Probability and significance

Question 1

Definition of Probability

Answer 1

• Definition: The likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain).
• Formula: Probability (P) = Number of favorable outcomes / Total number of possible outcomes.
• Example: The probability of flipping a coin and it landing on heads is P(heads) = 1/2 = 0.5.

Question 2

Types of Probability

Answer 2

1. Theoretical Probability
   o Based on the reasoning behind probability (e.g., flipping a fair coin).
   o Example: P(drawing an ace from a deck of cards) = 4/52 = 1/13.
2. Experimental Probability
   o Based on the actual results of an experiment.
   o Example: If in 100 coin flips, heads appears 45 times, the experimental probability is P(heads) = 45/100 = 0.45.
3. Subjective Probability
   o Based on personal judgment or experience rather than exact calculations.
   o Example: Estimating the probability of rain based on weather forecasts.

Question 3

Understanding Significance

Answer 3

• Definition of Statistical Significance: A result is statistically significant if it is unlikely to have occurred by chance alone, as determined by a predetermined significance level (usually set at p < 0.05).
• Purpose: To assess whether the observed effects in data are meaningful or could have happened due to random variability.

Question 4

Null Hypothesis (H0)

Answer 4

• Definition: A statement suggesting no effect or no difference; it is the hypothesis that researchers aim to test against.
• Example: H0: There is no difference in test scores between two teaching methods.

Question 5

Alternative Hypothesis (H1)

Answer 5

• Definition: A statement indicating the presence of an effect or difference; it represents the researcher’s expectation.
• Example: H1: There is a difference in test scores between two teaching methods.

Question 6

P-Value

Answer 6

• Definition: The p-value indicates the probability of observing the data, or something more extreme, if the null hypothesis is true.
• Interpretation:
  o If p < 0.05: Reject the null hypothesis (significant result).
  o If p ≥ 0.05: Fail to reject the null hypothesis (not significant).

Question 7

Type I and Type II Errors

Answer 7

1. Type I Error (False Positive)
   o Occurs when the null hypothesis is rejected when it is actually true.
   o Consequences: Concludes that there is an effect when there is none.
   o Example: Concluding a new medication is effective when it is not.
2. Type II Error (False Negative)
   o Occurs when the null hypothesis is not rejected when it is actually false.
   o Consequences: Fails to detect an effect that is present.
   o Example: Concluding a new medication is ineffective when it actually is effective.

Question 8

Effect Size

Answer 8

• Definition: A quantitative measure of the magnitude of a phenomenon or the strength of a relationship.
• Importance: Provides context for the significance result; a significant p-value does not imply a large effect size.
• Common Measures: Cohen’s d, Pearson’s r, and odds ratios.

Question 9

Confidence Intervals

Answer 9

• Definition: A range of values that is likely to contain the population parameter with a specified level of confidence (usually 95%).
• Interpretation: A 95% confidence interval means if the same study were repeated 100 times, 95 of the intervals would contain the true population parameter.
• Example: A 95% CI for a mean score of 80 ± 5 indicates that the true mean likely falls between 75 and 85.

Question 10

Importance of Probability and Significance

Answer 10

• Decision Making: Helps researchers make informed decisions based on data analysis.
• Research Integrity: Establishes a standard for evaluating findings and reduces the likelihood of erroneous conclusions.
• Communicating Results: Provides a framework for reporting and discussing research findings effectively.