Glossary
Alternative Hypothesis (Hₐ)
The statement that the researcher is trying to find evidence for, representing what is suspected to be true if the null hypothesis is false.
Example:
If the null hypothesis states a drug has no effect, the alternative hypothesis might state that the drug does reduce symptoms, or that it increases symptoms, or simply that it changes symptoms.
Confidence Intervals
A range of values, calculated from sample data, that is likely to contain the true value of an unknown population parameter with a certain level of confidence.
Example:
A 95% confidence interval for the average commute time in a city might be (25 minutes, 35 minutes), meaning we are 95% confident the true average commute time falls within this range.
Degrees of Freedom (df)
A parameter that specifies the shape of a t-distribution, typically calculated as n-1 for a one-sample t-test, representing the number of independent pieces of information available to estimate a parameter.
Example:
For a sample of 35 observations, the degrees of freedom for a t-test would be 34, influencing the specific shape of the t-distribution used.
Fail to Reject the Null Hypothesis
The decision made in a significance test when the p-value is greater than or equal to the chosen significance level (alpha), indicating insufficient evidence to conclude the alternative hypothesis is true.
Example:
If a p-value is 0.15 and alpha is 0.05, we fail to reject the null hypothesis, meaning we don't have enough evidence to support the alternative claim.
Independence Condition (10% Condition)
A condition for inference stating that when sampling without replacement, the population size must be at least 10 times the sample size to ensure observations are approximately independent.
Example:
If you sample 50 students from a high school, the independence condition requires that the school has at least 500 students (50 * 10) for valid inference.
Inference for Quantitative Data
The process of drawing conclusions or making predictions about a population's numerical characteristics (like means) based on sample data.
Example:
Using a sample of student GPAs to make an inference for quantitative data about the average GPA of all students at a large university.
Normal Condition (Central Limit Theorem)
A condition for inference requiring that the sampling distribution of the sample mean is approximately normal, which can be met if the population is normal, the sample size is large (n ≥ 30 by the Central Limit Theorem), or the sample data shows no strong skewness or outliers.
Example:
Even if individual student test scores are skewed, a large sample size (e.g., n=40) allows us to assume the sampling distribution of the mean test score is approximately normal due to the Central Limit Theorem.
Null Hypothesis (H₀)
The statement being tested in a significance test, typically a statement of no effect, no difference, or no change.
Example:
In a study testing a new teaching method, the null hypothesis might be that the new method has no effect on student test scores (i.e., the average score remains the same).
P-value
The probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming the null hypothesis is true.
Example:
If a p-value is 0.03, it means there's a 3% chance of getting results as extreme as observed if the null hypothesis were actually true, suggesting evidence against the null.
Population Mean
The true average value of a quantitative variable for an entire group of individuals or items being studied.
Example:
If we're studying the average height of all adult males in a country, the actual average height of every single male is the population mean.
Random Condition
A crucial condition for inference stating that the sample must be selected randomly from the population to ensure it is representative and unbiased.
Example:
To estimate the average income of households in a city, researchers must ensure they select a random sample of households to avoid bias.
Reject the Null Hypothesis
The decision made in a significance test when the p-value is less than the chosen significance level (alpha), indicating sufficient evidence against the null hypothesis.
Example:
If a study's p-value is 0.01 and alpha is 0.05, we reject the null hypothesis, concluding there's significant evidence for the alternative.
Sampling Distribution of the Sample Mean
The probability distribution of all possible sample means that could be obtained from samples of the same size drawn from a given population.
Example:
If you repeatedly take samples of 30 students and calculate their average height, the distribution of all those average heights would form the sampling distribution of the sample mean.
Significance Tests
A formal procedure for comparing observed data with a claim (hypothesis) whose truth we want to assess, using statistical evidence to determine if the data provides strong evidence against the claim.
Example:
A pharmaceutical company conducts a significance test to determine if a new drug is more effective than a placebo in reducing blood pressure.
Test Statistic (t-statistic)
A standardized value calculated from sample data that measures how far the sample result is from the null hypothesis parameter, in terms of standard errors.
Example:
A large absolute value of the t-statistic indicates that the observed sample mean is far from the hypothesized population mean, making the null hypothesis less plausible.
Two-Sample Inference
Statistical methods used to compare parameters (like means) from two independent populations or groups based on their respective sample data.
Example:
A study comparing the average test scores of students taught by two different methods would use two-sample inference.
Type I Error
Occurs when a true null hypothesis is incorrectly rejected, often referred to as a 'false positive'.
Example:
A company commits a Type I error if they conclude a new drug is effective when, in reality, it has no effect.
Type II Error
Occurs when a false null hypothesis is incorrectly failed to be rejected, often referred to as a 'false negative'.
Example:
A company commits a Type II error if they conclude a new drug is not effective when, in reality, it does have a positive effect.
t-distributions
A family of probability distributions used for inference about population means when the population standard deviation is unknown and estimated from the sample.
Example:
When analyzing the average battery life of a new phone model, since we don't know the true population standard deviation, we'd use a t-distribution to construct a confidence interval.
t-tests
Statistical hypothesis tests used to determine if there is a significant difference between a sample mean and a hypothesized population mean, or between two sample means, especially when the population standard deviation is unknown.
Example:
A researcher might use a t-test to see if a new fertilizer significantly increases the average yield of corn compared to the standard fertilizer.
z-test
A statistical test used for inference about population means when the population standard deviation (σ) is known and the population is normally distributed or the sample size is large.
Example:
If a manufacturer knows the exact standard deviation of the weight of their product, they might use a z-test to check if a new batch's average weight differs from the target.