probability and stat sig

Statistical tests conclude with a number, known as the calculated value.
This value is crucial for determining whether the research result is statistically significant, influencing the decision to accept the alternative or null hypothesis.
Understanding statistical tests necessitates grasping two key concepts: probability and significance.

Definition: A measure of the likelihood that a specific event will occur, where:
- 0 indicates statistical impossibility
- 1 indicates statistical certainty

Definition: A statistical term indicating the certainty that a difference or correlation exists.
A significant result enables researchers to reject the null hypothesis.

Definition: The numerical threshold within hypothesis testing that delineates acceptance from rejection regarding the null hypothesis.

Type I Error: Incorrect rejection of a true null hypothesis, commonly referred to as a false positive.
Type II Error: The failure to reject a false null hypothesis, referred to as a false negative.

Problem: What is the probability that two people at a football match share the same birthday?
- Participants: 23 (including the referee).
- Probability calculation:
1. Chance that any two individuals share a birthday: 1 in 365.
2. Total handshake combinations: 253 (pairs among 23 people).
3. Probability: $\frac{253}{965} \approx 0.69$ or 69%.
  - Conclusion: This results in surprising likelihood of shared birthdays among participants.

Researchers establish a hypothesis as the starting point for their investigation.
Example of a hypothesis:
- Null Hypothesis (H0): Claims no difference exists between two groups (e.g., participants who drink 300 ml of SpeedUpp and those who drink 300 ml of water).
- Alternative Hypothesis (H1): Indicates a prediction of a resulting difference (e.g., increased word count after drinking SpeedUpp compared to water).

Statistical tests work primarily on a probabilistic basis rather than absolute certainty.
A significance level is employed, serving as the threshold at which a researcher can claim the discovery of a significant difference or correlation.
The typical significance threshold in psychology is $p \leq 0.05$ (5% probability).
- This number defines the likelihood that the observed effect occurred by chance.

Once a statistical test is performed, it generates a calculated value (observed value).
For checking statistical significance, this must be compared against a critical value—determining whether the null hypothesis can be rejected. Each statistical test has a unique table of critical values that guides this comparison.

One-tailed or Two-tailed Test:
- If the hypothesis is directional, a one-tailed test is used.
- Non-directional hypotheses require a two-tailed test, which doubles the probability levels due to being a more conservative prediction.
Number of Participants:
- Represented as the N value in statistical tables. Degrees of freedom (df) may be relevant for certain tests.
Level of Significance:
- The default level is 0.05; however, some contexts may require a more stringent level (e.g., 0.01) for critical studies like drug testing.

The conventional level of significance in psychology is 0.05.
More stringent significance levels are occasionally employed, e.g., for drug trials to mitigate human costs, necessitating the assessment at a higher rigor prior to testing "$p \leq 0.05$".
- Significance thresholds offer an insight into the statistical likelihood of chance occurrence in research findings.

Type I Error: Occurs when researchers incorrectly reject a true null hypothesis, leading to a false conclusion of a significant effect.
Type II Error: Happens when researchers accept a false null hypothesis—a missed opportunity to identify a true effect or difference.
- Probability assessments in everyday life parallel these examples (e.g., 50% chance predictions).

Heightened significance levels can increase Type I error risks (e.g., using $p=0.1$ ).
Stringent premises invite risks of Type II errors when useful data is overlooked (e.g., significance levels of $p=0.01$ ).
The 5% threshold strikes a favorable balance in psychology research, minimizing both Type I and Type II error rates.

A researcher assesses the efficacy of a new medication (Anxocalm) against anxiety by contrasting two groups:
- Treatment group using Anxocalm
- Control group receiving a placebo.
Given potential side effects (e.g., headaches, nausea), testing on human participants must be carefully regulated to maintain ethical standards.
- Higher significance levels (e.g., 1%) may apply based on the critical value established at 5% before concluding.

The Rule of R: Identifies statistical tests where the calculated value must meet or exceed the critical value. Tests associated with an R (e.g., Pearson's r) are indicative of this requirement.
Conversely, tests without an R may operate on conditions requiring the calculated value to be less than the critical value.

Pregnancy tests are not completely reliable; thus, it is beneficial for women to undertake multiple tests to confirm pregnancy status.

Explain Type I and Type II Errors: Difference outlined, emphasizing acceptance and rejection implications.
Critical Value Definition: Provide insights into its role in statistical decision-making.
Accepted Significance Level: Reveal the standard psychological research threshold for understanding statistical significance (p = 0.05).