Glossary
2-PropZTest
A specific calculator function or statistical procedure used to perform a hypothesis test for the difference between two population proportions.
Example:
To quickly find the z-score and p-value for comparing the proportion of students who prefer online vs. in-person classes, an AP Stats student would use the 2-PropZTest on their graphing calculator.
Empirical Rule
Also known as the 68-95-99.7 rule, it describes the percentage of data that falls within one, two, and three standard deviations of the mean in a normal distribution.
Example:
According to the Empirical Rule, approximately 95% of data points in a normal distribution fall within two standard deviations of the mean.
Fail to Reject H₀
The decision made when the p-value is greater than or equal to the significance level, meaning there isn't enough evidence to conclude the null hypothesis is false.
Example:
With a p-value of 0.15 and α = 0.05, we Fail to Reject H₀, meaning the data does not provide sufficient evidence to support the alternative claim.
Hypothesis Testing for Two Proportions
A statistical method used to compare two population proportions to determine if there is a statistically significant difference between them.
Example:
A researcher uses Hypothesis Testing for Two Proportions to see if the success rate of a new teaching method is significantly different from an old one across two different schools.
Independence (Condition)
A condition for inference requiring that observations within each sample are independent of each other, and the two samples are independent of each other.
Example:
When comparing two groups of patients, the Independence condition means that one patient's response should not influence another's, and the treatment group should not affect the control group.
Large Enough Sample Sizes (Condition)
A condition for inference for proportions, requiring that both np ≥ 10 and n(1-p) ≥ 10 for each group to ensure the sampling distribution is approximately normal.
Example:
Before performing a z-test for proportions, a student checks the Large Enough Sample Sizes condition by verifying that at least 10 successes and 10 failures are expected in each sample.
Null Hypothesis (H₀)
A statement of no effect or no difference between population parameters, which is assumed true until evidence suggests otherwise.
Example:
In a taste test, the Null Hypothesis (H₀) might state that there is no preference between Brand A and Brand B of potato chips.
P-value
The probability of observing a sample difference as extreme as, or more extreme than, the one obtained, assuming the null hypothesis is true.
Example:
A p-value of 0.03 in a study on a new drug suggests there's only a 3% chance of seeing such a positive outcome if the drug actually had no effect.
Random Samples
A condition for inference requiring that data be collected from samples chosen randomly from their respective populations to ensure representativeness.
Example:
To compare voter preferences, researchers must ensure they collect Random Samples from different demographics, not just volunteers, to avoid bias.
Reject H₀
The decision made when the p-value is less than the significance level, indicating strong evidence against the null hypothesis.
Example:
If a study finds a p-value of 0.02 and α = 0.05, we Reject H₀, concluding there's significant evidence for the alternative hypothesis.
Significance Level (α)
The predetermined threshold for the p-value, below which the null hypothesis is rejected. Commonly set at 0.05.
Example:
Setting the Significance Level (α) at 0.01 means a researcher requires very strong evidence (a p-value less than 0.01) to reject the null hypothesis.
Type I Error
Occurs when the null hypothesis is true, but we incorrectly reject it. It's often called a 'false positive'.
Example:
A Type I Error would occur if a new, ineffective medication was concluded to be effective based on a statistical test.
Type II Error
Occurs when the null hypothesis is false, but we fail to reject it. It's often called a 'false negative'.
Example:
A Type II Error would occur if a truly effective new vaccine was concluded to be ineffective based on a statistical test.
Z-score
The test statistic in a hypothesis test for proportions, indicating how many standard deviations a sample difference is from the hypothesized mean (often zero).
Example:
If a z-score of 2.5 is calculated for a study comparing two types of fertilizer, it means the observed difference in plant growth is 2.5 standard deviations away from what would be expected if the fertilizers had no real difference.