Glossary
10% Condition
A rule stating that the sample size should be no more than 10% of the population size to ensure independence of observations when sampling without replacement.
Example:
If you're sampling 50 students from a high school, the 10% condition requires that the school has at least 500 students to assume independence.
Alternative Hypothesis (Ha)
A statement that contradicts the null hypothesis, proposing that there is a real difference or effect, which the researcher is trying to find evidence for.
Example:
If a researcher believes a new fertilizer increases crop yield, their alternative hypothesis would state that the average yield with the new fertilizer is greater than with the old one.
Conclusion (in context)
The final statement of a hypothesis test, which interprets the statistical decision (reject or fail to reject H0) in clear, non-technical language related to the original problem.
Example:
After analyzing the data, a conclusion in context might state: 'There is significant evidence to suggest that the new fertilizer increases crop yield, based on a p-value less than 0.05.'
Independence Condition
The requirement that observations within and between samples are independent of each other, meaning the outcome of one does not influence another.
Example:
When studying the effectiveness of two different advertisements, ensuring the independence condition means that one person's exposure to Ad A doesn't affect another person's response to Ad B.
Large Counts Condition
A condition for proportions that requires the expected number of successes and failures in each sample to be at least 10, ensuring the sampling distribution is approximately normal.
Example:
If you survey 100 people, the large counts condition means you need at least 10 'yes' responses and 10 'no' responses to use normal approximation for proportions.
Normal Condition
The requirement that the sampling distribution of the sample statistic (e.g., difference in proportions) is approximately normal, typically checked using the Large Counts Condition for proportions.
Example:
Before performing a z-test for proportions, we check the normal condition to ensure that the distribution of possible sample differences is bell-shaped, allowing us to use z-scores.
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:
For a new drug trial, the null hypothesis would state that the proportion of patients who recover with the new drug is equal to the proportion who recover with a placebo.
P-value
The probability of observing a sample result as extreme as, or more extreme than, the one obtained, assuming the null hypothesis is true.
Example:
If a study yields a p-value of 0.02, it means there's only a 2% chance of seeing such a difference in results if the null hypothesis (no difference) were actually true.
Parameters
Numerical values that describe a characteristic of an entire population, often estimated using sample statistics.
Example:
The true mean height of all adult males in a country is a parameter, while the mean height of a sample of 100 adult males is a statistic.
Pooled Proportion (pc-hat)
A combined estimate of the common population proportion, calculated by pooling the successes and total sample sizes from both groups, used when the null hypothesis assumes equal proportions.
Example:
When testing if two brands of popcorn have the same popping rate, you'd calculate the pooled proportion by combining the total popped kernels from both samples and dividing by the total kernels.
Population Proportions (p1 and p2)
The true, unknown proportions of a characteristic within two distinct populations that a significance test aims to compare.
Example:
In a study comparing voter preferences, population proportions and might represent the true percentage of all registered voters in two different states who support a particular candidate.
Random Assignment
The process of distributing subjects to different treatment groups in an experiment purely by chance, which helps create equivalent groups and ensures independence.
Example:
In a clinical trial, using random assignment means each patient has an equal chance of receiving the new drug or the placebo, minimizing confounding variables.
Random Condition
The requirement that samples must be randomly selected or subjects randomly assigned to treatments to ensure the data is representative and avoid bias.
Example:
To satisfy the random condition for a survey on student opinions, every student in the school should have an equal chance of being selected for the sample.
Sample Data
The observed information collected from a subset of a population, used to make inferences about the larger population.
Example:
If you survey 100 students about their favorite subject, the responses from those 100 students constitute your sample data.
Significance Level (alpha)
A predetermined threshold probability (e.g., 0.05) used to decide whether to reject the null hypothesis; if the p-value is less than this level, the result is considered statistically significant.
Example:
Setting a significance level of 0.05 means you are willing to accept a 5% chance of incorrectly rejecting a true null hypothesis (Type I error).
Significance Test
A formal procedure used to determine if an observed difference between sample statistics is statistically significant or likely due to random chance.
Example:
After collecting data on two different teaching methods, a significance test helps decide if one method truly leads to higher pass rates or if the observed difference is just random variation.
Test Statistic
A standardized value calculated from sample data that measures how far the observed sample result deviates from what is expected under the null hypothesis.
Example:
A high test statistic value in a study comparing two teaching methods would suggest that the observed difference in pass rates is unlikely to have occurred by chance if the methods were equally effective.