Glossary
Alternative Hypothesis (Ha)
A statement that contradicts the null hypothesis, proposing that there is a real effect or difference between population parameters.
Example:
The Alternative Hypothesis (Ha) might state that the new fertilizer leads to a greater mean corn yield than the standard fertilizer.
Central Limit Theorem
A fundamental theorem stating that the sampling distribution of the sample mean (or difference in means) will be approximately normal if the sample size is sufficiently large, regardless of the population's distribution.
Example:
Even if individual customer waiting times at a bank are skewed, the Central Limit Theorem tells us that the average waiting time from many large samples will form a normal distribution.
Critical Value
A threshold value from the t-distribution table, determined by the significance level and degrees of freedom, used to define the rejection region for a hypothesis test.
Example:
If the calculated t-score exceeds the Critical Value, it suggests the observed difference is extreme enough to reject the null hypothesis.
Degrees of Freedom (df)
A value related to the sample size(s) that determines the specific shape of the t-distribution, influencing the critical value and p-value.
Example:
When comparing two small groups of 8 and 10 students, the Degrees of Freedom (df) for the t-test would be 7 (the smaller sample size minus 1).
Fail to Reject the Null Hypothesis
The decision made when the p-value is greater than or equal to the significance level, indicating insufficient evidence to support the alternative hypothesis.
Example:
If the p-value is 0.15 and α is 0.05, we Fail to Reject the Null Hypothesis, meaning we don't have enough evidence to claim a significant difference.
Independence (Assumption)
An assumption stating that the observations within each sample and between the two samples must not influence each other.
Example:
When comparing the effectiveness of two different fertilizers, the plots of land must be far enough apart to ensure Independence of their yields.
Normality (Assumption)
An assumption that the populations from which the samples are drawn are approximately normally distributed, or that sample sizes are large enough for the Central Limit Theorem to apply.
Example:
Even if a small sample of student test scores isn't perfectly bell-shaped, we might assume Normality if the population of all student scores is known to be roughly normal.
Null Hypothesis (H0)
A statement of no effect or no difference between population parameters, which is assumed to be true until evidence suggests otherwise.
Example:
The Null Hypothesis (H0) for comparing two fertilizers would state that there is no difference in the mean corn yield between them.
P-value
The probability of observing a sample mean difference as extreme as, or more extreme than, the one obtained, assuming the null hypothesis is true.
Example:
A P-value of 0.03 means there's a 3% chance of seeing a difference this large or larger if there truly is no difference between the population means.
Randomness (Assumption)
An essential assumption for valid inference, requiring that data from both samples be collected randomly to ensure representativeness.
Example:
Before comparing two teaching methods, a teacher ensures Randomness by flipping a coin for each student to assign them to either the new method or the traditional one.
Reject the Null Hypothesis
The decision made when the p-value is less than the significance level, indicating sufficient evidence to support the alternative hypothesis.
Example:
If the p-value is 0.01 and α is 0.05, we Reject the Null Hypothesis, concluding there's a significant difference.
Significance Level (α)
A predetermined threshold (e.g., 0.05 or 0.01) that represents the maximum probability of making a Type I error (rejecting a true null hypothesis).
Example:
Setting the Significance Level (α) at 0.05 means a researcher is willing to accept a 5% chance of incorrectly concluding there's a difference when there isn't one.
Standard Error of the Difference
A measure of the typical variability or precision of the difference between two sample means, used in the denominator of the t-score formula.
Example:
A small Standard Error of the Difference indicates that the observed difference between two sample means is likely a more precise estimate of the true population difference.
Test Statistic (t-score)
A standardized value that measures how many standard errors the observed sample mean difference is from the hypothesized population mean difference (usually zero).
Example:
After calculating a Test Statistic (t-score) of 2.5, a student knows their observed difference in means is 2.5 standard errors away from what the null hypothesis predicts.
Two-Sample t-Tests
A statistical hypothesis test used to determine if there is a statistically significant difference between the means of two independent groups.
Example:
A researcher uses a Two-Sample t-Test to compare the average reaction times of students who drank coffee versus those who drank water before a test.