Exam 2- STAT 212

expected value

mean of a random variable, (n*E(x)) where n is sample and E(x) is the probability

z-score

how many standard deviations a data value is away from the mean

z-score formula

(x-mean or mu)/standard deviation

P(x\>\#)

probability that a variable is greater than a value

two tailed probability

what percentage of values are farther away from the mean than this value

uniform distributions: discrete

all outcomes are equally likely

uniform distributions: continuous

all equally-sized outcome ranges are equally likely to occur

population distribution

the entire distribution of a particular random variable

convergence in distribution

the larger your sample, the more your sample distribution will converge toward the population distribution in shape

the law of large numbers

more trials lead to more of the expected value (normal distribution)

convergence of an estimator

larger the sample the closer to the population mean it will be

X bar

sample mean

sampling distribution

distribution of a sample statistic, converge to a single point

sample distribution

converges to the population distribution

standard deviation of x bar

population SD/ sqrt(sample size)

central limit theorem

distribution of x bar will be normally distributed if sample size is large enough

standard error of sample mean

error we likely have in our sample stat, use SD of x bar formula

sigma

true SD of a population

S

standard deviation of a sample of data

p hat

sample proportion, almost always normally distributed

standard error of a proportion (SE p hat)

sqrt(p(1-p)/sqrt(n)

z score

p hat- p/SE p hat, how many SE off sample stat is away from the parameter

statistical inference

claim about a population based on sample data

hypothesis test

method of inference that examines whether a candidate parameter is plausible or not using information

p

proportion of sample we're testing for, candidate parameter

null hypothesis

the candidate parameter is true, H naught

alternative hypothesis

the parameter is some alternative value

p value

probability of getting this statistic or more extreme if null is true, positively correlated with null, doesn't tell magnitude of difference from null

reject the null

the null is unlikely, p value at or below 0.05

fail to reject null

the null is unlikely, not accept null bc could have small bias, above 0.05

alpha

significance level, usually 0.05

type 1 error

incorrectly rejecting the null hypothesis

type 2 error

incorrectly failing to reject the null hypothesis, more common in small samples

t distribution

corrected normal distribution, correction will be larger for larger sample sizes

increase in df means...

t score associated with each tail probability approaches the matching z score

test statistic

our sample statistic converted to a standardized value

do a z test when

p hat is normally distributed, have at least 10 of each response

do a t test when

x bar is normally distributed, n \> 30

point estimate

best guess for the parameter's value

confidence level

less confident in the margin of error, the smaller the interval

confidence interval for population proportion or Wald interval

p hat + or - (SE by % * SE of p hat)

margin of error formula

SE by % * SE p hat

Assumptions for Wald Intervals

have 10 of each answer if not then it would be too close to 0 or 100% so unreliable

if decrease confidence level the interval will

narrow

if decrease sample size the interval will

widen

higher confidence levels means

a higher margin of error

when building the confidence interval of an average change use

x bar instead of p hat

have a smaller margin of error due to

decreasing confidence %, increasing sample size

confidence intervals make a prediction of

the population parameter or mu

to approximate population mean confidence interval

use x bar and approximate standard error

Use a z-interval when:

sample is larger than 120, sampling distribution for the mean is normally distributed

Use a t-interval when:

sigma is being approximated by s, sampling distribution for the mean is normally distributed