Introduction to Inference

Null Hypothesis

• a statistical statement that assumes there is no effect, no difference, or no relationship between variables being studied
• the default position that any observed difference is due to random chance rather than a real effect
• the question is whether there is enough evidence to reject the null hypothesis

Hypothesis Testing

• goal: reject the null hypothesis in favor of an alternative hypothesis that suggests there is a real effect
• We only reject the null hypothesis if we find strong statistical evidence against it

Think of it like a criminal trial - the defendant is presumed innocent (null hypothesis) until proven guilty beyond a reasonable doubt. Similarly, we assume no effect exists until we have strong statistical evidence suggesting otherwise.

Null Hypothesis

\(H_0\) (Null Hypothesis): The variables region and happiness score are independent. The difference in scores across different regions was due to natural variability inherent in the population.

\(H_1\) (Alternative Hypothesis): The variables region and happiness score are not independent. The difference in scores across different regions was not due to natural variability.

Type I and Type II errors

Type I Error (False Positive):

• Rejecting the null hypothesis when the null hypothesis should not have been rejected

Type II Error (False Negative):

• Failing to reject the null hypothesis when it should have been rejected

A courtroom analogy:

• Type I Error: Convicting an innocent person
• Type II Error: Letting a guilty person go free

p-value

• the probability of type I error
• lower the p-value is, the lower the probability of getting that result if the null hypothesis were true
• alpha level α is usually 0.05 (often the alpha level is adjusted if more than one statistical test is run)

“p-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone” – American Statistical Association (ASA)