Topic 7 Statistical Tests Flashcards
True/False
The level of significance is denoted by 1−α.
False — that’s the level of confidence.
What distribution is used for confidence intervals on variance?
Chi-squared distribution
The hypothesis stating “no change” is known as the ________.
Null hypothesis
True/False
Accepting the null hypothesis means the hypothesis is definitely true.
False — it means we do not have enough evidence to reject it.
Which of the following is a valid alternative hypothesis?
a) H1: µB = µA
b) H1: µB < µA
c) H1: µB ≥ µA
d) H1: µB ± µA
b) H1: µB < µA
What is the test statistic for population B?
T = (X_B − μ_B) / (σ / √n)
When is a one-sided test used instead of a two-sided test?
When prior knowledge or interest is only in one direction of change.
What is the critical region in a statistical test?
The range of values for which the null hypothesis is rejected.
What are the critical regions for the two and one-sided tests?
Two-sided test critical region: |Z| > z_{α/2}
One-sided test critical region: Z > z_α
What is a p-value?
The probability of observing a test statistic as extreme or more extreme than the observed value under the null hypothesis.
How do you calculate the p-value for one-sided test?
One-sided: p = P(T > t₀) = 1 − Φ(t₀)
What is a Type I error?
Rejecting the null hypothesis when it is true.
True/False
Type I error is denoted by α.
True
What is a Type II error?
Accepting the null hypothesis when it is false.
The power of the test is defined as 1 − _____.
β
In symmetric testing, what do we assume about the samples?
a) Same size
b) One known mean
c) Equal importance
d) Same distribution
c) Equal importance
If σ is known in both populations, the test statistic follows a ________ distribution.
Normal
Formula for when sigmaA and sigmaB is known?
Z = [(X_A − X_B) − (μ_A − μ_B)] / √[(σ_A² / n_A) + (σ_B² / n_B)]
What is the formula for when variances are unknown but are equal?
T = (X_A − X_B) / [S_p * √(1/n_A + 1/n_B)] ∼ t(n_A + n_B − 2)
where
S_p² = [(n_A − 1) * S_A² + (n_B − 1) * S_B²] / (n_A + n_B − 2)
What is a goodness-of-fit test used for?
To determine whether a sample fits a specified distribution.
In a χ² test, what does
k represent?
The number of estimated parameters in the distribution (e.g. 2 for Normal).
What is the formula for Chi-squared goodness of Fit test?
Cₙ = ∑ from i=1 to l of ((Oᵢ − Eᵢ)² / Eᵢ)
Cₙ ∼ χ²(l − 1 − k)
True/False
You should avoid bins with expected frequencies below 5 in χ² tests.
True
For the Chi-sqaured test what is the critical region?
Cₙ > χ²_{l−1−k, 1−α}
What is the KS test used for?
Comparing empirical CDF with theoretical CDF to test goodness-of-fit.
What does the two-sample KS test check?
Whether two samples come from the same population without assuming a specific distribution.
What is the formula for two-sample KS test?
D_{m,n} = supₓ |Fₘ(x) − Fₙ(x)|
√(mn / (m + n)) · D_{m,n} → tabulated
