Podcast
Questions and Answers
What is the relationship between type 1 error (α) and type 2 error (β)?
What is the relationship between type 1 error (α) and type 2 error (β)?
- Increasing α decreases β (correct)
- Increasing α increases β
- Both α and β are independent
- Increasing β decreases α
Failing to reject a null hypothesis means that we can accept it as true.
Failing to reject a null hypothesis means that we can accept it as true.
False (B)
What does the p-value represent in hypothesis testing?
What does the p-value represent in hypothesis testing?
The probability of getting a test statistic as extreme as the observed value, given that the null hypothesis is true.
The probability of a type 2 error, β, increases when the null hypothesis is __________.
The probability of a type 2 error, β, increases when the null hypothesis is __________.
Match the terms with their corresponding definitions:
Match the terms with their corresponding definitions:
What is the primary purpose of a confidence interval?
What is the primary purpose of a confidence interval?
The t-distribution becomes more narrow as the degrees of freedom increase.
The t-distribution becomes more narrow as the degrees of freedom increase.
What does the null hypothesis (H0) represent in hypothesis testing?
What does the null hypothesis (H0) represent in hypothesis testing?
The probability of a type 1 error is denoted by the letter ______.
The probability of a type 1 error is denoted by the letter ______.
Match the types of errors with their descriptions:
Match the types of errors with their descriptions:
Which statement describes a rejection region?
Which statement describes a rejection region?
As the degrees of freedom increase, the t-distribution tends to become less similar to the normal distribution.
As the degrees of freedom increase, the t-distribution tends to become less similar to the normal distribution.
What is the decision if the null hypothesis is true and we reject it?
What is the decision if the null hypothesis is true and we reject it?
Flashcards
Type 1 Error
Type 1 Error
The probability of rejecting a true null hypothesis. It's the risk of making a wrong decision when the null hypothesis is actually correct.
Type 2 Error
Type 2 Error
The probability of failing to reject a false null hypothesis. It's the risk of missing a real effect.
Why we 'fail to reject' rather than 'accept' a null hypothesis
Why we 'fail to reject' rather than 'accept' a null hypothesis
A statistical test cannot prove the truth of a null hypothesis. It can only provide evidence to reject or fail to reject the null hypothesis.
P-Value
P-Value
Signup and view all the flashcards
P-Value Rule
P-Value Rule
Signup and view all the flashcards
Confidence Interval
Confidence Interval
Signup and view all the flashcards
t-Distribution
t-Distribution
Signup and view all the flashcards
Null Hypothesis (H0)
Null Hypothesis (H0)
Signup and view all the flashcards
Alternative Hypothesis (H1)
Alternative Hypothesis (H1)
Signup and view all the flashcards
Test Statistic
Test Statistic
Signup and view all the flashcards
Rejection Region
Rejection Region
Signup and view all the flashcards
Study Notes
Interval Estimation
- The distribution of b₂, the least squares estimator of B₂, is normal.
- A standardized normal random variable is obtained from b₂ by subtracting its mean and dividing by its standard deviation.
- The standardized random variable Z is normally distributed with mean 0 and variance 1.
- An interval (with a specific probability) contains the true parameter value b₂.
- 95% confidence example:
- P(−1.96 < Z < 1.96) = 0.95
The t-distribution
- Replacing σ² with s² creates a random variable with a t-distribution.
- The ratio b₂ − B₂ /se(b₂) follows a t-distribution with n − 2 degrees of freedom.
- In simple linear regression models, if SLR1-SLR6 assumptions hold, then (bₖ − Bₖ) / se(bₖ) follows a t(n − 2) distribution for k = 1, 2.
- The t-distribution is bell-shaped, centered at zero, and has larger variance and thicker tails compared to the standard normal distribution.
- The shape of the t-distribution is controlled by degrees of freedom. As degrees of freedom increases, the t-distribution approaches the standard normal distribution.
Obtaining Interval Estimates
- Use the cumulative distribution function (CDF) and quantiles (inverse CDF) of the t-distribution.
- The t(1-α/2, m) is the (1-α/2) percentile of the t-distribution with m degrees of freedom.
Hypothesis Testing
- Components of hypothesis testing include:
- Null hypothesis (H₀): Specifies a value for a regression parameter (e.g., H₀: bₖ = c).
- Alternative hypothesis (H₁): Specifies a different value or range of values for the parameter (e.g., H₁: bₖ < c, bₖ > c, bₖ ≠ c).
- Test statistic: The calculated value used to determine if the null hypothesis should be rejected.
- Critical value or p-value: Used to make the decision to reject or not reject the null hypothesis.
- We use t-statistics to perform the hypothesis test.
- The rejection region is a set of values for the test statistic that would lead to rejecting the null hypothesis.
- Statistical tests do not prove the truth of the null hypothesis but rather use evidence to either reject or fail to reject it.
- There is a trade-off between the probabilities of committing Type I and Type II errors.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Related Documents
Description
Explore the concepts of interval estimation and the t-distribution in this quiz. Understand how the least squares estimator behaves and how confidence intervals are constructed. Test your knowledge of these key statistical distributions and their applications in linear regression.
