9. Testing Hypotheses and Assessing Goodness of Fit Flashcards
Null Hypothesis
Definition
-the so-called null-hypothesis refers to some basic premise to which we will adhere unless evidence from the data causes us to abandon it
Null Hypothesis
In General
- in general:
- -we specify the pmf p(x;θ) [pdf f(x;θ)] but there is doubt about the value of parameter θ
- -θ is some element of a specified parameter space Θ
- -Ho specifies that θϵΘo⊂Θ
- -Ho is true if θϵΘo but false if θ∉Θo
Alternative Hypothesis
Definition
-as a convention,
-the compliment of Θo is denoted as:
Θ1 = Θ - Θo
Null Hypothesis and Alternative Hypothesis
Symbols
- the original hypothesis that θϵΘo is called the null hypothesis and denoted Ho
- the hypothesis that θϵΘ1 is referred to as the alternative hypothesis and denoted by H1
Hypothesis Test
Definition
- a hypothesis test is conducted using a test statistic with distribution known under the null hypothesis, Ho
- a hypothesis test is used to consider the likely truth of the null hypothesis as opposed to a stated alternative hypothesis
p-value
Definition
- the p-value (or significance level) is the probability of the test statistic taking a value at least as extreme as its observed value
- the p-value is calculated assuming that the test statistic is distributed as if the null hypothesis was true
t-test
Definition
-a t-test is used for observations independently drawn from a normal distribution N(μ,σ²) with unknown parameters
-given the sample mean X_ and the sample variance S², the test statistic is:
T = √n(X_-μo)/S² ~ t(n-1)
-under the null hypothesis Ho: μ=μo
z-test
Definition
-for observations drawn from a normal distribution N(μ,σ²) but with σ² known
-we use a Z-test of Ho: μ=μo with test statistic:
Z = √n(X_-μo)/σ ~ N(0,1)
-under Ho
Paired t-test
Definition
-suppose that we have pairs of random variables (Xi,Yi) and that Di=Xi-Yi, i=1,…,n is a random sample from a normal distribution N(μ,σ²) with unknown parameters
-we use the test statistic
√n(D_-μo)/Sd ~ t(n-1)
-under the null hypothesis Ho: μ=μo
-here Sd² is the sample variance of the difference Di
Two Sample t-test
Outline
-consider two random samples X1,…,X, and Y1,…,Yn which are independent, normally distributed with the same variance
-the null hypothesis is Ho: μx=μy
-under Ho we can construct a test statistic T such that:
T ~ t(m+n-2)
Two Sample t-test - Two Samples Same Variance
Definition
-use test statistic
T = (X_-Y_) / (S√[1/m + 1/n]) ~ t(m+n-2)
Critical Region
Definition
-suppose we have:
–data, x_ : x1,x2,…,xn
–we have set a decision rule in advance: rejecting the null hypothesis when the p-value is less than or equal to a fixed α
-C1 is called the critical region of the test and α is the significance level
-note that α is the probability of rejection of Ho given that it is true
-in other words:
P(x1,x2,…,xnϵC1 | Ho) = α
Type I Error
Definition
-the error of rejecting the null hypothesis when it is actually true
Type II Error
Definition
-the error of not rejecting the hypothesis when it is actually false
Probability of Type I Error
P(Type I error) = P(xϵC1 | Ho) = α
