- the one-sample t-statistic is distributed with N-1 degrees of freedom
- when we use s^2 to estimate the population variance, we must first estimate the mean
Effect size for related sample
measure of magnitude of mean difference
guidelines
small effect d=0.2
med. effect d=0.5
large effect d=0.8
Independent-samples t-test
test whether it is reasonable to believe that two sample means from independent samples came from a population with the same mean
Pooling Sample Variance
- typically assume that in the population, both sample standard deviations are equal
- best estimate of variance is known as the pooled variance estimate (weighted avg. when combined)
- sample variance estimates population variance, usually equal so you can estimate
- 1 estimate for 2 things
Effect size in independent-samples t-test
