Test Statistics Flashcards
What is the goal of hypothesis testing?
To make decisions about a population parameter based on sample data.
What does it mean to test ‘simple versus simple’ hypotheses?
To test two specific values for a parameter, such as H0: μ = 3 vs. HA: μ = 5.
What is a type I error?
Rejecting the null hypothesis when it is actually true.
What is the significance level (alpha) of a test?
The probability of making a type I error.
What are common choices for alpha?
0.05, 0.01, and 0.1.
What is the form of the test for H0: μ = 3 vs. HA: μ = 5?
Reject H0 if sample mean x̄ > c, where c is a critical value.
What is the form of the test for H0: μ = 5 vs. HA: μ = 3?
Reject H0 if sample mean x̄ < c, where c is a critical value.
What affects the probability of type II error?
The critical value c; increasing c makes type II error more likely.
What affects the probability of type I error?
The critical value c; decreasing c makes type I error more likely.
How is the sampling distribution of x̄ determined?
x̄ is normally distributed with mean μ and variance σ²/n.
What happens when you standardize x̄?
You convert it to a standard normal using Z = (x̄ - μ) / (σ/√n).
What is the role of the standard normal distribution in testing?
It is used to find critical values and rejection regions.
How do you find a critical value using R?
Use qnorm(1 - alpha) for right-tailed tests, or qnorm(alpha) for left-tailed.
What is the test statistic formula for x̄?
Z = (x̄ - μ) / (σ / √n).
What does a critical value represent?
A cutoff that defines the rejection region for a hypothesis test.
What is the conclusion rule in hypothesis testing?
Reject H0 if the test statistic falls in the rejection region.
What is the rejection region for alpha = 0.05 (left-tailed)?
x̄ < critical value found using qnorm(0.05).
Why is hypothesis testing not foolproof?
Because random samples can lead to wrong conclusions even when tests are correctly performed.
What is the test conclusion for H0: μ = 5 vs. HA: μ = 3 with n = 10, σ² = 4, α = 0.05?
Reject H0 if x̄ < 3.9596.
What is the value of the critical point Z0.05?
-1.645.
