Confidence Intervals Concept

Confidence Intervals (CI):

Definition: A CI is a range of values that likely contains the true population parameter (e.g., mean) with a certain level of confidence (e.g., 95%).

Interpretation: We can be 95% confident that the true population mean falls within the calculated interval. There's a 5% chance it falls outside.

Width: The width of the CI depends on:

Sample size: Larger samples lead to narrower CIs (more precise estimates).

Variability (standard deviation): Less variable data (lower SD) results in narrower CIs.

One-sided vs. Two-sided CI:

A two-sided 95% CI has a 2.5% chance of error in each direction (upper and lower limits).

A one-sided 95% CI can be derived from a two-sided 90% CI by focusing on only one direction (upper OR lower limit). This reduces the total error chance to 5%.

Statements in Statistics:

Statement 1: This refers to the conclusion drawn from the data, typically about a central tendency measure (mean, median) or other statistical summaries.

Statement 2: This addresses the reliability of the conclusion, assessed using the CI width (precision) or p-value (significance level).

Transforming Non-Normal Data:

Certain transformations can improve normality for statistical tests:

Log transformation: Suitable for data skewed to the right, leading to geometric means.

Square root or square transformation: Applicable for data with positive values and heavy tails.

Hypothesis Testing:

Assumptions:

Normally distributed data: Most statistical tests assume normality for accurate results.

Equal variances (homoscedasticity): The variances of the compared groups should be similar.

Null hypothesis (H₀): This hypothesis assumes no difference, no effect, or no relationship between variables. It essentially proposes the observed effects are due to random chance.

Alternative hypothesis (H₁): This hypothesis opposes the null hypothesis. It suggests a difference, an effect, or a relationship exists between variables.

Understanding CIs and Hypothesis Testing:

Zero values and CIs:

If the hypothesized value (often zero) falls within the 95% CI, we fail to reject the null hypothesis (H₀). This indicates non-significance, meaning we lack evidence to disprove the null hypothesis.

If the hypothesized value lies outside the 95% CI, we reject the null hypothesis (H₀) in favor of the alternative hypothesis (H₁). This suggests a significant effect, implying the observed difference is unlikely due to random chance.

Type 1 and Type 2 Errors:

Type 1 error (α): This is the risk of rejecting a true null hypothesis (false positive). The significance level (α) represents the acceptable risk of a type 1 error. Common values are 0.05 (5%) or 0.01 (1%).

Type 2 error (β): This is the risk of failing to reject a false null hypothesis (false negative). The power of a test (1 - β) reflects the ability to detect a true effect.

Significance and p-value:

Significance: A statistically significant result indicates a low probability (usually p < 0.05) of observing the effect by chance alone, strengthening the evidence against the null hypothesis.

p-value: This represents the probability of obtaining a result as extreme as the observed one, assuming the null hypothesis is true. A lower p-value indicates stronger evidence against the null hypothesis.

Sample Size Considerations:

Sample size is crucial for accurate and powerful analyses. It depends on:

Effect size: Smaller effects require larger samples for detection.

Variability: Higher variability necessitates larger samples.

Desired power: Higher power to avoid type 2 errors demands larger samples.

Beta (β) and Power:

Beta (β): This signifies the risk of committing a type 2 error (failing to detect a true effect).

Power: The power of a test is 1 minus the beta (power = 1 - β). It represents the ability to correctly identify a true effect. A higher power is desirable.