Statistical Hypothesis Testing

Statistical Hypothesis Testing

• Alpha Level (α) Specification

  • The alpha level should typically be specified before conducting the hypothesis test.
  • Commonly set at 0.05 or 5%.

• Rejection Criteria

  • If the p-value obtained is less than the alpha level (e.g., 5%), reject the null hypothesis.
  • Example in the transcript:
  • The peak value found is $0.000$, which is less than 5%. This leads to the conclusion to reject the null hypothesis.

• P-Value and Alpha Comparison

  • If the p-value is less than alpha, state: "Since the p-value is less than alpha, we reject the null hypothesis."

• Confidence Interval

  • Defines the range of values that do not fall into the rejection region.
  • If the hypothesized value falls within this range, do not reject the null hypothesis.

Regression Analysis

• Interpreting Coefficients

  • Each coefficient in a regression model needs interpretation.
  • E.g., beta coefficients (β) signify the relationship between predictor variables and the dependent variable (Y).
  • Suggestion: If one variable can be manipulated, it should be the number of children due to its significant impact.

• Significance of Coefficients

  • It’s essential to determine if all coefficients are significant or not.

Residuals and Homoscedasticity

• Residual Plotting

  • Residuals should be fitted against the predictor variables to assess model fit.
  • Analysis of residual plots is crucial to check for homoscedasticity (constant variance of residuals).
  • Observations showed that residuals were primarily around 10 and 15, suggesting randomness in the distributed values.

• Variable Consideration in Models

  • Low residual values might suggest insufficient variables are considered in the model.

Polynomial Regression Models

• Types of Regression Models

  • Multiple regression is a type of polynomial regression where variables can have various powers.
  • For a polynomial regression involving a predictor variable $x$, it can be represented as:
  • $y = \beta_0 + \beta_1 x + \beta_2 x^2 + \beta_3 x^3 + … + ext{error}$
  • Discussion of quadratic vs. cubic forms:
  • Quadratic (second order) reflects a U-shaped graph, while cubic might cross the x-axis three times, creating more complexity.

• Graphical Representation

  • When plotting, the form suggests whether the relationship is linear, quadratic, etc.
  • An interaction term should be considered if the effect of one variable depends on the level of another variable.

Modeling with Multiple Predictors

• Two Independent Variables

  • If $p = 1$ (linear), the model is:
  • $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ext{error}$

• Interaction Terms

  • When there are two independent variables, potential interaction terms might need to be included.
  • The model format becomes:
  • $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 (x_1 * x_2) + ext{error}$

Case Study: Regression Model for Fast Food Restaurant

• Family Income and Age of Children

  • Tasked to determine the relationship between family income and age of children on spending behavior at a fast-food restaurant.
  • Graph plotting revenue against income suggests a quadratic relationship:
  • Families with middle income levels tend to spend more than those at the extremes.

• Quadratic Model Form

  • The model must reflect this quadratic relationship:
  • $ext{Revenue} = \beta_0 + \beta_1 ( ext{Income}) + \beta_2 ( ext{Income}^2) + \beta_3 ( ext{Age}) + \beta_4 ( ext{Age}^2) + … + ext{error}$
  • Interaction terms can also be included where relevant, reflecting more complex behaviors.

Model Significance Testing

• Determining Significance

  • All coefficients should be tested for significance to confirm their relevance in the model.
  • High p-values (like p > 0.05) on coefficients indicate a lack of significance, possibly hinting at multicollinearity issues.

• Use of Indicator Variables

  • When categories are present (e.g., colors of cars), an indicator variable can be introduced to account for these categorical influences.