Glossary
Alternative Hypothesis
A statement that contradicts the null hypothesis, suggesting there is a difference, an effect, or that a population parameter is not equal to a specific value.
Example:
The alternative hypothesis for our medication study is that the new medication is more effective than the placebo.
Confidence Intervals for the Difference of Two Proportions
A range of plausible values that estimates the true difference between two population proportions, based on sample data.
Example:
We constructed a confidence interval for the difference of two proportions to estimate how much more likely Brand A users are to recommend it compared to Brand B users.
Confidence Level
The percentage of intervals, if the sampling process were repeated many times, that would contain the true population parameter.
Example:
A 95% confidence level means that if we took many samples and built an interval from each, about 95% of those intervals would capture the true population difference.
Confounding Variables
Variables that are not the primary focus of a study but can influence both the independent and dependent variables, potentially leading to misleading conclusions.
Example:
When comparing student performance in two schools, differences in teacher experience or student socioeconomic status could be confounding variables.
Context (in interpretation)
Relating statistical findings back to the specific real-world scenario, variables, and groups being studied, making the interpretation meaningful and understandable.
Example:
Always include the context by specifying that the interval refers to 'the true difference in the proportion of patients whose blood pressure decreased between the treatment and control groups.'
Critical Value
A value from a sampling distribution that defines the boundary of the rejection region in hypothesis testing or is used to calculate the margin of error for a confidence interval.
Example:
For a 95% confidence interval for proportions, the critical value (z*) is typically 1.96.
Hypothesis Testing with Confidence Intervals
A method to test a claim about a population parameter by observing whether a specific hypothesized value (often zero for differences) falls within a constructed confidence interval.
Example:
We used hypothesis testing with confidence intervals to determine if there was a statistically significant difference in the success rates of two different marketing campaigns.
Interpreting Confidence Intervals
The process of explaining what a confidence interval means in the context of the problem, including the confidence level, the population parameters being estimated, and the interval's numerical range.
Example:
When interpreting confidence intervals, remember to state, 'We are 95% confident that the true difference in proportions of students who prefer online learning and in-person learning is between 0.10 and 0.25.'
Null Hypothesis
A statement of no difference or no effect, often stating that a population parameter is equal to a specific value or that two population parameters are equal.
Example:
The null hypothesis for comparing two medications would state that there is no difference in their effectiveness.
Population Proportions
The true proportion of a characteristic within an entire population, which is typically unknown and estimated using sample data.
Example:
We are interested in the population proportions of all adults in City A versus City B who regularly exercise, not just those in our samples.
Sample Proportions
The proportion of individuals in a specific sample that possess a particular characteristic, calculated directly from observed data.
Example:
If 45 out of 100 patients in the treatment group showed a decrease in blood pressure, then 0.45 is the sample proportion for that group.
Sample Representativeness
The extent to which a sample accurately reflects the characteristics of the larger population from which it was drawn, crucial for generalizing findings.
Example:
To ensure sample representativeness, a researcher studying voter preferences should randomly select participants from diverse demographics across the entire region.
Standard Error
The standard deviation of a sampling distribution, indicating the typical amount of variability or error expected when estimating a population parameter from a sample.
Example:
A small standard error for the difference in proportions suggests that our sample difference is likely close to the true population difference.
Zero Inside the Interval
When the value zero is contained within a confidence interval for a difference, it suggests that a true difference of zero is plausible, leading to a failure to reject the null hypothesis.
Example:
If the confidence interval for the difference in proportions of coffee vs. tea preference is (-0.05, 0.15), then zero inside the interval means we cannot conclude a significant difference.
Zero Outside the Interval
When the value zero is not contained within a confidence interval for a difference, it suggests that a true difference of zero is not plausible, leading to the rejection of the null hypothesis.
Example:
Since the interval (0.063, 0.133) for the difference in shooting percentages has zero outside the interval, we can conclude there's a significant difference.