Inferential Statistics Fundamentals

Inferential Statistics

• Used to describe the population based on the observations of the sample
• Deals with making inferences about the population parameters (μ, σ) based on sample statistics (x, s)

Population and Sample Characteristics

• Population characteristics: parameters (μ = mean, σ = standard deviation)
• Sample characteristics: statistics (x = mean, s = standard deviation)

Central Limit Theorem (CLT)

• States that the distribution of sample means will be approximately normally distributed, regardless of the population's original shape, as long as the sample size is large enough

Hypothesis Testing

• Null hypothesis: states that the treatment has no effect, meaning any observed differences are due to random chance
• Alternative hypothesis: the treatment has an effect, indicating that observed differences are due to the treatment

Decision Making in Hypothesis Testing

• Retain the null hypothesis: if observed data are consistent with the null hypothesis, and the results can be reasonably attributed to chance
• Reject the null hypothesis: if the likelihood that the observed results are due to chance is very small, and the results are statistically significant

Significance Level and Alpha

• Significance level (alpha): probability threshold set by the researcher that defines the boundaries for rejecting or retaining the null hypothesis
• Typically set at 0.05

Rejection Region and Critical Value

• Rejection region: area in the tail(s) of the sampling distribution that represents the sample means that are highly unlikely to occur if the null hypothesis is true
• Critical value: threshold or cutoff point on the test statistic distribution that defines the boundaries of the rejection region

One-Tailed and Two-Tailed Tests

• One-tailed test: used when there is a directional alternative, and the rejection region is located entirely in one tail of the probability distribution
• Two-tailed test: used when the alternative hypothesis does not specify a direction of the effect, and tests for the possibility of an effect in both directions

Decision Errors

• Type I error: rejecting the null hypothesis when it is true
• Type II error: retaining the null hypothesis when it is false
• Both types of errors are mutually exclusive, and changes in one type of error have an effect on the other type

Minimizing Type I and Type II Errors

• To minimize type II error: increase alpha, increase N, use the most powerful statistical test, and have a good experimental design
• To minimize type I error: decrease alpha

Single Sample T-Test

• Used to determine whether the mean of a single sample differs significantly from a known or hypothesized population mean
• Used when the standard deviation of a population is unknown and needs to be estimated