Statistics and Hypothesis Testing

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Questions and Answers

What should you do if you encounter a bi-modal distribution for an outcome variable in a simple linear regression?

Discard the linear model (correct)
Transform the outcome variable
Use a non-linear model
Check for outliers

What can be inferred from a graph of the response variable that looks like a square root?

The data is normally distributed
The data is skewed to the right
The data is linear
A transformation is needed (correct)

What is the purpose of using boxplots in ANOVA?

To determine the statistical significance of group means
To identify outliers in the data
To visualize the distribution of each group (correct)
To compare the variability of each group

What is the null hypothesis in the F-test for equal means?

The means are the same (C) Signup and view all the answers

What is the objective of ANOVA?

To determine which groups have statistically significantly different means (D) Signup and view all the answers

What is the consequence of obtaining a lower p-value than the threshold in the F-test for equal means?

Reject the null hypothesis and conclude that the means are different (D) Signup and view all the answers

What is the purpose of the Box-Cox test?

To determine the best power transformation for the data (C) Signup and view all the answers

What is the assumption of a linear regression model?

The data is linear (C) Signup and view all the answers

What is the purpose of the Tukey method?

To compare two groups at a time (A) Signup and view all the answers

Which assumption of ANOVA is related to the relationship between the predictor and response variables?

Linearity (C) Signup and view all the answers

How can we check the constant variance assumption of ANOVA?

By plotting the residuals by treatment group (B) Signup and view all the answers

What is the purpose of a boxplot in exploratory analysis?

To visualize the distribution of a single variable (C) Signup and view all the answers

What does it mean when we say that purchasing power parity (PPP) is not 'robust' to unusual economic or political conditions?

PPP becomes weaker in predicting exchange rates in certain circumstances (A) Signup and view all the answers

What is the purpose of residual plots in regression analysis?

To check for constant variance (D) Signup and view all the answers

What is the main purpose of hypothesis testing?

To determine the significance of a predictor variable (B) Signup and view all the answers

What is the goal of model fit assessment?

To evaluate the goodness of fit of a model (A) Signup and view all the answers

What type of test is used when testing for equal means in ANOVA?

Two-tailed test with two parameters (D) Signup and view all the answers

What is the alternative hypothesis when testing for equal means in ANOVA?

At least one pair of means are statistically significantly different from each other (C) Signup and view all the answers

How do we check the assumption of independence in regression analysis?

By checking the residuals (C) Signup and view all the answers

What is the relationship between the prediction interval and the confidence interval in regression analysis?

The prediction interval is always wider than the confidence interval (B) Signup and view all the answers

What type of model is ANOVA?

A linear regression model with categorical variables (D) Signup and view all the answers

Which of the following statements is true about ANOVA and SLR?

ANOVA and SLR share all model assumptions in common except for constant variance (A) Signup and view all the answers

What is the purpose of a two-tailed test in hypothesis testing?

To test if the mean is different from a certain value (C) Signup and view all the answers

When conducting a one-tailed test using a Z distribution, what is the critical region?

One tail of the distribution (A) Signup and view all the answers

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Study Notes

Hypothesis Testing

A one-tailed test is conducted using a Z distribution or a t distribution.
A two-tailed test is conducted using a Z distribution or a t distribution.

Diagnostics and Model Evaluation

The assumption of independence is limited in checking, and it can be seen if it is broken by checking the residuals.

ANOVA

The F-test for equal means is a two-tailed test with two parameters (each one is a different measure for degrees of freedom).
In ANOVA, when testing for equal means, the alternative hypothesis is that at least one pair of means are statistically significantly different from each other.
ANOVA and SLR share all model assumptions in common except for constant variance.
ANOVA is a linear regression model with categorical variables.
Pairwise comparison can be done using the Tukey method.

Model Fit Assessment

Linearity is an assumption of ANOVA.
To check the constant variance assumption of ANOVA, residuals should be plotted by treatment group to assess whether the groups individually have different variability.

Data Example 1: Purchasing Power Parity (Part 1)

A boxplot can be used to show the response variables for different groups (developed and developing countries).
A visual difference in the response variables shown in the boxplots provides strong evidence that two simple linear regressions (one for Developing and one for Developed countries) would be better than a single model.

Data Example 1: The Theory of Purchasing Power Parity (Part 2)

The conclusion of the analysis is that purchasing power parity (PPP) is not 'robust' to unusual economic or political conditions.

Simple Linear Regression

A bi-modal distribution for an outcome variable does not necessarily mean the linear model should be discarded.
A graph of the response variable that resembles a square root (constantly increasing, but decreasing in the rate of increase) may suggest a transformation is needed.

ANOVA: Basic Concepts

Boxplots can be used to see whether group means are statistically significantly different from each other, but it is not definitive.

Test for Equal Means

The F-test for equal means has a null hypothesis that the means are the same.
If the p-value is lower than the threshold (normally 0.05), the null hypothesis is rejected, and it is concluded that the means tested as part of the F-test are not equal.