Goodness of Fit Tests Flashcards
If Y ∼ Bi(n, p), what is the equation for P[Y = k]?

What are the three conditions for the binomial distribution?
- n independent trials
- two outcomes, success or failure
- probability of success, p, is constant
If Y ∼ Bi(n, p) what is 𝔼[Y] and Var[Y]?
- 𝔼[Y] = np
- Var[Y] = np(1 - p)
If Y ∼ Bi(n, p), for large n what is the distrubtion of Y approximately equal to?
N(np, np(1 - p))
What do k, pi, Oi, and Ei, stand for in the Chi-square test?
- k - the number of outcomes or cells
- pi - the probability of outcome i
- Oi - the number of times outcome i is oberved
- Ei - the expected value of Oi, Ei = npi
What does Var[Oi] and E[Oi] equal?
- Var[Oi] = npi (1 - pi)
- E[Oi] = npi
What is the test statistic for the Chi-square goodness of fit test?

What is the distrubtion of Oi?
Oi ∼ Bi(n, pi)
What does E[𝒳2] equal ? And Why?

Finish this theorm:
Given H0 the sampling distribution of 𝒳2, with k cells, for large n, is approximately …
Given H0 the sampling distribution of 𝒳2, with k cells, for large n, is approximately a chi-squared distribution with k - 1 degrees of freedom.
What does each Ei have to be bigger than to justify a chi-squared approximation?
Ei ≧ 5
What do you do if Ei < 5 for some cells?
Pool the cells and add up the Oi’s
What are the degrees of freedom for a chi-squared distribution when we estimated s independent parameters?
k - s - 1
Do small or large values for 𝒳2 cast doubt on H0?
Large
How can you perform a significance test of H0 at level α, using the Chi-squared goodness of fit test?
- Calculate 𝒳2
- Identify the value Vα such that P[𝒳2 > Vα] = α
- The test rejects H0 at significance level α (or significant at level α) if 𝒳2 > Vα
- If 𝒳2 < Vα say test does not reject at level α (or result not significant at level α)
What is a p-value?
The probability that test statistic is larger than the observed value, if H0 is true
Are small or larger p-values evidence against H0? Provided?
Small, provided that there are other plausible hypotheses which would male the given outcome more likely
How do you calculate a p-value?
P[𝒳2 > Vα] = p٭
What two values do we typically use when pre-assignign the p-value?
0.01 or 0.05
What does nij, ni. and n.j stand for?
- nij is the number of observations for which R = i, C = j
- ni. is the number of observations with R = i
- n.j is the number of observations with C = j
What does πij, πi., π.j stand for?
- πij = P[R = i, C=j]
- πi. = P[R = i]
- π.j = = P[C=j]
In the Chi-squared test for independence what is the null hypothesis, H0?
πij = πi. x π.j
In the Chi-squared test for independence how do we estimate πi. , π.j and πij?

What is the test statistic in the Chi-squared test for independnce?

What is the degrees of freedom in the Chi-squared test for independence?
(r - 1)(c - 1)
If 𝒳2 < 𝒳2(r-1)(c-1) what does the say about the independence of the rows and columns?
They are independent
If 𝒳2 > 𝒳2(r-1)(c-1) what does the say about the independence of the rows and columns?
They are not independent