S2 DAPR1 lecture 3: hypothesis testing - critical values

what are the goals of hypothesis testing

estimating a population parameter, and test whether a hypothesised parameter value is possible

how to estimate the population parameter

use average of observations in sample (xbar as estimate)

what does standard error measure

the average distance of a sample mean from the population mean - the precision of how reliable this estimate is

confidence interval

gives a range of plausible values for the population mean

CI calculation

xbar +/- t\* x s/(sqrt(n))

R code for a 95% CI

qt(c(0.025, 0.975), df = n-1)

how to carry out testing whether a hypothesises parameter value is plausible

state null and alternative hypotheses; compute test statistic; for testing a population mean use the t-statistic

what is a test statistic used to measure

some sort of distance between the sample data and the null hypothesis

what is the t-statistic

distance of the sample mean from the hypothesised parameter value, measured in units of standard error

what form does the test statistic take when testing a mean

a distance between the observed and hypothesised mean, in units of the SE

the test statistic used for testing a mean is the

t-statistic or t-score

t statistic equation

t = (xbar - mu0)/(s/(sqrt(n))

after computing the t-statistic for the observed sample, we have to decide if this is surprising or not, how do we do this

we look at what all the possible values of the t-statistics are when H0 is true - i.e, what is the null distribution

how is a null distribution formed

take multiple samples of the same sample size, and for each one compute the average (xbar), SD (s) and t-statistic - the null distribution is the histogram of all these t statistics

description of a t distribution

has more variability than a standard normal

when H0 is true

the t-statistic follows a t(n-1) distribution

if the observed samples gives a t statistic that is unlikely when H0 is true

we begin to doubt the null hypothesis

how to calculate tobs

first compute the mean, and then doing : tobs

for a two-sided test, how to compute the p-value

as twice the area to the right of the observed value

what is the p value

the probability of observing a t-statistic having at least the same distance from 0 as the observed t-statistic when H0 is true

we then compare the p-value to the SL

if the p-value is less than the SL, we reject the null hypothesis, in favour of the alternative hypothesis

equivalent approach to p-values

critical values

how to use critical values

instead of comparing the area of alpha (SL) to the area of the p-value, we can compare the corresponding t-statistic along the x-axis

what are the critical values for a= 0.05

the t-values that cut an area of 0.025 to the left and 0.025 to right right

how are t values denoted

\+t and -t

when do we accept the null using CVs

when -t < t < +t (less extreme than CVs)

critical values method

compute the sample mean (xbar) and tobs; find the critical values for a t(n-1) distribution with SL of 96%; then check how extreme the observed t value is compared to the critical values

how to compute the pvalue using r

pvalue

how to compute observed t value using r

t_obs

how to compute t\*

tstar