stats exam 2

sample mean (xbar problems)

quantitative data

sample proportion (𝑝̂ problems)

categorical data (yes or no)

what does x represent

count of successes

what does 𝑝̂ represent in a sample

sample proportion of successes

what does 𝑝̂ estimate

p, the true population proportion of success given p

sampling distribution

distribution of a statistic

how can we study sampling distribution

look at cases where parameters are known and see what happens if we take repeated samples

mean of distribution of 𝑝̂

p

standard error of 𝑝̂

sqrt (p(1-p)/n)

standard error

standard deviation of a sampling distribution, to distinguish from standard deviation of a regular problem

conditions for calculating standard error for 𝑝̂

successes- np greater than or equal to 15

failures- n(1-p) greater than or equal to 15

how to represent sampling distribution

𝑝̂≈N (probability, standard error)

what does x bar estimate

the population mean

sampling distribution of xbar

same mean as original, smaller standard deviation than x, will become more normal as n increases

central limit theorem

for a srs with a large sample size, sampling distribution of the mean is approx normal with the same mean as the original distribution and the standard error of the original std/sqrt n

what are the conditions for clt

original distribution was normal or large n= (30). as n increases the sampling distribution of xbar will get closer to normal

point estimates

when you take a sample and compute the sample proportion, you wouldnt get the exact results. our best guess of the parameter but we need more info

when is a statistic biased

when it is not centered around the parameter

how do you reduce bias

random sampling

standard error

has to do with the spread of the sampling distribution. smaller spread means we have more values of the estimate closer to the parameter being estimated

how do you reduce standard error

increase sample size

confidence intervals to estimate population proportion

p̂ ± z sqrt (p̂-(1-p̂)/n)

confidence intervals of a population proportion requirementrs

at least 15 successes AND 15 failures, data is from a random sample

95% confidence z

1.96

90% confidence z

1.645

99% confidence z

2.58

confidence intervals for means

need to use a t table because we are estimating standard deviation with s

confidence interval for population requirements

data is normally distributed OR n is large. from a srs

confidence interval for population means

x̄ ± t(s/sqrt n)

degrees of freedom

n-1. as degrees of freedom increase, t gets closer to z. as it increases, it approaches a more normal distribution

t distributions

flatter than z curve but similar shape. symmetric and bell shaped centered at zero.

t table

read to the right

what to do if youre trying to construct a confidence interval for means but you have a small size

plot the data out. if its symmetrical and there arent any outliers, then you can continue with the problem

where are confidence intervals for mean centered around

x bar

bigger confidence

bigger interval

bigger sample size

smaller interval

how to have large confidence but small interval

increase the sample size. as sample size increases, confidence also increases

margin of error for ci

takes into account sampling variability but not errors due to biased samples.
t (s/sqrt n)

what do confidence intervals make statements about

population mean, not the sample mean or individuals

margin of error for determining sample size for estimating a population proportion

same as standard error

when is standard error the largest

when phat=0.5, the sample size needed for a particular margin of error will be largest at 0.5. only use when you have no idea what the true value of phat is

sample size for estimating a population proportion

always round up to an integer

solving for n determining sample size for estimating population mean

(σz/m)²

also must be an integer

what does a confidence interval give

gives a region that is likely to contain the parameter. just want to estimate it

significance tests

check to see which claim about the population is supported by the data. take a sample to see if the data supports the claim or what we believe is true

assumptions of a significance test

categorical, srs, minimum of 15 successes and failures

null and alternative hypothesis

statements about the unknown parameters, not the sample statistics

null hypothesis

H₀, states the parameter takes on a certain value. always uses = sign

Ho- p=#

alternative hypothesis

H_a, can either be 2 sided or 1 sided. sign is always determined from the story, not the data

one sided

only interested in deviations from Ho in one direction

p<# or p>#

two sided

want to show the parameter is different from the number claimed in the null hypothesis

p does not equal #

what type of p value to reject the nnull hypothesis

small p values

what are the significance levels

0.1, 0.05, 0.01

90%, 95%, 99%

when do you reject the null given the alphas

if p value is less than alpha

how to estimate standard deviation

range/standard deviations away, demonstrated by the confidence interval

how to check if something is normally distributed if it doesnt give you the normality

check to see if you can go 3 up and 3 down

what to do if your t value isnt on the table

its symmetric, so take 1-x and find it and make it negative

degrees of freedom not on the table

you can either make it less or infinity. you cant make it more

when to use t tables

population mean with small n

when to use z tables

population proportion or if you have a large n for mean

when it asks for probability of average

use z score but divide standard deviation by sqrtn

significance tests variable

p, not p hat

when it asks for a test statistic

asking for a z score

p value v z score

p values are always between 0 and 1 because its a probabulity

when do you do the +2 and +4 thing

for confidence intervals for PROPORTIONS only

x bar and mean being the same

the center for all x bars is the same as the population mean, but it varies for each sample