Glossary
Alternative Hypothesis (Ha)
The alternative hypothesis is the claim we are trying to find evidence to support, contradicting the null hypothesis. It can be one-sided (p < or p >) or two-sided (p ≠).
Example:
If we suspect the cereal boxes are actually underfilled, the Alternative Hypothesis (Ha) would be that the true mean weight is less than 16 ounces (μ < 16).
Independent Condition (10% Condition)
When sampling without replacement, this condition ensures that the observations are independent by requiring the population size to be at least 10 times the sample size. This prevents significant changes in population composition as items are sampled.
Example:
If you sample 50 candies from a large bag, the Independent Condition (10% condition) is met if the bag contains at least 500 candies, ensuring each pick doesn't significantly alter the remaining proportions.
Normal Condition (Large Counts Condition)
For proportions, this condition ensures the sampling distribution of the sample proportion is approximately normal. It requires that both the expected number of successes (np) and failures (n(1-p)) are at least 10.
Example:
Before using a Z-test for a proportion, you must check the Normal Condition; if you survey 100 people and expect 2% to have a rare trait, 100 * 0.02 = 2, which is less than 10, so the condition is not met.
Null Hypothesis (H0)
The null hypothesis is the initial statement about a population parameter that we assume to be true and attempt to find evidence against. It always includes an equals sign (p = [some value]).
Example:
A cereal company claims their boxes contain 16 ounces of cereal. The Null Hypothesis (H0) would be that the true mean weight is 16 ounces (μ = 16).
One-tailed test
A hypothesis test where the alternative hypothesis specifies a direction (either 'greater than' or 'less than'). This focuses the rejection region on one side of the sampling distribution.
Example:
Testing if a new fertilizer increases crop yield would involve a one-tailed test (Ha: μ > old yield).
P-Value
The probability of observing a sample statistic as extreme as, or more extreme than, the one obtained, assuming the null hypothesis is true. A small p-value provides evidence against the null hypothesis.
Example:
If a new drug shows a significant improvement in patients and the P-Value is 0.01, it means there's only a 1% chance of seeing such an improvement if the drug actually had no effect.
Random Condition
A crucial condition for inference, requiring that the sample data be collected using a random sampling method. This ensures the sample is representative of the population.
Example:
To generalize survey results about student opinions to the entire school, the students surveyed must be selected via a random condition, like a simple random sample.
Two-tailed test
A hypothesis test where the alternative hypothesis specifies that the parameter is simply 'not equal' to the hypothesized value. This means the rejection region is split between both tails of the sampling distribution.
Example:
Testing if a new drug has any effect on blood pressure (either increasing or decreasing it) would require a two-tailed test (Ha: μ ≠ normal blood pressure).
Type I Error
Occurs when the null hypothesis is true, but we incorrectly reject it. This is often referred to as a 'false positive'.
Example:
A company commits a Type I Error if they conclude their new product is better than the old one (reject H0) when, in reality, it performs just as well.
Type II Error
Occurs when the null hypothesis is false, but we fail to reject it. This is often referred to as a 'false negative'.
Example:
A medical test makes a Type II Error if it concludes a patient is healthy (fail to reject H0) when, in fact, they have the disease.
Z-Score (Test Statistic)
A standardized measure that indicates how many standard deviations a sample statistic (like a sample proportion) is from the hypothesized population parameter. It is calculated using a specific formula for proportions.
Example:
If your sample proportion of students who prefer online learning is much higher than the national average, your Z-Score would be a large positive number, indicating a significant difference.