Notes on Inferential statistics

Inferential statistics involves using statistical analysis to draw insights and infer conclusions about a population by testing specific hypotheses.
We take a sample from the population, collect data, and use statistical procedures to determine if there are differences between groups or changes in performance.
We infer any changes or differences observed in the sample to the larger population because the sample should represent the overall population.
This process is facilitated through null hypothesis testing, a formal method to test the evidence supporting a research hypothesis.
This helps determine if observed differences are due to chance, sampling error, or reflect a true effect of an intervention or a true difference between groups.

T-test: Compares two groups.
F-test: Compares multiple groups.
Correlation and Regression: Examines relationships between different variables (will be covered in the next module).

The null hypothesis ((H_0)) posits that there is no effect or no difference in the population.
The alternative hypothesis asserts that there is an effect or difference in the population.

Research Question: Does strength level differ between competition levels in rugby league players?
Research Hypothesis: NRL players are stronger in the 1RM bench press than players in Queensland Cup or Brisbane City competition.
Null Hypothesis ((H_0)): The means of the group strength levels do not differ.
Alternative Hypothesis: The means of the groups are not the same.

An F-test is used to analyze the data, providing an F statistic which is then compared against a critical value.
If the F statistic is greater than the critical value, the null hypothesis is rejected, supporting that there is evidence of a difference in means.
If the F statistic is less than the critical value, we fail to reject the null hypothesis, meaning there is not enough evidence to support a difference.
Failing to reject the null hypothesis doesn't confirm there is no difference in the population, it simply suggests there isn't enough evidence to prove that.

The P-value describes the probability of obtaining a test statistic as extreme as the observed results, assuming the null hypothesis is true.
Typically, the P-value is set at 0.05, representing a 5% chance of randomly obtaining an extreme F value.
If the P-value is below 0.05, we reject the null hypothesis.
This controls for Type I error, also known as a false positive, where the null hypothesis is rejected when it is actually true.

Type I error occurs when we reject the null hypothesis, but the null hypothesis is actually true.
A good decision would be not to reject the null hypothesis.
The P value controls the Type I error.
If the p-value is less than $0.05$ , the null hypothesis is rejected, and the difference or relationship is not due to sampling error or random chance.
If the p-value is greater than the cutoff for Type I error, we fail to reject the null hypothesis, and any differences or relationships are considered a result of sampling errors.

Even if the P-value is less than 0.05 and statistically significant, it is important not to rely solely on this value.
A statistically significant result does not always mean the performance change or difference is practically important.
Effect sizes should also be considered to inform the decision-making process, as they help determine the practical importance of the observed differences or changes.