Regression, Correlation and Hypothesis Testing

From Statistics 1

Interpret diagrams for single-variable data, including an understanding that an area in a histogram represents the frequency
Connect grouped frequency tables to probability distributions
Interpret scatter diagrams and regression lines for bivariate data, including recognition of scatter diagrams which include distinct sections of the population
Understand informal interpretation of correlation
Understand that correlation does not imply causation
Be able to calculate standard deviation, including from summary statistics
Recognise and interpret possible outliers in data sets and statistical diagrams
Be able to clean data, including dealing with missing data, errors and outliers

Correlation:

Coefficient Calculation: Calculate and interpret the product-moment correlation coefficient (Pearson’s r).
Causation vs. Correlation: Understand the difference between causation and correlation, and the dangers of concluding based solely on correlation.

Regression:

Linear Regression:
- Understand the principles behind linear regression to model relationships between two variables.
- Calculate the equation of the regression line of y on x.
- Interpret the gradient and y-intercept of the regression line in context.
- Use the regression line to make predictions and understand the limitations of extrapolation.
Exponential Models via Logarithmic Transformation:
- Identify when data appears to fit an exponential model.
- Understand how to transform the exponential model y=ab^x or y=ax^b using logarithms to achieve a linear form.
- Perform a logarithmic transformation and plot the transformed data.
- For the transformed data, calculate the linear regression line.
- Reverse-transform the linear regression equation back to its exponential form.

Correlation Hypothesis Testing:

Hypotheses Formulation: Formulate the null and alternative hypotheses for correlation testing.
Critical Value: Use statistical tables or technology to determine critical values for a given significance level.
Conduct Test: Perform a hypothesis test to ascertain the significance of the correlation between two variables, given a dataset.
Results Interpretation: Analyze and interpret the results of the hypothesis test in context.

Correlation:

Coefficient Calculation:
- Introduce the formula for Pearson’s r.
- Provide worked examples of its calculation.
Causation vs. Correlation:
- Discuss real-world examples where correlation does not imply causation.
- Emphasize the importance of external factors and confounding variables.

Regression:

Linear Regression:
- Define the terms “gradient” and “y-intercept.”
- Derive and explain the formula for linear regression.
- Provide exercises that involve making predictions based on a regression line.
Exponential Models via Logarithmic Transformation:
- Introduce the properties of logarithms and their application to data transformation.
- Demonstrate the transformation of exponential models to linear models.
- Provide examples of reverse transformation to retrieve the exponential model after regression analysis on transformed data.

Correlation Hypothesis Testing:

Hypotheses Formulation:
- Define the null hypothesis (H₀) and alternative hypothesis (H₁).
- Discuss scenarios where one would want to test the significance of a correlation.
Critical Value:
- Show students how to find critical values using statistical tables.
- Introduce how calculators can aid in this process.
Conduct Test:
- Walk through the steps of conducting a hypothesis test, emphasizing the importance of each step.
- Offer varied examples and practice problems for students to try.
Results Interpretation:
- Discuss the potential outcomes of a hypothesis test and their implications.

Correlation:

All Correlation is Causation: Students often believe that if two variables are correlated, one must cause the other.
Strength of Correlation: Students might think a correlation coefficient of r=0.5 means one variable is “half caused” by the other.
Perfect Correlation: Some believe that a correlation coefficient of exactly 1 or -1 means the data points fall perfectly on a straight line.

Regression:

Extrapolation: Students might believe that a regression model is equally accurate everywhere, even beyond the range of observed data.
Intercept Significance: The y-intercept always has real-world significance.
Linearity Assumption: Linear regression is suitable for all types of data distributions.
Transformed Regression: After transforming exponential data to linear form using logarithms and creating a linear model, the data is now “linear”.

Correlation Hypothesis Testing:

Significance = Importance: A statistically significant result means the finding has practical or real-world significance.
P-value Misunderstanding: A p-value represents the probability that the null hypothesis is true.