A First Test Flashcards
What is the setup for a simple vs. simple hypothesis test in a normal distribution?
A random sample of size n from a normal distribution with unknown mean μ and known variance σ².
What estimator is used to test hypotheses about the mean μ?
The sample mean ( \bar{X} ).
When μ₀ < μ₁, what is the form of the test?
Reject H₀ if ( \bar{X} > c ), where c is a critical value.
What is Alpha (α) in hypothesis testing?
The probability of a Type I error: rejecting H₀ when H₀ is true.
How do you determine c (the critical value) when μ₀ < μ₁?
Use the critical value ( Z_α ) from the standard normal: ( c = μ₀ + Z_α \cdot \frac{σ}{\sqrt{n}} ).
What distribution does ( \bar{X} ) follow when μ = μ₀?
Normal with mean μ₀ and variance ( σ²/n ).
What is the standardized form of ( \bar{X} )?
( Z = \frac{\bar{X} - μ}{σ/\sqrt{n}} )
What is the test rule when μ₀ > μ₁?
Reject H₀ if ( \bar{X} < c ), where ( c = μ₀ + Z_{1−α} \cdot \frac{σ}{\sqrt{n}} )
Why does Z_α change to Z_{1−α} when switching the direction of the test?
Because the rejection region flips from upper tail to lower tail.
What are Type I and Type II errors?
Type I: Rejecting H₀ when it’s true. Type II: Failing to reject H₀ when it’s false.
Why are these examples called simple vs. simple hypotheses?
Both H₀ and H₁ specify exact values for μ.
What is a composite hypothesis?
A hypothesis that specifies a range or set of values for the parameter, not just one value.
What topic is introduced next after this video content?
Type II error, composite vs. composite hypotheses, and p-values.
