Statistics - Hypothesis Testing Flashcards
what type of situations would you apply hypothesis testing
is a population parameter different from a predicted value, difference between two observations on each of the elements of a population (paired measurements), difference between two groups of a population
what is a hypothesis
a statement about the value of a population parameter or probability distribution
what is H0 and what does it typically indicate
the null hypotheses which typically indicates no difference
what is H1 and what does it typically indicate
the alternative hypothesis which typically indicates some difference eg <, >, ≠
what is the test idea
take a random sample, test if there is sufficient evidence to reject the null hypothesis in favour of the alternative hypothesis
what is a type I error
rejecting H0 when it should have been accepted
what is a type II error
accepting H0 when it should have been rejected
what is the meaning of the confidence level in hypothesis testing
α is the maximum probability with which we are willing to risk a type I error, ie rejecting H0 when it is correct
what are the seven general steps to a hypothesis test
- decide null hypothesis
- decide alternative hypothesis
- decide significance level
- calculate observed value of test statistic
- find tabulated test statistic
- compare values from 4 and 5
- state conclusion
what is μ0
the predicted mean
for a one sample t-test what is the calculated test statistic Calc T
(xbar - μ0)/(s/sqrt(n))
for a one sample t-test what is the tabulated test statistic Tab T for one sided tests
tn-1;α
for a one sample t-test what is the tabulated test statistic for a two sided test
tn-1;α/2
for a one sided t-test what is the rejection criteria for H0 if H1 is μ > μ0
Calc T>Tab T
for a one sided t-test what is the rejection criteria for H0 if H1 is μ < μ0
Calc T<-Tab T
for a two sided t-test what is the rejection criteria for H0 if H1 is μ ≠ μ0
|Calc T|>Tab T
what is the p-value of the test
assuming H0 is true, the probability that the calculated test statistic will take on values as extreme or more extreme than the observed test statistic computed from the sample data
a small p-value computed from the sample data shows
strong evidence against H0
what is the p-value for the one sided t-test H1:μ > μ0
P(T>Calc T)
what is the p-value for the one sided t-test H1:μ < μ0
P(T<-CalcT)
what is the p-value for the two sided t-test
P(T<-Calc T)+P(T>CalcT)
what is the rejection criteria in terms of the p-value
H0 is rejected if p-value>α, where α is the confidence level
Paired data (X1,Y1),…,(Xn,Yn) are pairs of random variables satisfying what criteria
Xi and Yi can be related to each other and Xi is independent of all the rest of the variables
how do we go about a t-test for paired data
reduce it to a two sided one sample t-test by calculating a new r.v which shows the difference, calculate xbar and s from this set of values
what is the null hypothesis for paired data t-test
H0: μd = 0
what is the alternative hypothesis for paired data t-test
H1 : μd ≠ 0
what is being tested for in a two sample t-test
a difference in the population means
what is the null hypothesis for a two sample t-test
H0: μ1 = μ2
what is the one sided alternative hypotheses for a two sample t-test
H1: μ1 > μ2
what is the two sided alternative hypothesis for a two sample t-test
H1 : μ1 ≠ μ2
for a two sample t-test how do you obtain the case μ1< μ2
by exchanging X1 and X2
what is the calculated test statistic Calc T for a two sample t-test
(xbar1-xbar2)/(sp*sqrt(1/n1+1/n2))
what is sp^2
the pooled variance
what is the pooled variance
the weighted average of the two sample variances s1^2 and s2^2
for a two sample t-test what does T~tn1+n2-2 mean
the samples of X1 and X2 are normally distributed and the population variances σ1^2 and σ2^2 are equal and they have a t-distribution with v=n1+n2-2 degrees of freedom
for a two sample t-test how can we know if σ1^2 and σ2^2 are equal
we can assume this if 1/2<s1^2/s2^2<2
what is the tabulated test statistic Tab T for a two sample, one-sided t-test
tn1+n2-2;α
what is the tabulated test statistic Tab T for a two sample, two-sided t-test
tn1+n2-2;α/2
what are the rejection criteria for H0 for two sample t-tests
the same as 1 sample t-tests if Calc T or |Calc T| are >= Tab T
when would we use a z-test
when the degrees of freedom is more than 30
what is the only difference between the t-test and the z-test
the tabulated statistic Tab Z
what is the tabulated statistic for the z-test
the critical values for the standard normal which is the critical values of the t distribution with infinite degrees of freedom
what is Tab Z for one sided z-tests
zα
what is Tab Z for two sided z-tests
zα/2
does Tab Z change for a two sample z-test
no, Tab Z does not depend on sample size
